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  22.    <div id="projectname">3DEX&#160;<span id="projectnumber">1.0</span></div>
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  23.    <div id="projectbrief">Three-dimensional Fourier-Bessel decomposition</div>
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  62. <div class="title">/Users/bl/Dropbox/3DEX/src/f90/external/quadpack.f90</div>  </div>
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    65. 

  65. <a href="quadpack_8f90.html">Go to the documentation of this file.</a><div class="fragment"><pre class="fragment"><a name="l00001"></a><a class="code" href="quadpack_8f90.html#a44906a25a31588f7e4f41f0e5253193a">00001</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a44906a25a31588f7e4f41f0e5253193a">qag</a> ( f, a, b, epsabs, epsrel, key, result, abserr, neval, ier )
    66. 

  66. <a name="l00002"></a>00002 
    67. 

  67. <a name="l00003"></a>00003 <span class="comment">!*****************************************************************************80</span>
    68. 

  68. <a name="l00004"></a>00004 <span class="comment">!</span>
    69. 

  69. <a name="l00005"></a>00005 <span class="comment">!! QAG approximates an integral over a finite interval.</span>
    70. 

  70. <a name="l00006"></a>00006 <span class="comment">!</span>
    71. 

  71. <a name="l00007"></a>00007 <span class="comment">!  Discussion:</span>
    72. 

  72. <a name="l00008"></a>00008 <span class="comment">!</span>
    73. 

  73. <a name="l00009"></a>00009 <span class="comment">!    The routine calculates an approximation RESULT to a definite integral   </span>
    74. 

  74. <a name="l00010"></a>00010 <span class="comment">!      I = integral of F over (A,B),</span>
    75. 

  75. <a name="l00011"></a>00011 <span class="comment">!    hopefully satisfying</span>
    76. 

  76. <a name="l00012"></a>00012 <span class="comment">!      || I - RESULT || &lt;= max ( EPSABS, EPSREL * ||I|| ).</span>
    77. 

  77. <a name="l00013"></a>00013 <span class="comment">!</span>
    78. 

  78. <a name="l00014"></a>00014 <span class="comment">!    QAG is a simple globally adaptive integrator using the strategy of </span>
    79. 

  79. <a name="l00015"></a>00015 <span class="comment">!    Aind (Piessens, 1973).  It is possible to choose between 6 pairs of</span>
    80. 

  80. <a name="l00016"></a>00016 <span class="comment">!    Gauss-Kronrod quadrature formulae for the rule evaluation component. </span>
    81. 

  81. <a name="l00017"></a>00017 <span class="comment">!    The pairs of high degree of precision are suitable for handling</span>
    82. 

  82. <a name="l00018"></a>00018 <span class="comment">!    integration difficulties due to a strongly oscillating integrand.</span>
    83. 

  83. <a name="l00019"></a>00019 <span class="comment">!</span>
    84. 

  84. <a name="l00020"></a>00020 <span class="comment">!  Author:</span>
    85. 

  85. <a name="l00021"></a>00021 <span class="comment">!</span>
    86. 

  86. <a name="l00022"></a>00022 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    87. 

  87. <a name="l00023"></a>00023 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    88. 

  88. <a name="l00024"></a>00024 <span class="comment">!</span>
    89. 

  89. <a name="l00025"></a>00025 <span class="comment">!  Reference:</span>
    90. 

  90. <a name="l00026"></a>00026 <span class="comment">!</span>
    91. 

  91. <a name="l00027"></a>00027 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    92. 

  92. <a name="l00028"></a>00028 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    93. 

  93. <a name="l00029"></a>00029 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    94. 

  94. <a name="l00030"></a>00030 <span class="comment">!    Springer Verlag, 1983</span>
    95. 

  95. <a name="l00031"></a>00031 <span class="comment">!</span>
    96. 

  96. <a name="l00032"></a>00032 <span class="comment">!  Parameters:</span>
    97. 

  97. <a name="l00033"></a>00033 <span class="comment">!</span>
    98. 

  98. <a name="l00034"></a>00034 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    99. 

  99. <a name="l00035"></a>00035 <span class="comment">!      function f ( x )</span>
    100. 

  100. <a name="l00036"></a>00036 <span class="comment">!      real f</span>
    101. 

  101. <a name="l00037"></a>00037 <span class="comment">!      real x</span>
    102. 

  102. <a name="l00038"></a>00038 <span class="comment">!    which evaluates the integrand function.</span>
    103. 

  103. <a name="l00039"></a>00039 <span class="comment">!</span>
    104. 

  104. <a name="l00040"></a>00040 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    105. 

  105. <a name="l00041"></a>00041 <span class="comment">!</span>
    106. 

  106. <a name="l00042"></a>00042 <span class="comment">!    Input, real EPSABS, EPSREL, the absolute and relative accuracy requested.</span>
    107. 

  107. <a name="l00043"></a>00043 <span class="comment">!</span>
    108. 

  108. <a name="l00044"></a>00044 <span class="comment">!    Input, integer KEY, chooses the order of the local integration rule:</span>
    109. 

  109. <a name="l00045"></a>00045 <span class="comment">!    1,  7 Gauss points, 15 Gauss-Kronrod points,</span>
    110. 

  110. <a name="l00046"></a>00046 <span class="comment">!    2, 10 Gauss points, 21 Gauss-Kronrod points,</span>
    111. 

  111. <a name="l00047"></a>00047 <span class="comment">!    3, 15 Gauss points, 31 Gauss-Kronrod points,</span>
    112. 

  112. <a name="l00048"></a>00048 <span class="comment">!    4, 20 Gauss points, 41 Gauss-Kronrod points,</span>
    113. 

  113. <a name="l00049"></a>00049 <span class="comment">!    5, 25 Gauss points, 51 Gauss-Kronrod points,</span>
    114. 

  114. <a name="l00050"></a>00050 <span class="comment">!    6, 30 Gauss points, 61 Gauss-Kronrod points.</span>
    115. 

  115. <a name="l00051"></a>00051 <span class="comment">!</span>
    116. 

  116. <a name="l00052"></a>00052 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    117. 

  117. <a name="l00053"></a>00053 <span class="comment">!</span>
    118. 

  118. <a name="l00054"></a>00054 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    119. 

  119. <a name="l00055"></a>00055 <span class="comment">!</span>
    120. 

  120. <a name="l00056"></a>00056 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    121. 

  121. <a name="l00057"></a>00057 <span class="comment">!</span>
    122. 

  122. <a name="l00058"></a>00058 <span class="comment">!    Output, integer IER, return code.</span>
    123. 

  123. <a name="l00059"></a>00059 <span class="comment">!    0, normal and reliable termination of the routine.  It is assumed that the </span>
    124. 

  124. <a name="l00060"></a>00060 <span class="comment">!      requested accuracy has been achieved.</span>
    125. 

  125. <a name="l00061"></a>00061 <span class="comment">!    1, maximum number of subdivisions allowed has been achieved.  One can </span>
    126. 

  126. <a name="l00062"></a>00062 <span class="comment">!      allow more subdivisions by increasing the value of LIMIT in QAG. </span>
    127. 

  127. <a name="l00063"></a>00063 <span class="comment">!      However, if this yields no improvement it is advised to analyze the</span>
    128. 

  128. <a name="l00064"></a>00064 <span class="comment">!      integrand to determine the integration difficulties.  If the position</span>
    129. 

  129. <a name="l00065"></a>00065 <span class="comment">!      of a local difficulty can be determined, such as a singularity or</span>
    130. 

  130. <a name="l00066"></a>00066 <span class="comment">!      discontinuity within the interval) one will probably gain from </span>
    131. 

  131. <a name="l00067"></a>00067 <span class="comment">!      splitting up the interval at this point and calling the integrator </span>
    132. 

  132. <a name="l00068"></a>00068 <span class="comment">!      on the subranges.  If possible, an appropriate special-purpose </span>
    133. 

  133. <a name="l00069"></a>00069 <span class="comment">!      integrator should be used which is designed for handling the type </span>
    134. 

  134. <a name="l00070"></a>00070 <span class="comment">!      of difficulty involved.</span>
    135. 

  135. <a name="l00071"></a>00071 <span class="comment">!    2, the occurrence of roundoff error is detected, which prevents the</span>
    136. 

  136. <a name="l00072"></a>00072 <span class="comment">!      requested tolerance from being achieved.</span>
    137. 

  137. <a name="l00073"></a>00073 <span class="comment">!    3, extremely bad integrand behavior occurs at some points of the</span>
    138. 

  138. <a name="l00074"></a>00074 <span class="comment">!      integration interval.</span>
    139. 

  139. <a name="l00075"></a>00075 <span class="comment">!    6, the input is invalid, because EPSABS &lt; 0 and EPSREL &lt; 0.</span>
    140. 

  140. <a name="l00076"></a>00076 <span class="comment">!</span>
    141. 

  141. <a name="l00077"></a>00077 <span class="comment">!  Local parameters:</span>
    142. 

  142. <a name="l00078"></a>00078 <span class="comment">!</span>
    143. 

  143. <a name="l00079"></a>00079 <span class="comment">!    LIMIT is the maximum number of subintervals allowed in</span>
    144. 

  144. <a name="l00080"></a>00080 <span class="comment">!    the subdivision process of QAGE.</span>
    145. 

  145. <a name="l00081"></a>00081 <span class="comment">!</span>
    146. 

  146. <a name="l00082"></a>00082   <span class="keyword">implicit none</span>
    147. 

  147. <a name="l00083"></a>00083 
    148. 

  148. <a name="l00084"></a>00084   <span class="keywordtype">integer</span>, <span class="keywordtype">parameter</span> :: limit = 500
    149. 

  149. <a name="l00085"></a>00085 
    150. 

  150. <a name="l00086"></a>00086   <span class="keywordtype">real</span> a
    151. 

  151. <a name="l00087"></a>00087   <span class="keywordtype">real</span> abserr
    152. 

  152. <a name="l00088"></a>00088   <span class="keywordtype">real</span> alist(limit)
    153. 

  153. <a name="l00089"></a>00089   <span class="keywordtype">real</span> b
    154. 

  154. <a name="l00090"></a>00090   <span class="keywordtype">real</span> blist(limit)
    155. 

  155. <a name="l00091"></a>00091   <span class="keywordtype">real</span> elist(limit)
    156. 

  156. <a name="l00092"></a>00092   <span class="keywordtype">real</span> epsabs
    157. 

  157. <a name="l00093"></a>00093   <span class="keywordtype">real</span> epsrel
    158. 

  158. <a name="l00094"></a>00094   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    159. 

  159. <a name="l00095"></a>00095   <span class="keywordtype">integer</span> ier
    160. 

  160. <a name="l00096"></a>00096   <span class="keywordtype">integer</span> iord(limit)
    161. 

  161. <a name="l00097"></a>00097   <span class="keywordtype">integer</span> key
    162. 

  162. <a name="l00098"></a>00098   <span class="keywordtype">integer</span> last
    163. 

  163. <a name="l00099"></a>00099   <span class="keywordtype">integer</span> neval
    164. 

  164. <a name="l00100"></a>00100   <span class="keywordtype">real</span> result
    165. 

  165. <a name="l00101"></a>00101   <span class="keywordtype">real</span> rlist(limit)
    166. 

  166. <a name="l00102"></a>00102 
    167. 

  167. <a name="l00103"></a>00103   call <a class="code" href="quadpack_8f90.html#ab602437c218a2c74d6a13f9462f98854">qage </a>( f, a, b, epsabs, epsrel, key, limit, result, abserr, neval, &amp;
    168. 

  168. <a name="l00104"></a>00104     ier, alist, blist, rlist, elist, iord, last )
    169. 

  169. <a name="l00105"></a>00105 
    170. 

  170. <a name="l00106"></a>00106   return
    171. 

  171. <a name="l00107"></a>00107 <span class="keyword">end</span>
    172. 

  172. <a name="l00108"></a><a class="code" href="quadpack_8f90.html#ab602437c218a2c74d6a13f9462f98854">00108</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#ab602437c218a2c74d6a13f9462f98854">qage</a> ( f, a, b, epsabs, epsrel, key, limit, result, abserr, neval, &amp;
    173. 

  173. <a name="l00109"></a>00109   ier, alist, blist, rlist, elist, iord, last )
    174. 

  174. <a name="l00110"></a>00110 
    175. 

  175. <a name="l00111"></a>00111 <span class="comment">!*****************************************************************************80</span>
    176. 

  176. <a name="l00112"></a>00112 <span class="comment">!</span>
    177. 

  177. <a name="l00113"></a>00113 <span class="comment">!! QAGE estimates a definite integral.</span>
    178. 

  178. <a name="l00114"></a>00114 <span class="comment">!</span>
    179. 

  179. <a name="l00115"></a>00115 <span class="comment">!  Discussion:</span>
    180. 

  180. <a name="l00116"></a>00116 <span class="comment">!</span>
    181. 

  181. <a name="l00117"></a>00117 <span class="comment">!    The routine calculates an approximation RESULT to a definite integral   </span>
    182. 

  182. <a name="l00118"></a>00118 <span class="comment">!      I = integral of F over (A,B),</span>
    183. 

  183. <a name="l00119"></a>00119 <span class="comment">!    hopefully satisfying</span>
    184. 

  184. <a name="l00120"></a>00120 <span class="comment">!      || I - RESULT || &lt;= max ( EPSABS, EPSREL * ||I|| ).</span>
    185. 

  185. <a name="l00121"></a>00121 <span class="comment">!</span>
    186. 

  186. <a name="l00122"></a>00122 <span class="comment">!  Author:</span>
    187. 

  187. <a name="l00123"></a>00123 <span class="comment">!</span>
    188. 

  188. <a name="l00124"></a>00124 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    189. 

  189. <a name="l00125"></a>00125 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    190. 

  190. <a name="l00126"></a>00126 <span class="comment">!</span>
    191. 

  191. <a name="l00127"></a>00127 <span class="comment">!  Reference:</span>
    192. 

  192. <a name="l00128"></a>00128 <span class="comment">!</span>
    193. 

  193. <a name="l00129"></a>00129 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    194. 

  194. <a name="l00130"></a>00130 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    195. 

  195. <a name="l00131"></a>00131 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    196. 

  196. <a name="l00132"></a>00132 <span class="comment">!    Springer Verlag, 1983</span>
    197. 

  197. <a name="l00133"></a>00133 <span class="comment">!</span>
    198. 

  198. <a name="l00134"></a>00134 <span class="comment">!  Parameters:</span>
    199. 

  199. <a name="l00135"></a>00135 <span class="comment">!</span>
    200. 

  200. <a name="l00136"></a>00136 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    201. 

  201. <a name="l00137"></a>00137 <span class="comment">!      function f ( x )</span>
    202. 

  202. <a name="l00138"></a>00138 <span class="comment">!      real f</span>
    203. 

  203. <a name="l00139"></a>00139 <span class="comment">!      real x</span>
    204. 

  204. <a name="l00140"></a>00140 <span class="comment">!    which evaluates the integrand function.</span>
    205. 

  205. <a name="l00141"></a>00141 <span class="comment">!</span>
    206. 

  206. <a name="l00142"></a>00142 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    207. 

  207. <a name="l00143"></a>00143 <span class="comment">!</span>
    208. 

  208. <a name="l00144"></a>00144 <span class="comment">!    Input, real EPSABS, EPSREL, the absolute and relative accuracy requested.</span>
    209. 

  209. <a name="l00145"></a>00145 <span class="comment">!</span>
    210. 

  210. <a name="l00146"></a>00146 <span class="comment">!    Input, integer KEY, chooses the order of the local integration rule:</span>
    211. 

  211. <a name="l00147"></a>00147 <span class="comment">!    1,  7 Gauss points, 15 Gauss-Kronrod points,</span>
    212. 

  212. <a name="l00148"></a>00148 <span class="comment">!    2, 10 Gauss points, 21 Gauss-Kronrod points,</span>
    213. 

  213. <a name="l00149"></a>00149 <span class="comment">!    3, 15 Gauss points, 31 Gauss-Kronrod points,</span>
    214. 

  214. <a name="l00150"></a>00150 <span class="comment">!    4, 20 Gauss points, 41 Gauss-Kronrod points,</span>
    215. 

  215. <a name="l00151"></a>00151 <span class="comment">!    5, 25 Gauss points, 51 Gauss-Kronrod points,</span>
    216. 

  216. <a name="l00152"></a>00152 <span class="comment">!    6, 30 Gauss points, 61 Gauss-Kronrod points.</span>
    217. 

  217. <a name="l00153"></a>00153 <span class="comment">!</span>
    218. 

  218. <a name="l00154"></a>00154 <span class="comment">!    Input, integer LIMIT, the maximum number of subintervals that</span>
    219. 

  219. <a name="l00155"></a>00155 <span class="comment">!    can be used.</span>
    220. 

  220. <a name="l00156"></a>00156 <span class="comment">!</span>
    221. 

  221. <a name="l00157"></a>00157 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    222. 

  222. <a name="l00158"></a>00158 <span class="comment">!</span>
    223. 

  223. <a name="l00159"></a>00159 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    224. 

  224. <a name="l00160"></a>00160 <span class="comment">!</span>
    225. 

  225. <a name="l00161"></a>00161 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    226. 

  226. <a name="l00162"></a>00162 <span class="comment">!</span>
    227. 

  227. <a name="l00163"></a>00163 <span class="comment">!    Output, integer IER, return code.</span>
    228. 

  228. <a name="l00164"></a>00164 <span class="comment">!    0, normal and reliable termination of the routine.  It is assumed that the </span>
    229. 

  229. <a name="l00165"></a>00165 <span class="comment">!      requested accuracy has been achieved.</span>
    230. 

  230. <a name="l00166"></a>00166 <span class="comment">!    1, maximum number of subdivisions allowed has been achieved.  One can </span>
    231. 

  231. <a name="l00167"></a>00167 <span class="comment">!      allow more subdivisions by increasing the value of LIMIT in QAG. </span>
    232. 

  232. <a name="l00168"></a>00168 <span class="comment">!      However, if this yields no improvement it is advised to analyze the</span>
    233. 

  233. <a name="l00169"></a>00169 <span class="comment">!      integrand to determine the integration difficulties.  If the position</span>
    234. 

  234. <a name="l00170"></a>00170 <span class="comment">!      of a local difficulty can be determined, such as a singularity or</span>
    235. 

  235. <a name="l00171"></a>00171 <span class="comment">!      discontinuity within the interval) one will probably gain from </span>
    236. 

  236. <a name="l00172"></a>00172 <span class="comment">!      splitting up the interval at this point and calling the integrator </span>
    237. 

  237. <a name="l00173"></a>00173 <span class="comment">!      on the subranges.  If possible, an appropriate special-purpose </span>
    238. 

  238. <a name="l00174"></a>00174 <span class="comment">!      integrator should be used which is designed for handling the type </span>
    239. 

  239. <a name="l00175"></a>00175 <span class="comment">!      of difficulty involved.</span>
    240. 

  240. <a name="l00176"></a>00176 <span class="comment">!    2, the occurrence of roundoff error is detected, which prevents the</span>
    241. 

  241. <a name="l00177"></a>00177 <span class="comment">!      requested tolerance from being achieved.</span>
    242. 

  242. <a name="l00178"></a>00178 <span class="comment">!    3, extremely bad integrand behavior occurs at some points of the</span>
    243. 

  243. <a name="l00179"></a>00179 <span class="comment">!      integration interval.</span>
    244. 

  244. <a name="l00180"></a>00180 <span class="comment">!    6, the input is invalid, because EPSABS &lt; 0 and EPSREL &lt; 0.</span>
    245. 

  245. <a name="l00181"></a>00181 <span class="comment">!</span>
    246. 

  246. <a name="l00182"></a>00182 <span class="comment">!    Workspace, real ALIST(LIMIT), BLIST(LIMIT), contains in entries 1 </span>
    247. 

  247. <a name="l00183"></a>00183 <span class="comment">!    through LAST the left and right ends of the partition subintervals.</span>
    248. 

  248. <a name="l00184"></a>00184 <span class="comment">!</span>
    249. 

  249. <a name="l00185"></a>00185 <span class="comment">!    Workspace, real RLIST(LIMIT), contains in entries 1 through LAST</span>
    250. 

  250. <a name="l00186"></a>00186 <span class="comment">!    the integral approximations on the subintervals.</span>
    251. 

  251. <a name="l00187"></a>00187 <span class="comment">!</span>
    252. 

  252. <a name="l00188"></a>00188 <span class="comment">!    Workspace, real ELIST(LIMIT), contains in entries 1 through LAST</span>
    253. 

  253. <a name="l00189"></a>00189 <span class="comment">!    the absolute error estimates on the subintervals.</span>
    254. 

  254. <a name="l00190"></a>00190 <span class="comment">!</span>
    255. 

  255. <a name="l00191"></a>00191 <span class="comment">!    Output, integer IORD(LIMIT), the first K elements of which are pointers </span>
    256. 

  256. <a name="l00192"></a>00192 <span class="comment">!    to the error estimates over the subintervals, such that</span>
    257. 

  257. <a name="l00193"></a>00193 <span class="comment">!    elist(iord(1)), ..., elist(iord(k)) form a decreasing sequence, with</span>
    258. 

  258. <a name="l00194"></a>00194 <span class="comment">!    k = last if last &lt;= (limit/2+2), and k = limit+1-last otherwise.</span>
    259. 

  259. <a name="l00195"></a>00195 <span class="comment">!</span>
    260. 

  260. <a name="l00196"></a>00196 <span class="comment">!    Output, integer LAST, the number of subintervals actually produced </span>
    261. 

  261. <a name="l00197"></a>00197 <span class="comment">!    in the subdivision process.</span>
    262. 

  262. <a name="l00198"></a>00198 <span class="comment">!</span>
    263. 

  263. <a name="l00199"></a>00199 <span class="comment">!  Local parameters:</span>
    264. 

  264. <a name="l00200"></a>00200 <span class="comment">!</span>
    265. 

  265. <a name="l00201"></a>00201 <span class="comment">!    alist     - list of left end points of all subintervals</span>
    266. 

  266. <a name="l00202"></a>00202 <span class="comment">!                       considered up to now</span>
    267. 

  267. <a name="l00203"></a>00203 <span class="comment">!    blist     - list of right end points of all subintervals</span>
    268. 

  268. <a name="l00204"></a>00204 <span class="comment">!                       considered up to now</span>
    269. 

  269. <a name="l00205"></a>00205 <span class="comment">!    elist(i)  - error estimate applying to rlist(i)</span>
    270. 

  270. <a name="l00206"></a>00206 <span class="comment">!    maxerr    - pointer to the interval with largest error estimate</span>
    271. 

  271. <a name="l00207"></a>00207 <span class="comment">!    errmax    - elist(maxerr)</span>
    272. 

  272. <a name="l00208"></a>00208 <span class="comment">!    area      - sum of the integrals over the subintervals</span>
    273. 

  273. <a name="l00209"></a>00209 <span class="comment">!    errsum    - sum of the errors over the subintervals</span>
    274. 

  274. <a name="l00210"></a>00210 <span class="comment">!    errbnd    - requested accuracy max(epsabs,epsrel*abs(result))</span>
    275. 

  275. <a name="l00211"></a>00211 <span class="comment">!    *****1    - variable for the left subinterval</span>
    276. 

  276. <a name="l00212"></a>00212 <span class="comment">!    *****2    - variable for the right subinterval</span>
    277. 

  277. <a name="l00213"></a>00213 <span class="comment">!    last      - index for subdivision</span>
    278. 

  278. <a name="l00214"></a>00214 <span class="comment">!</span>
    279. 

  279. <a name="l00215"></a>00215   <span class="keyword">implicit none</span>
    280. 

  280. <a name="l00216"></a>00216 
    281. 

  281. <a name="l00217"></a>00217   <span class="keywordtype">integer</span> limit
    282. 

  282. <a name="l00218"></a>00218 
    283. 

  283. <a name="l00219"></a>00219   <span class="keywordtype">real</span> a
    284. 

  284. <a name="l00220"></a>00220   <span class="keywordtype">real</span> abserr
    285. 

  285. <a name="l00221"></a>00221   <span class="keywordtype">real</span> alist(limit)
    286. 

  286. <a name="l00222"></a>00222   <span class="keywordtype">real</span> area
    287. 

  287. <a name="l00223"></a>00223   <span class="keywordtype">real</span> area1
    288. 

  288. <a name="l00224"></a>00224   <span class="keywordtype">real</span> area12
    289. 

  289. <a name="l00225"></a>00225   <span class="keywordtype">real</span> area2
    290. 

  290. <a name="l00226"></a>00226   <span class="keywordtype">real</span> a1
    291. 

  291. <a name="l00227"></a>00227   <span class="keywordtype">real</span> a2
    292. 

  292. <a name="l00228"></a>00228   <span class="keywordtype">real</span> b
    293. 

  293. <a name="l00229"></a>00229   <span class="keywordtype">real</span> blist(limit)
    294. 

  294. <a name="l00230"></a>00230   <span class="keywordtype">real</span> b1
    295. 

  295. <a name="l00231"></a>00231   <span class="keywordtype">real</span> b2
    296. 

  296. <a name="l00232"></a>00232   <span class="keywordtype">real</span> c
    297. 

  297. <a name="l00233"></a>00233   <span class="keywordtype">real</span> defabs
    298. 

  298. <a name="l00234"></a>00234   <span class="keywordtype">real</span> defab1
    299. 

  299. <a name="l00235"></a>00235   <span class="keywordtype">real</span> defab2
    300. 

  300. <a name="l00236"></a>00236   <span class="keywordtype">real</span> elist(limit)
    301. 

  301. <a name="l00237"></a>00237   <span class="keywordtype">real</span> epsabs
    302. 

  302. <a name="l00238"></a>00238   <span class="keywordtype">real</span> epsrel
    303. 

  303. <a name="l00239"></a>00239   <span class="keywordtype">real</span> errbnd
    304. 

  304. <a name="l00240"></a>00240   <span class="keywordtype">real</span> errmax
    305. 

  305. <a name="l00241"></a>00241   <span class="keywordtype">real</span> error1
    306. 

  306. <a name="l00242"></a>00242   <span class="keywordtype">real</span> error2
    307. 

  307. <a name="l00243"></a>00243   <span class="keywordtype">real</span> erro12
    308. 

  308. <a name="l00244"></a>00244   <span class="keywordtype">real</span> errsum
    309. 

  309. <a name="l00245"></a>00245   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    310. 

  310. <a name="l00246"></a>00246   <span class="keywordtype">integer</span> ier
    311. 

  311. <a name="l00247"></a>00247   <span class="keywordtype">integer</span> iord(limit)
    312. 

  312. <a name="l00248"></a>00248   <span class="keywordtype">integer</span> iroff1
    313. 

  313. <a name="l00249"></a>00249   <span class="keywordtype">integer</span> iroff2
    314. 

  314. <a name="l00250"></a>00250   <span class="keywordtype">integer</span> key
    315. 

  315. <a name="l00251"></a>00251   <span class="keywordtype">integer</span> keyf
    316. 

  316. <a name="l00252"></a>00252   <span class="keywordtype">integer</span> last
    317. 

  317. <a name="l00253"></a>00253   <span class="keywordtype">integer</span> maxerr
    318. 

  318. <a name="l00254"></a>00254   <span class="keywordtype">integer</span> neval
    319. 

  319. <a name="l00255"></a>00255   <span class="keywordtype">integer</span> nrmax
    320. 

  320. <a name="l00256"></a>00256   <span class="keywordtype">real</span> resabs
    321. 

  321. <a name="l00257"></a>00257   <span class="keywordtype">real</span> result
    322. 

  322. <a name="l00258"></a>00258   <span class="keywordtype">real</span> rlist(limit)
    323. 

  323. <a name="l00259"></a>00259 <span class="comment">!</span>
    324. 

  324. <a name="l00260"></a>00260 <span class="comment">!  Test on validity of parameters.</span>
    325. 

  325. <a name="l00261"></a>00261 <span class="comment">!</span>
    326. 

  326. <a name="l00262"></a>00262   ier = 0
    327. 

  327. <a name="l00263"></a>00263   neval = 0
    328. 

  328. <a name="l00264"></a>00264   last = 0
    329. 

  329. <a name="l00265"></a>00265   result = 0.0e+00
    330. 

  330. <a name="l00266"></a>00266   abserr = 0.0e+00
    331. 

  331. <a name="l00267"></a>00267   alist(1) = a
    332. 

  332. <a name="l00268"></a>00268   blist(1) = b
    333. 

  333. <a name="l00269"></a>00269   rlist(1) = 0.0e+00
    334. 

  334. <a name="l00270"></a>00270   elist(1) = 0.0e+00
    335. 

  335. <a name="l00271"></a>00271   iord(1) = 0
    336. 

  336. <a name="l00272"></a>00272 
    337. 

  337. <a name="l00273"></a>00273   <span class="keyword">if</span> ( epsabs &lt; 0.0e+00 .and. epsrel &lt; 0.0e+00 ) <span class="keyword">then</span>
    338. 

  338. <a name="l00274"></a>00274     ier = 6
    339. 

  339. <a name="l00275"></a>00275     return
    340. 

  340. <a name="l00276"></a>00276   <span class="keyword">end if</span>
    341. 

  341. <a name="l00277"></a>00277 <span class="comment">!</span>
    342. 

  342. <a name="l00278"></a>00278 <span class="comment">!  First approximation to the integral.</span>
    343. 

  343. <a name="l00279"></a>00279 <span class="comment">!</span>
    344. 

  344. <a name="l00280"></a>00280   keyf = key
    345. 

  345. <a name="l00281"></a>00281   keyf = max ( keyf, 1 )
    346. 

  346. <a name="l00282"></a>00282   keyf = min ( keyf, 6 )
    347. 

  347. <a name="l00283"></a>00283 
    348. 

  348. <a name="l00284"></a>00284   c = keyf
    349. 

  349. <a name="l00285"></a>00285   neval = 0
    350. 

  350. <a name="l00286"></a>00286 
    351. 

  351. <a name="l00287"></a>00287   <span class="keyword">if</span> ( keyf == 1 ) <span class="keyword">then</span>
    352. 

  352. <a name="l00288"></a>00288     call <a class="code" href="quadpack_8f90.html#a1722ad5ba07cec52d38c9ebf9df80a2d">qk15 </a>( f, a, b, result, abserr, defabs, resabs )
    353. 

  353. <a name="l00289"></a>00289   <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 2 ) <span class="keyword">then</span>
    354. 

  354. <a name="l00290"></a>00290     call <a class="code" href="quadpack_8f90.html#a27241a527b249e9de59a5ed6bee5f805">qk21 </a>( f, a, b, result, abserr, defabs, resabs )
    355. 

  355. <a name="l00291"></a>00291   <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 3 ) <span class="keyword">then</span>
    356. 

  356. <a name="l00292"></a>00292     call <a class="code" href="quadpack_8f90.html#aded2e8dd2218fbd159b78c0e8975a4cd">qk31 </a>( f, a, b, result, abserr, defabs, resabs )
    357. 

  357. <a name="l00293"></a>00293   <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 4 ) <span class="keyword">then</span>
    358. 

  358. <a name="l00294"></a>00294     call <a class="code" href="quadpack_8f90.html#aface4edf24710a0b323f5aaeb6bdec34">qk41 </a>( f, a, b, result, abserr, defabs, resabs )
    359. 

  359. <a name="l00295"></a>00295   <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 5 ) <span class="keyword">then</span>
    360. 

  360. <a name="l00296"></a>00296     call <a class="code" href="quadpack_8f90.html#a73edb4987a87a40ebf4731ab63d7f03e">qk51 </a>( f, a, b, result, abserr, defabs, resabs )
    361. 

  361. <a name="l00297"></a>00297   <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 6 ) <span class="keyword">then</span>
    362. 

  362. <a name="l00298"></a>00298     call <a class="code" href="quadpack_8f90.html#acb4a48f5e54a2c5f951d0828e8f8146d">qk61 </a>( f, a, b, result, abserr, defabs, resabs )
    363. 

  363. <a name="l00299"></a>00299   <span class="keyword">end if</span>
    364. 

  364. <a name="l00300"></a>00300 
    365. 

  365. <a name="l00301"></a>00301   last = 1
    366. 

  366. <a name="l00302"></a>00302   rlist(1) = result
    367. 

  367. <a name="l00303"></a>00303   elist(1) = abserr
    368. 

  368. <a name="l00304"></a>00304   iord(1) = 1
    369. 

  369. <a name="l00305"></a>00305 <span class="comment">!</span>
    370. 

  370. <a name="l00306"></a>00306 <span class="comment">!  Test on accuracy.</span>
    371. 

  371. <a name="l00307"></a>00307 <span class="comment">!</span>
    372. 

  372. <a name="l00308"></a>00308   errbnd = max ( epsabs, epsrel * abs ( result ) )
    373. 

  373. <a name="l00309"></a>00309 
    374. 

  374. <a name="l00310"></a>00310   <span class="keyword">if</span> ( abserr &lt;= 5.0e+01 * epsilon ( defabs ) * defabs .and. &amp;
    375. 

  375. <a name="l00311"></a>00311     errbnd &lt; abserr ) <span class="keyword">then</span>
    376. 

  376. <a name="l00312"></a>00312     ier = 2
    377. 

  377. <a name="l00313"></a>00313   <span class="keyword">end if</span>
    378. 

  378. <a name="l00314"></a>00314 
    379. 

  379. <a name="l00315"></a>00315   <span class="keyword">if</span> ( limit == 1 ) <span class="keyword">then</span>
    380. 

  380. <a name="l00316"></a>00316     ier = 1
    381. 

  381. <a name="l00317"></a>00317   <span class="keyword">end if</span>
    382. 

  382. <a name="l00318"></a>00318 
    383. 

  383. <a name="l00319"></a>00319   <span class="keyword">if</span> ( ier /= 0 .or. &amp;
    384. 

  384. <a name="l00320"></a>00320     ( abserr &lt;= errbnd .and. abserr /= resabs ) .or. &amp;
    385. 

  385. <a name="l00321"></a>00321     abserr == 0.0e+00 ) <span class="keyword">then</span>
    386. 

  386. <a name="l00322"></a>00322 
    387. 

  387. <a name="l00323"></a>00323     <span class="keyword">if</span> ( keyf /= 1 ) <span class="keyword">then</span>
    388. 

  388. <a name="l00324"></a>00324       neval = (10*keyf+1) * (2*neval+1)
    389. 

  389. <a name="l00325"></a>00325     <span class="keyword">else</span>
    390. 

  390. <a name="l00326"></a>00326       neval = 30 * neval + 15
    391. 

  391. <a name="l00327"></a>00327     <span class="keyword">end if</span>
    392. 

  392. <a name="l00328"></a>00328 
    393. 

  393. <a name="l00329"></a>00329     return
    394. 

  394. <a name="l00330"></a>00330 
    395. 

  395. <a name="l00331"></a>00331   <span class="keyword">end if</span>
    396. 

  396. <a name="l00332"></a>00332 <span class="comment">!</span>
    397. 

  397. <a name="l00333"></a>00333 <span class="comment">!  Initialization.</span>
    398. 

  398. <a name="l00334"></a>00334 <span class="comment">!</span>
    399. 

  399. <a name="l00335"></a>00335   errmax = abserr
    400. 

  400. <a name="l00336"></a>00336   maxerr = 1
    401. 

  401. <a name="l00337"></a>00337   area = result
    402. 

  402. <a name="l00338"></a>00338   errsum = abserr
    403. 

  403. <a name="l00339"></a>00339   nrmax = 1
    404. 

  404. <a name="l00340"></a>00340   iroff1 = 0
    405. 

  405. <a name="l00341"></a>00341   iroff2 = 0
    406. 

  406. <a name="l00342"></a>00342 
    407. 

  407. <a name="l00343"></a>00343   <span class="keyword">do</span> last = 2, limit
    408. 

  408. <a name="l00344"></a>00344 <span class="comment">!</span>
    409. 

  409. <a name="l00345"></a>00345 <span class="comment">!  Bisect the subinterval with the largest error estimate.</span>
    410. 

  410. <a name="l00346"></a>00346 <span class="comment">!</span>
    411. 

  411. <a name="l00347"></a>00347     a1 = alist(maxerr)
    412. 

  412. <a name="l00348"></a>00348     b1 = 0.5E+00 * ( alist(maxerr) + blist(maxerr) )
    413. 

  413. <a name="l00349"></a>00349     a2 = b1
    414. 

  414. <a name="l00350"></a>00350     b2 = blist(maxerr)
    415. 

  415. <a name="l00351"></a>00351 
    416. 

  416. <a name="l00352"></a>00352     <span class="keyword">if</span> ( keyf == 1 ) <span class="keyword">then</span>
    417. 

  417. <a name="l00353"></a>00353       call <a class="code" href="quadpack_8f90.html#a1722ad5ba07cec52d38c9ebf9df80a2d">qk15 </a>( f, a1, b1, area1, error1, resabs, defab1 )
    418. 

  418. <a name="l00354"></a>00354     <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 2 ) <span class="keyword">then</span>
    419. 

  419. <a name="l00355"></a>00355       call <a class="code" href="quadpack_8f90.html#a27241a527b249e9de59a5ed6bee5f805">qk21 </a>( f, a1, b1, area1, error1, resabs, defab1 )
    420. 

  420. <a name="l00356"></a>00356     <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 3 ) <span class="keyword">then</span>
    421. 

  421. <a name="l00357"></a>00357       call <a class="code" href="quadpack_8f90.html#aded2e8dd2218fbd159b78c0e8975a4cd">qk31 </a>( f, a1, b1, area1, error1, resabs, defab1 )
    422. 

  422. <a name="l00358"></a>00358     <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 4 ) <span class="keyword">then</span>
    423. 

  423. <a name="l00359"></a>00359       call <a class="code" href="quadpack_8f90.html#aface4edf24710a0b323f5aaeb6bdec34">qk41 </a>( f, a1, b1, area1, error1, resabs, defab1)
    424. 

  424. <a name="l00360"></a>00360     <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 5 ) <span class="keyword">then</span>
    425. 

  425. <a name="l00361"></a>00361       call <a class="code" href="quadpack_8f90.html#a73edb4987a87a40ebf4731ab63d7f03e">qk51 </a>( f, a1, b1, area1, error1, resabs, defab1 )
    426. 

  426. <a name="l00362"></a>00362     <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 6 ) <span class="keyword">then</span>
    427. 

  427. <a name="l00363"></a>00363       call <a class="code" href="quadpack_8f90.html#acb4a48f5e54a2c5f951d0828e8f8146d">qk61 </a>( f, a1, b1, area1, error1, resabs, defab1 )
    428. 

  428. <a name="l00364"></a>00364     <span class="keyword">end if</span>
    429. 

  429. <a name="l00365"></a>00365 
    430. 

  430. <a name="l00366"></a>00366     <span class="keyword">if</span> ( keyf == 1 ) <span class="keyword">then</span>
    431. 

  431. <a name="l00367"></a>00367       call <a class="code" href="quadpack_8f90.html#a1722ad5ba07cec52d38c9ebf9df80a2d">qk15 </a>( f, a2, b2, area2, error2, resabs, defab2 )
    432. 

  432. <a name="l00368"></a>00368     <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 2 ) <span class="keyword">then</span>
    433. 

  433. <a name="l00369"></a>00369       call <a class="code" href="quadpack_8f90.html#a27241a527b249e9de59a5ed6bee5f805">qk21 </a>( f, a2, b2, area2, error2, resabs, defab2 )
    434. 

  434. <a name="l00370"></a>00370     <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 3 ) <span class="keyword">then</span>
    435. 

  435. <a name="l00371"></a>00371       call <a class="code" href="quadpack_8f90.html#aded2e8dd2218fbd159b78c0e8975a4cd">qk31 </a>( f, a2, b2, area2, error2, resabs, defab2 )
    436. 

  436. <a name="l00372"></a>00372     <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 4 ) <span class="keyword">then</span>
    437. 

  437. <a name="l00373"></a>00373       call <a class="code" href="quadpack_8f90.html#aface4edf24710a0b323f5aaeb6bdec34">qk41 </a>( f, a2, b2, area2, error2, resabs, defab2 )
    438. 

  438. <a name="l00374"></a>00374     <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 5 ) <span class="keyword">then</span>
    439. 

  439. <a name="l00375"></a>00375       call <a class="code" href="quadpack_8f90.html#a73edb4987a87a40ebf4731ab63d7f03e">qk51 </a>( f, a2, b2, area2, error2, resabs, defab2 )
    440. 

  440. <a name="l00376"></a>00376     <span class="keyword">else</span> <span class="keyword">if</span> ( keyf == 6 ) <span class="keyword">then</span>
    441. 

  441. <a name="l00377"></a>00377       call <a class="code" href="quadpack_8f90.html#acb4a48f5e54a2c5f951d0828e8f8146d">qk61 </a>( f, a2, b2, area2, error2, resabs, defab2 )
    442. 

  442. <a name="l00378"></a>00378     <span class="keyword">end if</span>
    443. 

  443. <a name="l00379"></a>00379 <span class="comment">!</span>
    444. 

  444. <a name="l00380"></a>00380 <span class="comment">!  Improve previous approximations to integral and error and</span>
    445. 

  445. <a name="l00381"></a>00381 <span class="comment">!  test for accuracy.</span>
    446. 

  446. <a name="l00382"></a>00382 <span class="comment">!</span>
    447. 

  447. <a name="l00383"></a>00383     neval = neval + 1
    448. 

  448. <a name="l00384"></a>00384     area12 = area1 + area2
    449. 

  449. <a name="l00385"></a>00385     erro12 = error1 + error2
    450. 

  450. <a name="l00386"></a>00386     errsum = errsum + erro12 - errmax
    451. 

  451. <a name="l00387"></a>00387     area = area + area12 - rlist(maxerr)
    452. 

  452. <a name="l00388"></a>00388 
    453. 

  453. <a name="l00389"></a>00389     <span class="keyword">if</span> ( defab1 /= error1 .and. defab2 /= error2 ) <span class="keyword">then</span>
    454. 

  454. <a name="l00390"></a>00390 
    455. 

  455. <a name="l00391"></a>00391       <span class="keyword">if</span> ( abs ( rlist(maxerr) - area12 ) &lt;= 1.0e-05 * abs ( area12 ) &amp;
    456. 

  456. <a name="l00392"></a>00392         .and. 9.9e-01 * errmax &lt;= erro12 ) <span class="keyword">then</span>
    457. 

  457. <a name="l00393"></a>00393         iroff1 = iroff1 + 1
    458. 

  458. <a name="l00394"></a>00394       <span class="keyword">end if</span>
    459. 

  459. <a name="l00395"></a>00395 
    460. 

  460. <a name="l00396"></a>00396       <span class="keyword">if</span> ( 10 &lt; last .and. errmax &lt; erro12 ) <span class="keyword">then</span>
    461. 

  461. <a name="l00397"></a>00397         iroff2 = iroff2 + 1
    462. 

  462. <a name="l00398"></a>00398       <span class="keyword">end if</span>
    463. 

  463. <a name="l00399"></a>00399 
    464. 

  464. <a name="l00400"></a>00400     <span class="keyword">end if</span>
    465. 

  465. <a name="l00401"></a>00401 
    466. 

  466. <a name="l00402"></a>00402     rlist(maxerr) = area1
    467. 

  467. <a name="l00403"></a>00403     rlist(last) = area2
    468. 

  468. <a name="l00404"></a>00404     errbnd = max ( epsabs, epsrel * abs ( area ) )
    469. 

  469. <a name="l00405"></a>00405 <span class="comment">!</span>
    470. 

  470. <a name="l00406"></a>00406 <span class="comment">!  Test for roundoff error and eventually set error flag.</span>
    471. 

  471. <a name="l00407"></a>00407 <span class="comment">!</span>
    472. 

  472. <a name="l00408"></a>00408     <span class="keyword">if</span> ( errbnd &lt; errsum ) <span class="keyword">then</span>
    473. 

  473. <a name="l00409"></a>00409 
    474. 

  474. <a name="l00410"></a>00410       <span class="keyword">if</span> ( 6 &lt;= iroff1 .or. 20 &lt;= iroff2 ) <span class="keyword">then</span>
    475. 

  475. <a name="l00411"></a>00411         ier = 2
    476. 

  476. <a name="l00412"></a>00412       <span class="keyword">end if</span>
    477. 

  477. <a name="l00413"></a>00413 <span class="comment">!</span>
    478. 

  478. <a name="l00414"></a>00414 <span class="comment">!  Set error flag in the case that the number of subintervals</span>
    479. 

  479. <a name="l00415"></a>00415 <span class="comment">!  equals limit.</span>
    480. 

  480. <a name="l00416"></a>00416 <span class="comment">!</span>
    481. 

  481. <a name="l00417"></a>00417       <span class="keyword">if</span> ( last == limit ) <span class="keyword">then</span>
    482. 

  482. <a name="l00418"></a>00418         ier = 1
    483. 

  483. <a name="l00419"></a>00419       <span class="keyword">end if</span>
    484. 

  484. <a name="l00420"></a>00420 <span class="comment">!</span>
    485. 

  485. <a name="l00421"></a>00421 <span class="comment">!  Set error flag in the case of bad integrand behavior</span>
    486. 

  486. <a name="l00422"></a>00422 <span class="comment">!  at a point of the integration range.</span>
    487. 

  487. <a name="l00423"></a>00423 <span class="comment">!</span>
    488. 

  488. <a name="l00424"></a>00424       <span class="keyword">if</span> ( max ( abs ( a1 ), abs ( b2 ) ) &lt;= ( 1.0e+00 + c * 1.0e+03 * &amp;
    489. 

  489. <a name="l00425"></a>00425         epsilon ( a1 ) ) * ( abs ( a2 ) + 1.0e+04 * tiny ( a2 ) ) ) <span class="keyword">then</span>
    490. 

  490. <a name="l00426"></a>00426         ier = 3
    491. 

  491. <a name="l00427"></a>00427       <span class="keyword">end if</span>
    492. 

  492. <a name="l00428"></a>00428 
    493. 

  493. <a name="l00429"></a>00429     <span class="keyword">end if</span>
    494. 

  494. <a name="l00430"></a>00430 <span class="comment">!</span>
    495. 

  495. <a name="l00431"></a>00431 <span class="comment">!  Append the newly-created intervals to the list.</span>
    496. 

  496. <a name="l00432"></a>00432 <span class="comment">!</span>
    497. 

  497. <a name="l00433"></a>00433     <span class="keyword">if</span> ( error2 &lt;= error1 ) <span class="keyword">then</span>
    498. 

  498. <a name="l00434"></a>00434       alist(last) = a2
    499. 

  499. <a name="l00435"></a>00435       blist(maxerr) = b1
    500. 

  500. <a name="l00436"></a>00436       blist(last) = b2
    501. 

  501. <a name="l00437"></a>00437       elist(maxerr) = error1
    502. 

  502. <a name="l00438"></a>00438       elist(last) = error2
    503. 

  503. <a name="l00439"></a>00439     <span class="keyword">else</span>
    504. 

  504. <a name="l00440"></a>00440       alist(maxerr) = a2
    505. 

  505. <a name="l00441"></a>00441       alist(last) = a1
    506. 

  506. <a name="l00442"></a>00442       blist(last) = b1
    507. 

  507. <a name="l00443"></a>00443       rlist(maxerr) = area2
    508. 

  508. <a name="l00444"></a>00444       rlist(last) = area1
    509. 

  509. <a name="l00445"></a>00445       elist(maxerr) = error2
    510. 

  510. <a name="l00446"></a>00446       elist(last) = error1
    511. 

  511. <a name="l00447"></a>00447     <span class="keyword">end if</span>
    512. 

  512. <a name="l00448"></a>00448 <span class="comment">!</span>
    513. 

  513. <a name="l00449"></a>00449 <span class="comment">!  Call QSORT to maintain the descending ordering</span>
    514. 

  514. <a name="l00450"></a>00450 <span class="comment">!  in the list of error estimates and select the subinterval</span>
    515. 

  515. <a name="l00451"></a>00451 <span class="comment">!  with the largest error estimate (to be bisected next).</span>
    516. 

  516. <a name="l00452"></a>00452 <span class="comment">!</span>
    517. 

  517. <a name="l00453"></a>00453     call <a class="code" href="quadpack_8f90.html#a55e08a684c5a6315fb37dd0fdc66d8e6">qsort </a>( limit, last, maxerr, errmax, elist, iord, nrmax )
    518. 

  518. <a name="l00454"></a>00454  
    519. 

  519. <a name="l00455"></a>00455     <span class="keyword">if</span> ( ier /= 0 .or. errsum &lt;= errbnd ) <span class="keyword">then</span>
    520. 

  520. <a name="l00456"></a>00456       exit
    521. 

  521. <a name="l00457"></a>00457     <span class="keyword">end if</span>
    522. 

  522. <a name="l00458"></a>00458 
    523. 

  523. <a name="l00459"></a>00459   <span class="keyword">end do</span>
    524. 

  524. <a name="l00460"></a>00460 <span class="comment">!</span>
    525. 

  525. <a name="l00461"></a>00461 <span class="comment">!  Compute final result.</span>
    526. 

  526. <a name="l00462"></a>00462 <span class="comment">!</span>
    527. 

  527. <a name="l00463"></a>00463   result = sum ( rlist(1:last) )
    528. 

  528. <a name="l00464"></a>00464 
    529. 

  529. <a name="l00465"></a>00465   abserr = errsum
    530. 

  530. <a name="l00466"></a>00466 
    531. 

  531. <a name="l00467"></a>00467   <span class="keyword">if</span> ( keyf /= 1 ) <span class="keyword">then</span>
    532. 

  532. <a name="l00468"></a>00468     neval = ( 10 * keyf + 1 ) * ( 2 * neval + 1 )
    533. 

  533. <a name="l00469"></a>00469   <span class="keyword">else</span>
    534. 

  534. <a name="l00470"></a>00470     neval = 30 * neval + 15
    535. 

  535. <a name="l00471"></a>00471   <span class="keyword">end if</span>
    536. 

  536. <a name="l00472"></a>00472 
    537. 

  537. <a name="l00473"></a>00473   return
    538. 

  538. <a name="l00474"></a>00474 <span class="keyword">end</span>
    539. 

  539. <a name="l00475"></a><a class="code" href="quadpack_8f90.html#ac59eaf7c56c1d421d129425895fa0107">00475</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#ac59eaf7c56c1d421d129425895fa0107">qagi</a> ( f, bound, inf, epsabs, epsrel, result, abserr, neval, ier )
    540. 

  540. <a name="l00476"></a>00476 
    541. 

  541. <a name="l00477"></a>00477 <span class="comment">!*****************************************************************************80</span>
    542. 

  542. <a name="l00478"></a>00478 <span class="comment">!</span>
    543. 

  543. <a name="l00479"></a>00479 <span class="comment">!! QAGI estimates an integral over a semi-infinite or infinite interval.</span>
    544. 

  544. <a name="l00480"></a>00480 <span class="comment">!</span>
    545. 

  545. <a name="l00481"></a>00481 <span class="comment">!  Discussion:</span>
    546. 

  546. <a name="l00482"></a>00482 <span class="comment">!</span>
    547. 

  547. <a name="l00483"></a>00483 <span class="comment">!    The routine calculates an approximation RESULT to a definite integral   </span>
    548. 

  548. <a name="l00484"></a>00484 <span class="comment">!      I = integral of F over (A, +Infinity), </span>
    549. 

  549. <a name="l00485"></a>00485 <span class="comment">!    or </span>
    550. 

  550. <a name="l00486"></a>00486 <span class="comment">!      I = integral of F over (-Infinity,A)</span>
    551. 

  551. <a name="l00487"></a>00487 <span class="comment">!    or </span>
    552. 

  552. <a name="l00488"></a>00488 <span class="comment">!      I = integral of F over (-Infinity,+Infinity),</span>
    553. 

  553. <a name="l00489"></a>00489 <span class="comment">!    hopefully satisfying</span>
    554. 

  554. <a name="l00490"></a>00490 <span class="comment">!      || I - RESULT || &lt;= max ( EPSABS, EPSREL * ||I|| ).</span>
    555. 

  555. <a name="l00491"></a>00491 <span class="comment">!</span>
    556. 

  556. <a name="l00492"></a>00492 <span class="comment">!  Author:</span>
    557. 

  557. <a name="l00493"></a>00493 <span class="comment">!</span>
    558. 

  558. <a name="l00494"></a>00494 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    559. 

  559. <a name="l00495"></a>00495 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    560. 

  560. <a name="l00496"></a>00496 <span class="comment">!</span>
    561. 

  561. <a name="l00497"></a>00497 <span class="comment">!  Reference:</span>
    562. 

  562. <a name="l00498"></a>00498 <span class="comment">!</span>
    563. 

  563. <a name="l00499"></a>00499 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    564. 

  564. <a name="l00500"></a>00500 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    565. 

  565. <a name="l00501"></a>00501 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    566. 

  566. <a name="l00502"></a>00502 <span class="comment">!    Springer Verlag, 1983</span>
    567. 

  567. <a name="l00503"></a>00503 <span class="comment">!</span>
    568. 

  568. <a name="l00504"></a>00504 <span class="comment">!  Parameters:</span>
    569. 

  569. <a name="l00505"></a>00505 <span class="comment">!</span>
    570. 

  570. <a name="l00506"></a>00506 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    571. 

  571. <a name="l00507"></a>00507 <span class="comment">!      function f ( x )</span>
    572. 

  572. <a name="l00508"></a>00508 <span class="comment">!      real f</span>
    573. 

  573. <a name="l00509"></a>00509 <span class="comment">!      real x</span>
    574. 

  574. <a name="l00510"></a>00510 <span class="comment">!    which evaluates the integrand function.</span>
    575. 

  575. <a name="l00511"></a>00511 <span class="comment">!</span>
    576. 

  576. <a name="l00512"></a>00512 <span class="comment">!    Input, real BOUND, the value of the finite endpoint of the integration</span>
    577. 

  577. <a name="l00513"></a>00513 <span class="comment">!    range, if any, that is, if INF is 1 or -1.</span>
    578. 

  578. <a name="l00514"></a>00514 <span class="comment">!</span>
    579. 

  579. <a name="l00515"></a>00515 <span class="comment">!    Input, integer INF, indicates the type of integration range.</span>
    580. 

  580. <a name="l00516"></a>00516 <span class="comment">!    1:  (  BOUND,    +Infinity),</span>
    581. 

  581. <a name="l00517"></a>00517 <span class="comment">!    -1: ( -Infinity,  BOUND),</span>
    582. 

  582. <a name="l00518"></a>00518 <span class="comment">!    2:  ( -Infinity, +Infinity).</span>
    583. 

  583. <a name="l00519"></a>00519 <span class="comment">!</span>
    584. 

  584. <a name="l00520"></a>00520 <span class="comment">!    Input, real EPSABS, EPSREL, the absolute and relative accuracy requested.</span>
    585. 

  585. <a name="l00521"></a>00521 <span class="comment">!</span>
    586. 

  586. <a name="l00522"></a>00522 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    587. 

  587. <a name="l00523"></a>00523 <span class="comment">!</span>
    588. 

  588. <a name="l00524"></a>00524 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    589. 

  589. <a name="l00525"></a>00525 <span class="comment">!</span>
    590. 

  590. <a name="l00526"></a>00526 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    591. 

  591. <a name="l00527"></a>00527 <span class="comment">!</span>
    592. 

  592. <a name="l00528"></a>00528 <span class="comment">!    Output, integer IER, error indicator.</span>
    593. 

  593. <a name="l00529"></a>00529 <span class="comment">!    0, normal and reliable termination of the routine.  It is assumed that </span>
    594. 

  594. <a name="l00530"></a>00530 <span class="comment">!      the requested accuracy has been achieved.</span>
    595. 

  595. <a name="l00531"></a>00531 <span class="comment">!    &gt; 0,  abnormal termination of the routine.  The estimates for result</span>
    596. 

  596. <a name="l00532"></a>00532 <span class="comment">!      and error are less reliable.  It is assumed that the requested</span>
    597. 

  597. <a name="l00533"></a>00533 <span class="comment">!      accuracy has not been achieved.</span>
    598. 

  598. <a name="l00534"></a>00534 <span class="comment">!    1, maximum number of subdivisions allowed has been achieved.  One can </span>
    599. 

  599. <a name="l00535"></a>00535 <span class="comment">!      allow more subdivisions by increasing the data value of LIMIT in QAGI</span>
    600. 

  600. <a name="l00536"></a>00536 <span class="comment">!      (and taking the according dimension adjustments into account).</span>
    601. 

  601. <a name="l00537"></a>00537 <span class="comment">!      However, if this yields no improvement it is advised to analyze the</span>
    602. 

  602. <a name="l00538"></a>00538 <span class="comment">!      integrand in order to determine the integration difficulties.  If the</span>
    603. 

  603. <a name="l00539"></a>00539 <span class="comment">!      position of a local difficulty can be determined (e.g. singularity,</span>
    604. 

  604. <a name="l00540"></a>00540 <span class="comment">!      discontinuity within the interval) one will probably gain from</span>
    605. 

  605. <a name="l00541"></a>00541 <span class="comment">!      splitting up the interval at this point and calling the integrator </span>
    606. 

  606. <a name="l00542"></a>00542 <span class="comment">!      on the subranges.  If possible, an appropriate special-purpose </span>
    607. 

  607. <a name="l00543"></a>00543 <span class="comment">!      integrator should be used, which is designed for handling the type</span>
    608. 

  608. <a name="l00544"></a>00544 <span class="comment">!      of difficulty involved.</span>
    609. 

  609. <a name="l00545"></a>00545 <span class="comment">!    2, the occurrence of roundoff error is detected, which prevents the</span>
    610. 

  610. <a name="l00546"></a>00546 <span class="comment">!      requested tolerance from being achieved.  The error may be</span>
    611. 

  611. <a name="l00547"></a>00547 <span class="comment">!      under-estimated.</span>
    612. 

  612. <a name="l00548"></a>00548 <span class="comment">!    3, extremely bad integrand behavior occurs at some points of the</span>
    613. 

  613. <a name="l00549"></a>00549 <span class="comment">!      integration interval.</span>
    614. 

  614. <a name="l00550"></a>00550 <span class="comment">!    4, the algorithm does not converge.  Roundoff error is detected in the</span>
    615. 

  615. <a name="l00551"></a>00551 <span class="comment">!      extrapolation table.  It is assumed that the requested tolerance</span>
    616. 

  616. <a name="l00552"></a>00552 <span class="comment">!      cannot be achieved, and that the returned result is the best which </span>
    617. 

  617. <a name="l00553"></a>00553 <span class="comment">!      can be obtained.</span>
    618. 

  618. <a name="l00554"></a>00554 <span class="comment">!    5, the integral is probably divergent, or slowly convergent.  It must </span>
    619. 

  619. <a name="l00555"></a>00555 <span class="comment">!      be noted that divergence can occur with any other value of IER.</span>
    620. 

  620. <a name="l00556"></a>00556 <span class="comment">!    6, the input is invalid, because INF /= 1 and INF /= -1 and INF /= 2, or</span>
    621. 

  621. <a name="l00557"></a>00557 <span class="comment">!      epsabs &lt; 0 and epsrel &lt; 0.  result, abserr, neval are set to zero.</span>
    622. 

  622. <a name="l00558"></a>00558 <span class="comment">!</span>
    623. 

  623. <a name="l00559"></a>00559 <span class="comment">!  Local parameters:</span>
    624. 

  624. <a name="l00560"></a>00560 <span class="comment">!</span>
    625. 

  625. <a name="l00561"></a>00561 <span class="comment">!            the dimension of rlist2 is determined by the value of</span>
    626. 

  626. <a name="l00562"></a>00562 <span class="comment">!            limexp in QEXTR.</span>
    627. 

  627. <a name="l00563"></a>00563 <span class="comment">!</span>
    628. 

  628. <a name="l00564"></a>00564 <span class="comment">!           alist     - list of left end points of all subintervals</span>
    629. 

  629. <a name="l00565"></a>00565 <span class="comment">!                       considered up to now</span>
    630. 

  630. <a name="l00566"></a>00566 <span class="comment">!           blist     - list of right end points of all subintervals</span>
    631. 

  631. <a name="l00567"></a>00567 <span class="comment">!                       considered up to now</span>
    632. 

  632. <a name="l00568"></a>00568 <span class="comment">!           rlist(i)  - approximation to the integral over</span>
    633. 

  633. <a name="l00569"></a>00569 <span class="comment">!                       (alist(i),blist(i))</span>
    634. 

  634. <a name="l00570"></a>00570 <span class="comment">!           rlist2    - array of dimension at least (limexp+2),</span>
    635. 

  635. <a name="l00571"></a>00571 <span class="comment">!                       containing the part of the epsilon table</span>
    636. 

  636. <a name="l00572"></a>00572 <span class="comment">!                       which is still needed for further computations</span>
    637. 

  637. <a name="l00573"></a>00573 <span class="comment">!           elist(i)  - error estimate applying to rlist(i)</span>
    638. 

  638. <a name="l00574"></a>00574 <span class="comment">!           maxerr    - pointer to the interval with largest error</span>
    639. 

  639. <a name="l00575"></a>00575 <span class="comment">!                       estimate</span>
    640. 

  640. <a name="l00576"></a>00576 <span class="comment">!           errmax    - elist(maxerr)</span>
    641. 

  641. <a name="l00577"></a>00577 <span class="comment">!           erlast    - error on the interval currently subdivided</span>
    642. 

  642. <a name="l00578"></a>00578 <span class="comment">!                       (before that subdivision has taken place)</span>
    643. 

  643. <a name="l00579"></a>00579 <span class="comment">!           area      - sum of the integrals over the subintervals</span>
    644. 

  644. <a name="l00580"></a>00580 <span class="comment">!           errsum    - sum of the errors over the subintervals</span>
    645. 

  645. <a name="l00581"></a>00581 <span class="comment">!           errbnd    - requested accuracy max(epsabs,epsrel*</span>
    646. 

  646. <a name="l00582"></a>00582 <span class="comment">!                       abs(result))</span>
    647. 

  647. <a name="l00583"></a>00583 <span class="comment">!           *****1    - variable for the left subinterval</span>
    648. 

  648. <a name="l00584"></a>00584 <span class="comment">!           *****2    - variable for the right subinterval</span>
    649. 

  649. <a name="l00585"></a>00585 <span class="comment">!           last      - index for subdivision</span>
    650. 

  650. <a name="l00586"></a>00586 <span class="comment">!           nres      - number of calls to the extrapolation routine</span>
    651. 

  651. <a name="l00587"></a>00587 <span class="comment">!           numrl2    - number of elements currently in rlist2. if an</span>
    652. 

  652. <a name="l00588"></a>00588 <span class="comment">!                       appropriate approximation to the compounded</span>
    653. 

  653. <a name="l00589"></a>00589 <span class="comment">!                       integral has been obtained, it is put in</span>
    654. 

  654. <a name="l00590"></a>00590 <span class="comment">!                       rlist2(numrl2) after numrl2 has been increased</span>
    655. 

  655. <a name="l00591"></a>00591 <span class="comment">!                       by one.</span>
    656. 

  656. <a name="l00592"></a>00592 <span class="comment">!           small     - length of the smallest interval considered up</span>
    657. 

  657. <a name="l00593"></a>00593 <span class="comment">!                       to now, multiplied by 1.5</span>
    658. 

  658. <a name="l00594"></a>00594 <span class="comment">!           erlarg    - sum of the errors over the intervals larger</span>
    659. 

  659. <a name="l00595"></a>00595 <span class="comment">!                       than the smallest interval considered up to now</span>
    660. 

  660. <a name="l00596"></a>00596 <span class="comment">!           extrap    - logical variable denoting that the routine</span>
    661. 

  661. <a name="l00597"></a>00597 <span class="comment">!                       is attempting to perform extrapolation. i.e.</span>
    662. 

  662. <a name="l00598"></a>00598 <span class="comment">!                       before subdividing the smallest interval we</span>
    663. 

  663. <a name="l00599"></a>00599 <span class="comment">!                       try to decrease the value of erlarg.</span>
    664. 

  664. <a name="l00600"></a>00600 <span class="comment">!           noext     - logical variable denoting that extrapolation</span>
    665. 

  665. <a name="l00601"></a>00601 <span class="comment">!                       is no longer allowed (true-value)</span>
    666. 

  666. <a name="l00602"></a>00602 <span class="comment">!</span>
    667. 

  667. <a name="l00603"></a>00603   <span class="keyword">implicit none</span>
    668. 

  668. <a name="l00604"></a>00604 
    669. 

  669. <a name="l00605"></a>00605   <span class="keywordtype">integer</span>, <span class="keywordtype">parameter</span> :: limit = 500
    670. 

  670. <a name="l00606"></a>00606 
    671. 

  671. <a name="l00607"></a>00607   <span class="keywordtype">real</span> abseps
    672. 

  672. <a name="l00608"></a>00608   <span class="keywordtype">real</span> abserr
    673. 

  673. <a name="l00609"></a>00609   <span class="keywordtype">real</span> alist(limit)
    674. 

  674. <a name="l00610"></a>00610   <span class="keywordtype">real</span> area
    675. 

  675. <a name="l00611"></a>00611   <span class="keywordtype">real</span> area1
    676. 

  676. <a name="l00612"></a>00612   <span class="keywordtype">real</span> area12
    677. 

  677. <a name="l00613"></a>00613   <span class="keywordtype">real</span> area2
    678. 

  678. <a name="l00614"></a>00614   <span class="keywordtype">real</span> a1
    679. 

  679. <a name="l00615"></a>00615   <span class="keywordtype">real</span> a2
    680. 

  680. <a name="l00616"></a>00616   <span class="keywordtype">real</span> blist(limit)
    681. 

  681. <a name="l00617"></a>00617   <span class="keywordtype">real</span> boun
    682. 

  682. <a name="l00618"></a>00618   <span class="keywordtype">real</span> bound
    683. 

  683. <a name="l00619"></a>00619   <span class="keywordtype">real</span> b1
    684. 

  684. <a name="l00620"></a>00620   <span class="keywordtype">real</span> b2
    685. 

  685. <a name="l00621"></a>00621   <span class="keywordtype">real</span> correc
    686. 

  686. <a name="l00622"></a>00622   <span class="keywordtype">real</span> defabs
    687. 

  687. <a name="l00623"></a>00623   <span class="keywordtype">real</span> defab1
    688. 

  688. <a name="l00624"></a>00624   <span class="keywordtype">real</span> defab2
    689. 

  689. <a name="l00625"></a>00625   <span class="keywordtype">real</span> dres
    690. 

  690. <a name="l00626"></a>00626   <span class="keywordtype">real</span> elist(limit)
    691. 

  691. <a name="l00627"></a>00627   <span class="keywordtype">real</span> epsabs
    692. 

  692. <a name="l00628"></a>00628   <span class="keywordtype">real</span> epsrel
    693. 

  693. <a name="l00629"></a>00629   <span class="keywordtype">real</span> erlarg
    694. 

  694. <a name="l00630"></a>00630   <span class="keywordtype">real</span> erlast
    695. 

  695. <a name="l00631"></a>00631   <span class="keywordtype">real</span> errbnd
    696. 

  696. <a name="l00632"></a>00632   <span class="keywordtype">real</span> errmax
    697. 

  697. <a name="l00633"></a>00633   <span class="keywordtype">real</span> error1
    698. 

  698. <a name="l00634"></a>00634   <span class="keywordtype">real</span> error2
    699. 

  699. <a name="l00635"></a>00635   <span class="keywordtype">real</span> erro12
    700. 

  700. <a name="l00636"></a>00636   <span class="keywordtype">real</span> errsum
    701. 

  701. <a name="l00637"></a>00637   <span class="keywordtype">real</span> ertest
    702. 

  702. <a name="l00638"></a>00638   <span class="keywordtype">logical</span> extrap
    703. 

  703. <a name="l00639"></a>00639   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    704. 

  704. <a name="l00640"></a>00640   <span class="keywordtype">integer</span> id
    705. 

  705. <a name="l00641"></a>00641   <span class="keywordtype">integer</span> ier
    706. 

  706. <a name="l00642"></a>00642   <span class="keywordtype">integer</span> ierro
    707. 

  707. <a name="l00643"></a>00643   <span class="keywordtype">integer</span> inf
    708. 

  708. <a name="l00644"></a>00644   <span class="keywordtype">integer</span> iord(limit)
    709. 

  709. <a name="l00645"></a>00645   <span class="keywordtype">integer</span> iroff1
    710. 

  710. <a name="l00646"></a>00646   <span class="keywordtype">integer</span> iroff2
    711. 

  711. <a name="l00647"></a>00647   <span class="keywordtype">integer</span> iroff3
    712. 

  712. <a name="l00648"></a>00648   <span class="keywordtype">integer</span> jupbnd
    713. 

  713. <a name="l00649"></a>00649   <span class="keywordtype">integer</span> k
    714. 

  714. <a name="l00650"></a>00650   <span class="keywordtype">integer</span> ksgn
    715. 

  715. <a name="l00651"></a>00651   <span class="keywordtype">integer</span> ktmin
    716. 

  716. <a name="l00652"></a>00652   <span class="keywordtype">integer</span> last
    717. 

  717. <a name="l00653"></a>00653   <span class="keywordtype">integer</span> maxerr
    718. 

  718. <a name="l00654"></a>00654   <span class="keywordtype">integer</span> neval
    719. 

  719. <a name="l00655"></a>00655   <span class="keywordtype">logical</span> noext
    720. 

  720. <a name="l00656"></a>00656   <span class="keywordtype">integer</span> nres
    721. 

  721. <a name="l00657"></a>00657   <span class="keywordtype">integer</span> nrmax
    722. 

  722. <a name="l00658"></a>00658   <span class="keywordtype">integer</span> numrl2
    723. 

  723. <a name="l00659"></a>00659   <span class="keywordtype">real</span> resabs
    724. 

  724. <a name="l00660"></a>00660   <span class="keywordtype">real</span> reseps
    725. 

  725. <a name="l00661"></a>00661   <span class="keywordtype">real</span> result
    726. 

  726. <a name="l00662"></a>00662   <span class="keywordtype">real</span> res3la(3)
    727. 

  727. <a name="l00663"></a>00663   <span class="keywordtype">real</span> rlist(limit)
    728. 

  728. <a name="l00664"></a>00664   <span class="keywordtype">real</span> rlist2(52)
    729. 

  729. <a name="l00665"></a>00665   <span class="keywordtype">real</span> small
    730. 

  730. <a name="l00666"></a>00666 <span class="comment">!</span>
    731. 

  731. <a name="l00667"></a>00667 <span class="comment">!  Test on validity of parameters.</span>
    732. 

  732. <a name="l00668"></a>00668 <span class="comment">!</span>
    733. 

  733. <a name="l00669"></a>00669   ier = 0
    734. 

  734. <a name="l00670"></a>00670   neval = 0
    735. 

  735. <a name="l00671"></a>00671   last = 0
    736. 

  736. <a name="l00672"></a>00672   result = 0.0e+00
    737. 

  737. <a name="l00673"></a>00673   abserr = 0.0e+00
    738. 

  738. <a name="l00674"></a>00674   alist(1) = 0.0e+00
    739. 

  739. <a name="l00675"></a>00675   blist(1) = 1.0e+00
    740. 

  740. <a name="l00676"></a>00676   rlist(1) = 0.0e+00
    741. 

  741. <a name="l00677"></a>00677   elist(1) = 0.0e+00
    742. 

  742. <a name="l00678"></a>00678   iord(1) = 0
    743. 

  743. <a name="l00679"></a>00679 
    744. 

  744. <a name="l00680"></a>00680   <span class="keyword">if</span> ( epsabs &lt; 0.0e+00 .and. epsrel &lt; 0.0e+00 ) <span class="keyword">then</span>
    745. 

  745. <a name="l00681"></a>00681     ier = 6
    746. 

  746. <a name="l00682"></a>00682     return
    747. 

  747. <a name="l00683"></a>00683   <span class="keyword">end if</span>
    748. 

  748. <a name="l00684"></a>00684 <span class="comment">!</span>
    749. 

  749. <a name="l00685"></a>00685 <span class="comment">!  First approximation to the integral.</span>
    750. 

  750. <a name="l00686"></a>00686 <span class="comment">!</span>
    751. 

  751. <a name="l00687"></a>00687 <span class="comment">!  Determine the interval to be mapped onto (0,1).</span>
    752. 

  752. <a name="l00688"></a>00688 <span class="comment">!  If INF = 2 the integral is computed as i = i1+i2, where</span>
    753. 

  753. <a name="l00689"></a>00689 <span class="comment">!  i1 = integral of f over (-infinity,0),</span>
    754. 

  754. <a name="l00690"></a>00690 <span class="comment">!  i2 = integral of f over (0,+infinity).</span>
    755. 

  755. <a name="l00691"></a>00691 <span class="comment">!</span>
    756. 

  756. <a name="l00692"></a>00692   <span class="keyword">if</span> ( inf == 2 ) <span class="keyword">then</span>
    757. 

  757. <a name="l00693"></a>00693     boun = 0.0e+00
    758. 

  758. <a name="l00694"></a>00694   <span class="keyword">else</span>
    759. 

  759. <a name="l00695"></a>00695     boun = bound
    760. 

  760. <a name="l00696"></a>00696   <span class="keyword">end if</span>
    761. 

  761. <a name="l00697"></a>00697 
    762. 

  762. <a name="l00698"></a>00698   call <a class="code" href="quadpack_8f90.html#a59164415fc33f2f3bf4ebc4ee2220f7e">qk15i </a>( f, boun, inf, 0.0e+00, 1.0e+00, result, abserr, defabs, resabs )
    763. 

  763. <a name="l00699"></a>00699 <span class="comment">!</span>
    764. 

  764. <a name="l00700"></a>00700 <span class="comment">!  Test on accuracy.</span>
    765. 

  765. <a name="l00701"></a>00701 <span class="comment">!</span>
    766. 

  766. <a name="l00702"></a>00702   last = 1
    767. 

  767. <a name="l00703"></a>00703   rlist(1) = result
    768. 

  768. <a name="l00704"></a>00704   elist(1) = abserr
    769. 

  769. <a name="l00705"></a>00705   iord(1) = 1
    770. 

  770. <a name="l00706"></a>00706   dres = abs ( result )
    771. 

  771. <a name="l00707"></a>00707   errbnd = max ( epsabs, epsrel * dres )
    772. 

  772. <a name="l00708"></a>00708 
    773. 

  773. <a name="l00709"></a>00709   <span class="keyword">if</span> ( abserr &lt;= 100.0E+00 * epsilon ( defabs ) * defabs .and. &amp;
    774. 

  774. <a name="l00710"></a>00710     errbnd &lt; abserr ) <span class="keyword">then</span>
    775. 

  775. <a name="l00711"></a>00711     ier = 2
    776. 

  776. <a name="l00712"></a>00712   <span class="keyword">end if</span>
    777. 

  777. <a name="l00713"></a>00713 
    778. 

  778. <a name="l00714"></a>00714   <span class="keyword">if</span> ( limit == 1 ) <span class="keyword">then</span>
    779. 

  779. <a name="l00715"></a>00715     ier = 1
    780. 

  780. <a name="l00716"></a>00716   <span class="keyword">end if</span>
    781. 

  781. <a name="l00717"></a>00717 
    782. 

  782. <a name="l00718"></a>00718   <span class="keyword">if</span> ( ier /= 0 .or. (abserr &lt;= errbnd .and. abserr /= resabs ) .or. &amp;
    783. 

  783. <a name="l00719"></a>00719     abserr == 0.0e+00 ) go to 130
    784. 

  784. <a name="l00720"></a>00720 <span class="comment">!</span>
    785. 

  785. <a name="l00721"></a>00721 <span class="comment">!  Initialization.</span>
    786. 

  786. <a name="l00722"></a>00722 <span class="comment">!</span>
    787. 

  787. <a name="l00723"></a>00723   rlist2(1) = result
    788. 

  788. <a name="l00724"></a>00724   errmax = abserr
    789. 

  789. <a name="l00725"></a>00725   maxerr = 1
    790. 

  790. <a name="l00726"></a>00726   area = result
    791. 

  791. <a name="l00727"></a>00727   errsum = abserr
    792. 

  792. <a name="l00728"></a>00728   abserr = huge ( abserr )
    793. 

  793. <a name="l00729"></a>00729   nrmax = 1
    794. 

  794. <a name="l00730"></a>00730   nres = 0
    795. 

  795. <a name="l00731"></a>00731   ktmin = 0
    796. 

  796. <a name="l00732"></a>00732   numrl2 = 2
    797. 

  797. <a name="l00733"></a>00733   extrap = .false.
    798. 

  798. <a name="l00734"></a>00734   noext = .false.
    799. 

  799. <a name="l00735"></a>00735   ierro = 0
    800. 

  800. <a name="l00736"></a>00736   iroff1 = 0
    801. 

  801. <a name="l00737"></a>00737   iroff2 = 0
    802. 

  802. <a name="l00738"></a>00738   iroff3 = 0
    803. 

  803. <a name="l00739"></a>00739 
    804. 

  804. <a name="l00740"></a>00740   <span class="keyword">if</span> ( ( 1.0e+00 - 5.0e+01 * epsilon ( defabs ) ) * defabs &lt;= dres ) <span class="keyword">then</span>
    805. 

  805. <a name="l00741"></a>00741     ksgn = 1
    806. 

  806. <a name="l00742"></a>00742   <span class="keyword">else</span>
    807. 

  807. <a name="l00743"></a>00743     ksgn = -1
    808. 

  808. <a name="l00744"></a>00744   <span class="keyword">end if</span>
    809. 

  809. <a name="l00745"></a>00745 
    810. 

  810. <a name="l00746"></a>00746   <span class="keyword">do</span> last = 2, limit
    811. 

  811. <a name="l00747"></a>00747 <span class="comment">!</span>
    812. 

  812. <a name="l00748"></a>00748 <span class="comment">!  Bisect the subinterval with nrmax-th largest error estimate.</span>
    813. 

  813. <a name="l00749"></a>00749 <span class="comment">!</span>
    814. 

  814. <a name="l00750"></a>00750     a1 = alist(maxerr)
    815. 

  815. <a name="l00751"></a>00751     b1 = 5.0e-01 * ( alist(maxerr) + blist(maxerr) )
    816. 

  816. <a name="l00752"></a>00752     a2 = b1
    817. 

  817. <a name="l00753"></a>00753     b2 = blist(maxerr)
    818. 

  818. <a name="l00754"></a>00754     erlast = errmax
    819. 

  819. <a name="l00755"></a>00755     call <a class="code" href="quadpack_8f90.html#a59164415fc33f2f3bf4ebc4ee2220f7e">qk15i </a>( f, boun, inf, a1, b1, area1, error1, resabs, defab1 )
    820. 

  820. <a name="l00756"></a>00756     call <a class="code" href="quadpack_8f90.html#a59164415fc33f2f3bf4ebc4ee2220f7e">qk15i </a>( f, boun, inf, a2, b2, area2, error2, resabs, defab2 )
    821. 

  821. <a name="l00757"></a>00757 <span class="comment">!</span>
    822. 

  822. <a name="l00758"></a>00758 <span class="comment">!  Improve previous approximations to integral and error</span>
    823. 

  823. <a name="l00759"></a>00759 <span class="comment">!  and test for accuracy.</span>
    824. 

  824. <a name="l00760"></a>00760 <span class="comment">!</span>
    825. 

  825. <a name="l00761"></a>00761     area12 = area1 + area2
    826. 

  826. <a name="l00762"></a>00762     erro12 = error1 + error2
    827. 

  827. <a name="l00763"></a>00763     errsum = errsum + erro12 - errmax
    828. 

  828. <a name="l00764"></a>00764     area = area + area12 - rlist(maxerr)
    829. 

  829. <a name="l00765"></a>00765 
    830. 

  830. <a name="l00766"></a>00766     <span class="keyword">if</span> ( defab1 /= error1 .and. defab2 /= error2 ) <span class="keyword">then</span>
    831. 

  831. <a name="l00767"></a>00767 
    832. 

  832. <a name="l00768"></a>00768       <span class="keyword">if</span> ( abs ( rlist(maxerr) - area12 ) &lt;= 1.0e-05 * abs ( area12 ) &amp;
    833. 

  833. <a name="l00769"></a>00769         .and. 9.9e-01 * errmax &lt;= erro12 ) <span class="keyword">then</span>
    834. 

  834. <a name="l00770"></a>00770 
    835. 

  835. <a name="l00771"></a>00771         <span class="keyword">if</span> ( extrap ) <span class="keyword">then</span>
    836. 

  836. <a name="l00772"></a>00772           iroff2 = iroff2 + 1
    837. 

  837. <a name="l00773"></a>00773         <span class="keyword">end if</span>
    838. 

  838. <a name="l00774"></a>00774 
    839. 

  839. <a name="l00775"></a>00775         <span class="keyword">if</span> ( .not. extrap ) <span class="keyword">then</span>
    840. 

  840. <a name="l00776"></a>00776           iroff1 = iroff1 + 1
    841. 

  841. <a name="l00777"></a>00777         <span class="keyword">end if</span>
    842. 

  842. <a name="l00778"></a>00778 
    843. 

  843. <a name="l00779"></a>00779       <span class="keyword">end if</span>
    844. 

  844. <a name="l00780"></a>00780 
    845. 

  845. <a name="l00781"></a>00781       <span class="keyword">if</span> ( 10 &lt; last .and. errmax &lt; erro12 ) <span class="keyword">then</span>
    846. 

  846. <a name="l00782"></a>00782         iroff3 = iroff3 + 1
    847. 

  847. <a name="l00783"></a>00783       <span class="keyword">end if</span>
    848. 

  848. <a name="l00784"></a>00784 
    849. 

  849. <a name="l00785"></a>00785     <span class="keyword">end if</span>
    850. 

  850. <a name="l00786"></a>00786 
    851. 

  851. <a name="l00787"></a>00787     rlist(maxerr) = area1
    852. 

  852. <a name="l00788"></a>00788     rlist(last) = area2
    853. 

  853. <a name="l00789"></a>00789     errbnd = max ( epsabs, epsrel * abs ( area ) )
    854. 

  854. <a name="l00790"></a>00790 <span class="comment">!</span>
    855. 

  855. <a name="l00791"></a>00791 <span class="comment">!  Test for roundoff error and eventually set error flag.</span>
    856. 

  856. <a name="l00792"></a>00792 <span class="comment">!</span>
    857. 

  857. <a name="l00793"></a>00793     <span class="keyword">if</span> ( 10 &lt;= iroff1 + iroff2 .or. 20 &lt;= iroff3 ) <span class="keyword">then</span>
    858. 

  858. <a name="l00794"></a>00794       ier = 2
    859. 

  859. <a name="l00795"></a>00795     <span class="keyword">end if</span>
    860. 

  860. <a name="l00796"></a>00796 
    861. 

  861. <a name="l00797"></a>00797     <span class="keyword">if</span> ( 5 &lt;= iroff2 ) <span class="keyword">then</span>
    862. 

  862. <a name="l00798"></a>00798       ierro = 3
    863. 

  863. <a name="l00799"></a>00799     <span class="keyword">end if</span>
    864. 

  864. <a name="l00800"></a>00800 <span class="comment">!</span>
    865. 

  865. <a name="l00801"></a>00801 <span class="comment">!  Set error flag in the case that the number of subintervals equals LIMIT.</span>
    866. 

  866. <a name="l00802"></a>00802 <span class="comment">!</span>
    867. 

  867. <a name="l00803"></a>00803     <span class="keyword">if</span> ( last == limit ) <span class="keyword">then</span>
    868. 

  868. <a name="l00804"></a>00804       ier = 1
    869. 

  869. <a name="l00805"></a>00805     <span class="keyword">end if</span>
    870. 

  870. <a name="l00806"></a>00806 <span class="comment">!</span>
    871. 

  871. <a name="l00807"></a>00807 <span class="comment">!  Set error flag in the case of bad integrand behavior</span>
    872. 

  872. <a name="l00808"></a>00808 <span class="comment">!  at some points of the integration range.</span>
    873. 

  873. <a name="l00809"></a>00809 <span class="comment">!</span>
    874. 

  874. <a name="l00810"></a>00810     <span class="keyword">if</span> ( max ( abs(a1), abs(b2) ) &lt;= (1.0e+00 + 1.0e+03 * epsilon ( a1 ) ) * &amp;
    875. 

  875. <a name="l00811"></a>00811     ( abs(a2) + 1.0e+03 * tiny ( a2 ) )) <span class="keyword">then</span>
    876. 

  876. <a name="l00812"></a>00812       ier = 4
    877. 

  877. <a name="l00813"></a>00813     <span class="keyword">end if</span>
    878. 

  878. <a name="l00814"></a>00814 <span class="comment">!</span>
    879. 

  879. <a name="l00815"></a>00815 <span class="comment">!  Append the newly-created intervals to the list.</span>
    880. 

  880. <a name="l00816"></a>00816 <span class="comment">!</span>
    881. 

  881. <a name="l00817"></a>00817     <span class="keyword">if</span> ( error2 &lt;= error1 ) <span class="keyword">then</span>
    882. 

  882. <a name="l00818"></a>00818       alist(last) = a2
    883. 

  883. <a name="l00819"></a>00819       blist(maxerr) = b1
    884. 

  884. <a name="l00820"></a>00820       blist(last) = b2
    885. 

  885. <a name="l00821"></a>00821       elist(maxerr) = error1
    886. 

  886. <a name="l00822"></a>00822       elist(last) = error2
    887. 

  887. <a name="l00823"></a>00823     <span class="keyword">else</span>
    888. 

  888. <a name="l00824"></a>00824       alist(maxerr) = a2
    889. 

  889. <a name="l00825"></a>00825       alist(last) = a1
    890. 

  890. <a name="l00826"></a>00826       blist(last) = b1
    891. 

  891. <a name="l00827"></a>00827       rlist(maxerr) = area2
    892. 

  892. <a name="l00828"></a>00828       rlist(last) = area1
    893. 

  893. <a name="l00829"></a>00829       elist(maxerr) = error2
    894. 

  894. <a name="l00830"></a>00830       elist(last) = error1
    895. 

  895. <a name="l00831"></a>00831     <span class="keyword">end if</span>
    896. 

  896. <a name="l00832"></a>00832 <span class="comment">!</span>
    897. 

  897. <a name="l00833"></a>00833 <span class="comment">!  Call QSORT to maintain the descending ordering</span>
    898. 

  898. <a name="l00834"></a>00834 <span class="comment">!  in the list of error estimates and select the subinterval</span>
    899. 

  899. <a name="l00835"></a>00835 <span class="comment">!  with NRMAX-th largest error estimate (to be bisected next).</span>
    900. 

  900. <a name="l00836"></a>00836 <span class="comment">!</span>
    901. 

  901. <a name="l00837"></a>00837     call <a class="code" href="quadpack_8f90.html#a55e08a684c5a6315fb37dd0fdc66d8e6">qsort </a>( limit, last, maxerr, errmax, elist, iord, nrmax )
    902. 

  902. <a name="l00838"></a>00838 
    903. 

  903. <a name="l00839"></a>00839     <span class="keyword">if</span> ( errsum &lt;= errbnd ) go to 115
    904. 

  904. <a name="l00840"></a>00840 
    905. 

  905. <a name="l00841"></a>00841     <span class="keyword">if</span> ( ier /= 0 ) <span class="keyword">then</span>
    906. 

  906. <a name="l00842"></a>00842       exit
    907. 

  907. <a name="l00843"></a>00843     <span class="keyword">end if</span>
    908. 

  908. <a name="l00844"></a>00844 
    909. 

  909. <a name="l00845"></a>00845     <span class="keyword">if</span> ( last == 2 ) <span class="keyword">then</span>
    910. 

  910. <a name="l00846"></a>00846       small = 3.75e-01
    911. 

  911. <a name="l00847"></a>00847       erlarg = errsum
    912. 

  912. <a name="l00848"></a>00848       ertest = errbnd
    913. 

  913. <a name="l00849"></a>00849       rlist2(2) = area
    914. 

  914. <a name="l00850"></a>00850       cycle
    915. 

  915. <a name="l00851"></a>00851     <span class="keyword">end if</span>
    916. 

  916. <a name="l00852"></a>00852 
    917. 

  917. <a name="l00853"></a>00853     <span class="keyword">if</span> ( noext ) <span class="keyword">then</span>
    918. 

  918. <a name="l00854"></a>00854       cycle
    919. 

  919. <a name="l00855"></a>00855     <span class="keyword">end if</span>
    920. 

  920. <a name="l00856"></a>00856 
    921. 

  921. <a name="l00857"></a>00857     erlarg = erlarg - erlast
    922. 

  922. <a name="l00858"></a>00858 
    923. 

  923. <a name="l00859"></a>00859     <span class="keyword">if</span> ( small &lt; abs ( b1 - a1 ) ) <span class="keyword">then</span>
    924. 

  924. <a name="l00860"></a>00860       erlarg = erlarg + erro12
    925. 

  925. <a name="l00861"></a>00861     <span class="keyword">end if</span>
    926. 

  926. <a name="l00862"></a>00862 <span class="comment">!</span>
    927. 

  927. <a name="l00863"></a>00863 <span class="comment">!  Test whether the interval to be bisected next is the</span>
    928. 

  928. <a name="l00864"></a>00864 <span class="comment">!  smallest interval.</span>
    929. 

  929. <a name="l00865"></a>00865 <span class="comment">!</span>
    930. 

  930. <a name="l00866"></a>00866     <span class="keyword">if</span> ( .not. extrap ) <span class="keyword">then</span>
    931. 

  931. <a name="l00867"></a>00867 
    932. 

  932. <a name="l00868"></a>00868       <span class="keyword">if</span> ( small &lt; abs ( blist(maxerr) - alist(maxerr) ) ) <span class="keyword">then</span>
    933. 

  933. <a name="l00869"></a>00869         cycle
    934. 

  934. <a name="l00870"></a>00870       <span class="keyword">end if</span>
    935. 

  935. <a name="l00871"></a>00871 
    936. 

  936. <a name="l00872"></a>00872       extrap = .true.
    937. 

  937. <a name="l00873"></a>00873       nrmax = 2
    938. 

  938. <a name="l00874"></a>00874 
    939. 

  939. <a name="l00875"></a>00875     <span class="keyword">end if</span>
    940. 

  940. <a name="l00876"></a>00876 
    941. 

  941. <a name="l00877"></a>00877     <span class="keyword">if</span> ( ierro == 3 .or. erlarg &lt;= ertest ) <span class="keyword">then</span>
    942. 

  942. <a name="l00878"></a>00878       go to 60
    943. 

  943. <a name="l00879"></a>00879     <span class="keyword">end if</span>
    944. 

  944. <a name="l00880"></a>00880 <span class="comment">!</span>
    945. 

  945. <a name="l00881"></a>00881 <span class="comment">!  The smallest interval has the largest error.</span>
    946. 

  946. <a name="l00882"></a>00882 <span class="comment">!  before bisecting decrease the sum of the errors over the</span>
    947. 

  947. <a name="l00883"></a>00883 <span class="comment">!  larger intervals (erlarg) and perform extrapolation.</span>
    948. 

  948. <a name="l00884"></a>00884 <span class="comment">!</span>
    949. 

  949. <a name="l00885"></a>00885     id = nrmax
    950. 

  950. <a name="l00886"></a>00886     jupbnd = last
    951. 

  951. <a name="l00887"></a>00887 
    952. 

  952. <a name="l00888"></a>00888     <span class="keyword">if</span> ( (2+limit/2) &lt; last ) <span class="keyword">then</span>
    953. 

  953. <a name="l00889"></a>00889       jupbnd = limit + 3 - last
    954. 

  954. <a name="l00890"></a>00890     <span class="keyword">end if</span>
    955. 

  955. <a name="l00891"></a>00891 
    956. 

  956. <a name="l00892"></a>00892     <span class="keyword">do</span> k = id, jupbnd
    957. 

  957. <a name="l00893"></a>00893       maxerr = iord(nrmax)
    958. 

  958. <a name="l00894"></a>00894       errmax = elist(maxerr)
    959. 

  959. <a name="l00895"></a>00895       <span class="keyword">if</span> ( small &lt; abs ( blist(maxerr) - alist(maxerr) ) ) <span class="keyword">then</span>
    960. 

  960. <a name="l00896"></a>00896         go to 90
    961. 

  961. <a name="l00897"></a>00897       <span class="keyword">end if</span>
    962. 

  962. <a name="l00898"></a>00898       nrmax = nrmax + 1
    963. 

  963. <a name="l00899"></a>00899     <span class="keyword">end do</span>
    964. 

  964. <a name="l00900"></a>00900 <span class="comment">!</span>
    965. 

  965. <a name="l00901"></a>00901 <span class="comment">!  Extrapolate.</span>
    966. 

  966. <a name="l00902"></a>00902 <span class="comment">!</span>
    967. 

  967. <a name="l00903"></a>00903 60  continue
    968. 

  968. <a name="l00904"></a>00904 
    969. 

  969. <a name="l00905"></a>00905     numrl2 = numrl2 + 1
    970. 

  970. <a name="l00906"></a>00906     rlist2(numrl2) = area
    971. 

  971. <a name="l00907"></a>00907     call <a class="code" href="quadpack_8f90.html#a5a75101d080f224c63adde98a0e64386">qextr </a>( numrl2, rlist2, reseps, abseps, res3la, nres ) 
    972. 

  972. <a name="l00908"></a>00908     ktmin = ktmin+1
    973. 

  973. <a name="l00909"></a>00909 
    974. 

  974. <a name="l00910"></a>00910     <span class="keyword">if</span> ( 5 &lt; ktmin .and. abserr &lt; 1.0e-03 * errsum ) <span class="keyword">then</span>
    975. 

  975. <a name="l00911"></a>00911       ier = 5
    976. 

  976. <a name="l00912"></a>00912     <span class="keyword">end if</span>
    977. 

  977. <a name="l00913"></a>00913 
    978. 

  978. <a name="l00914"></a>00914     <span class="keyword">if</span> ( abseps &lt; abserr ) <span class="keyword">then</span>
    979. 

  979. <a name="l00915"></a>00915 
    980. 

  980. <a name="l00916"></a>00916       ktmin = 0
    981. 

  981. <a name="l00917"></a>00917       abserr = abseps
    982. 

  982. <a name="l00918"></a>00918       result = reseps
    983. 

  983. <a name="l00919"></a>00919       correc = erlarg
    984. 

  984. <a name="l00920"></a>00920       ertest = max ( epsabs, epsrel * abs(reseps) )
    985. 

  985. <a name="l00921"></a>00921 
    986. 

  986. <a name="l00922"></a>00922       <span class="keyword">if</span> ( abserr &lt;= ertest ) <span class="keyword">then</span>
    987. 

  987. <a name="l00923"></a>00923         exit
    988. 

  988. <a name="l00924"></a>00924       <span class="keyword">end if</span>
    989. 

  989. <a name="l00925"></a>00925 
    990. 

  990. <a name="l00926"></a>00926     <span class="keyword">end if</span>
    991. 

  991. <a name="l00927"></a>00927 <span class="comment">!</span>
    992. 

  992. <a name="l00928"></a>00928 <span class="comment">!  Prepare bisection of the smallest interval.</span>
    993. 

  993. <a name="l00929"></a>00929 <span class="comment">!</span>
    994. 

  994. <a name="l00930"></a>00930     <span class="keyword">if</span> ( numrl2 == 1 ) <span class="keyword">then</span>
    995. 

  995. <a name="l00931"></a>00931       noext = .true.
    996. 

  996. <a name="l00932"></a>00932     <span class="keyword">end if</span>
    997. 

  997. <a name="l00933"></a>00933 
    998. 

  998. <a name="l00934"></a>00934     <span class="keyword">if</span> ( ier == 5 ) <span class="keyword">then</span>
    999. 

  999. <a name="l00935"></a>00935       exit
    1000. 

  1000. <a name="l00936"></a>00936     <span class="keyword">end if</span>
    1001. 

  1001. <a name="l00937"></a>00937 
    1002. 

  1002. <a name="l00938"></a>00938     maxerr = iord(1)
    1003. 

  1003. <a name="l00939"></a>00939     errmax = elist(maxerr)
    1004. 

  1004. <a name="l00940"></a>00940     nrmax = 1
    1005. 

  1005. <a name="l00941"></a>00941     extrap = .false.
    1006. 

  1006. <a name="l00942"></a>00942     small = small * 5.0e-01
    1007. 

  1007. <a name="l00943"></a>00943     erlarg = errsum
    1008. 

  1008. <a name="l00944"></a>00944 
    1009. 

  1009. <a name="l00945"></a>00945 90  continue
    1010. 

  1010. <a name="l00946"></a>00946 
    1011. 

  1011. <a name="l00947"></a>00947   <span class="keyword">end do</span>
    1012. 

  1012. <a name="l00948"></a>00948 <span class="comment">!</span>
    1013. 

  1013. <a name="l00949"></a>00949 <span class="comment">!  Set final result and error estimate.</span>
    1014. 

  1014. <a name="l00950"></a>00950 <span class="comment">!</span>
    1015. 

  1015. <a name="l00951"></a>00951   <span class="keyword">if</span> ( abserr == huge ( abserr ) ) <span class="keyword">then</span>
    1016. 

  1016. <a name="l00952"></a>00952     go to 115
    1017. 

  1017. <a name="l00953"></a>00953   <span class="keyword">end if</span>
    1018. 

  1018. <a name="l00954"></a>00954 
    1019. 

  1019. <a name="l00955"></a>00955   <span class="keyword">if</span> ( ( ier + ierro ) == 0 ) <span class="keyword">then</span>
    1020. 

  1020. <a name="l00956"></a>00956     go to 110
    1021. 

  1021. <a name="l00957"></a>00957   <span class="keyword">end if</span>
    1022. 

  1022. <a name="l00958"></a>00958 
    1023. 

  1023. <a name="l00959"></a>00959   <span class="keyword">if</span> ( ierro == 3 ) <span class="keyword">then</span>
    1024. 

  1024. <a name="l00960"></a>00960     abserr = abserr + correc
    1025. 

  1025. <a name="l00961"></a>00961   <span class="keyword">end if</span>
    1026. 

  1026. <a name="l00962"></a>00962 
    1027. 

  1027. <a name="l00963"></a>00963   <span class="keyword">if</span> ( ier == 0 ) <span class="keyword">then</span>
    1028. 

  1028. <a name="l00964"></a>00964     ier = 3
    1029. 

  1029. <a name="l00965"></a>00965   <span class="keyword">end if</span>
    1030. 

  1030. <a name="l00966"></a>00966 
    1031. 

  1031. <a name="l00967"></a>00967   <span class="keyword">if</span> ( result /= 0.0e+00 .and. area /= 0.0e+00) <span class="keyword">then</span>
    1032. 

  1032. <a name="l00968"></a>00968     go to 105
    1033. 

  1033. <a name="l00969"></a>00969   <span class="keyword">end if</span>
    1034. 

  1034. <a name="l00970"></a>00970 
    1035. 

  1035. <a name="l00971"></a>00971   <span class="keyword">if</span> ( errsum &lt; abserr ) <span class="keyword">then</span>
    1036. 

  1036. <a name="l00972"></a>00972     go to 115
    1037. 

  1037. <a name="l00973"></a>00973   <span class="keyword">end if</span>
    1038. 

  1038. <a name="l00974"></a>00974 
    1039. 

  1039. <a name="l00975"></a>00975   <span class="keyword">if</span> ( area == 0.0e+00 ) <span class="keyword">then</span>
    1040. 

  1040. <a name="l00976"></a>00976     go to 130
    1041. 

  1041. <a name="l00977"></a>00977   <span class="keyword">end if</span>
    1042. 

  1042. <a name="l00978"></a>00978 
    1043. 

  1043. <a name="l00979"></a>00979   go to 110
    1044. 

  1044. <a name="l00980"></a>00980 
    1045. 

  1045. <a name="l00981"></a>00981 105 continue
    1046. 

  1046. <a name="l00982"></a>00982 
    1047. 

  1047. <a name="l00983"></a>00983   <span class="keyword">if</span> ( errsum / abs ( area ) &lt; abserr / abs ( result )  ) <span class="keyword">then</span>
    1048. 

  1048. <a name="l00984"></a>00984     go to 115
    1049. 

  1049. <a name="l00985"></a>00985   <span class="keyword">end if</span>
    1050. 

  1050. <a name="l00986"></a>00986 <span class="comment">!</span>
    1051. 

  1051. <a name="l00987"></a>00987 <span class="comment">!  Test on divergence</span>
    1052. 

  1052. <a name="l00988"></a>00988 <span class="comment">!</span>
    1053. 

  1053. <a name="l00989"></a>00989 110 continue
    1054. 

  1054. <a name="l00990"></a>00990 
    1055. 

  1055. <a name="l00991"></a>00991   <span class="keyword">if</span> ( ksgn == (-1) .and. &amp;
    1056. 

  1056. <a name="l00992"></a>00992   max ( abs(result), abs(area) ) &lt;=  defabs * 1.0e-02) go to 130
    1057. 

  1057. <a name="l00993"></a>00993 
    1058. 

  1058. <a name="l00994"></a>00994   <span class="keyword">if</span> ( 1.0e-02 &gt; (result/area) .or. &amp;
    1059. 

  1059. <a name="l00995"></a>00995     (result/area) &gt; 1.0e+02 .or. &amp;
    1060. 

  1060. <a name="l00996"></a>00996     errsum &gt; abs(area)) <span class="keyword">then</span>
    1061. 

  1061. <a name="l00997"></a>00997     ier = 6
    1062. 

  1062. <a name="l00998"></a>00998   <span class="keyword">end if</span>
    1063. 

  1063. <a name="l00999"></a>00999 
    1064. 

  1064. <a name="l01000"></a>01000   go to 130
    1065. 

  1065. <a name="l01001"></a>01001 <span class="comment">!</span>
    1066. 

  1066. <a name="l01002"></a>01002 <span class="comment">!  Compute global integral sum.</span>
    1067. 

  1067. <a name="l01003"></a>01003 <span class="comment">!</span>
    1068. 

  1068. <a name="l01004"></a>01004   115 continue
    1069. 

  1069. <a name="l01005"></a>01005 
    1070. 

  1070. <a name="l01006"></a>01006   result = sum ( rlist(1:last) )
    1071. 

  1071. <a name="l01007"></a>01007 
    1072. 

  1072. <a name="l01008"></a>01008   abserr = errsum
    1073. 

  1073. <a name="l01009"></a>01009   130 continue
    1074. 

  1074. <a name="l01010"></a>01010 
    1075. 

  1075. <a name="l01011"></a>01011   neval = 30 * last - 15
    1076. 

  1076. <a name="l01012"></a>01012   <span class="keyword">if</span> ( inf == 2 ) <span class="keyword">then</span>
    1077. 

  1077. <a name="l01013"></a>01013     neval = 2 * neval
    1078. 

  1078. <a name="l01014"></a>01014   <span class="keyword">end if</span>
    1079. 

  1079. <a name="l01015"></a>01015 
    1080. 

  1080. <a name="l01016"></a>01016   <span class="keyword">if</span> ( 2 &lt; ier ) <span class="keyword">then</span>
    1081. 

  1081. <a name="l01017"></a>01017     ier = ier - 1
    1082. 

  1082. <a name="l01018"></a>01018   <span class="keyword">end if</span>
    1083. 

  1083. <a name="l01019"></a>01019 
    1084. 

  1084. <a name="l01020"></a>01020   return
    1085. 

  1085. <a name="l01021"></a>01021 <span class="keyword">end</span>
    1086. 

  1086. <a name="l01022"></a><a class="code" href="quadpack_8f90.html#a99cf2a02a14029fad4762555f04cac0e">01022</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a99cf2a02a14029fad4762555f04cac0e">qagp</a> ( f, a, b, npts2, points, epsabs, epsrel, result, abserr, &amp;
    1087. 

  1087. <a name="l01023"></a>01023   neval, ier )
    1088. 

  1088. <a name="l01024"></a>01024 
    1089. 

  1089. <a name="l01025"></a>01025 <span class="comment">!*****************************************************************************80</span>
    1090. 

  1090. <a name="l01026"></a>01026 <span class="comment">!</span>
    1091. 

  1091. <a name="l01027"></a>01027 <span class="comment">!! QAGP computes a definite integral.</span>
    1092. 

  1092. <a name="l01028"></a>01028 <span class="comment">!</span>
    1093. 

  1093. <a name="l01029"></a>01029 <span class="comment">!  Discussion:</span>
    1094. 

  1094. <a name="l01030"></a>01030 <span class="comment">!</span>
    1095. 

  1095. <a name="l01031"></a>01031 <span class="comment">!    The routine calculates an approximation RESULT to a definite integral   </span>
    1096. 

  1096. <a name="l01032"></a>01032 <span class="comment">!      I = integral of F over (A,B),</span>
    1097. 

  1097. <a name="l01033"></a>01033 <span class="comment">!    hopefully satisfying</span>
    1098. 

  1098. <a name="l01034"></a>01034 <span class="comment">!      || I - RESULT || &lt;= max ( EPSABS, EPSREL * ||I|| ).</span>
    1099. 

  1099. <a name="l01035"></a>01035 <span class="comment">!</span>
    1100. 

  1100. <a name="l01036"></a>01036 <span class="comment">!    Interior break points of the integration interval,</span>
    1101. 

  1101. <a name="l01037"></a>01037 <span class="comment">!    where local difficulties of the integrand may occur, such as</span>
    1102. 

  1102. <a name="l01038"></a>01038 <span class="comment">!    singularities or discontinuities, are provided by the user.</span>
    1103. 

  1103. <a name="l01039"></a>01039 <span class="comment">!</span>
    1104. 

  1104. <a name="l01040"></a>01040 <span class="comment">!  Author:</span>
    1105. 

  1105. <a name="l01041"></a>01041 <span class="comment">!</span>
    1106. 

  1106. <a name="l01042"></a>01042 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    1107. 

  1107. <a name="l01043"></a>01043 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    1108. 

  1108. <a name="l01044"></a>01044 <span class="comment">!</span>
    1109. 

  1109. <a name="l01045"></a>01045 <span class="comment">!  Reference:</span>
    1110. 

  1110. <a name="l01046"></a>01046 <span class="comment">!</span>
    1111. 

  1111. <a name="l01047"></a>01047 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    1112. 

  1112. <a name="l01048"></a>01048 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    1113. 

  1113. <a name="l01049"></a>01049 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    1114. 

  1114. <a name="l01050"></a>01050 <span class="comment">!    Springer Verlag, 1983</span>
    1115. 

  1115. <a name="l01051"></a>01051 <span class="comment">!</span>
    1116. 

  1116. <a name="l01052"></a>01052 <span class="comment">!  Parameters:</span>
    1117. 

  1117. <a name="l01053"></a>01053 <span class="comment">!</span>
    1118. 

  1118. <a name="l01054"></a>01054 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    1119. 

  1119. <a name="l01055"></a>01055 <span class="comment">!      function f ( x )</span>
    1120. 

  1120. <a name="l01056"></a>01056 <span class="comment">!      real f</span>
    1121. 

  1121. <a name="l01057"></a>01057 <span class="comment">!      real x</span>
    1122. 

  1122. <a name="l01058"></a>01058 <span class="comment">!    which evaluates the integrand function.</span>
    1123. 

  1123. <a name="l01059"></a>01059 <span class="comment">!</span>
    1124. 

  1124. <a name="l01060"></a>01060 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    1125. 

  1125. <a name="l01061"></a>01061 <span class="comment">!</span>
    1126. 

  1126. <a name="l01062"></a>01062 <span class="comment">!    Input, integer NPTS2, the number of user-supplied break points within </span>
    1127. 

  1127. <a name="l01063"></a>01063 <span class="comment">!    the integration range, plus 2.  NPTS2 must be at least 2.</span>
    1128. 

  1128. <a name="l01064"></a>01064 <span class="comment">!</span>
    1129. 

  1129. <a name="l01065"></a>01065 <span class="comment">!    Input/output, real POINTS(NPTS2), contains the user provided interior</span>
    1130. 

  1130. <a name="l01066"></a>01066 <span class="comment">!    breakpoints in entries 1 through NPTS2-2.  If these points are not</span>
    1131. 

  1131. <a name="l01067"></a>01067 <span class="comment">!    in ascending order on input, they will be sorted.</span>
    1132. 

  1132. <a name="l01068"></a>01068 <span class="comment">!</span>
    1133. 

  1133. <a name="l01069"></a>01069 <span class="comment">!    Input, real EPSABS, EPSREL, the absolute and relative accuracy requested.</span>
    1134. 

  1134. <a name="l01070"></a>01070 <span class="comment">!</span>
    1135. 

  1135. <a name="l01071"></a>01071 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    1136. 

  1136. <a name="l01072"></a>01072 <span class="comment">!</span>
    1137. 

  1137. <a name="l01073"></a>01073 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    1138. 

  1138. <a name="l01074"></a>01074 <span class="comment">!</span>
    1139. 

  1139. <a name="l01075"></a>01075 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    1140. 

  1140. <a name="l01076"></a>01076 <span class="comment">!</span>
    1141. 

  1141. <a name="l01077"></a>01077 <span class="comment">!    Output, integer IER, return flag.</span>
    1142. 

  1142. <a name="l01078"></a>01078 <span class="comment">!                     ier = 0 normal and reliable termination of the</span>
    1143. 

  1143. <a name="l01079"></a>01079 <span class="comment">!                             routine. it is assumed that the requested</span>
    1144. 

  1144. <a name="l01080"></a>01080 <span class="comment">!                             accuracy has been achieved.</span>
    1145. 

  1145. <a name="l01081"></a>01081 <span class="comment">!                     ier &gt; 0 abnormal termination of the routine.</span>
    1146. 

  1146. <a name="l01082"></a>01082 <span class="comment">!                             the estimates for integral and error are</span>
    1147. 

  1147. <a name="l01083"></a>01083 <span class="comment">!                             less reliable. it is assumed that the</span>
    1148. 

  1148. <a name="l01084"></a>01084 <span class="comment">!                             requested accuracy has not been achieved.</span>
    1149. 

  1149. <a name="l01085"></a>01085 <span class="comment">!                     ier = 1 maximum number of subdivisions allowed</span>
    1150. 

  1150. <a name="l01086"></a>01086 <span class="comment">!                             has been achieved. one can allow more</span>
    1151. 

  1151. <a name="l01087"></a>01087 <span class="comment">!                             subdivisions by increasing the data value</span>
    1152. 

  1152. <a name="l01088"></a>01088 <span class="comment">!                             of limit in qagp(and taking the according</span>
    1153. 

  1153. <a name="l01089"></a>01089 <span class="comment">!                             dimension adjustments into account).</span>
    1154. 

  1154. <a name="l01090"></a>01090 <span class="comment">!                             however, if this yields no improvement</span>
    1155. 

  1155. <a name="l01091"></a>01091 <span class="comment">!                             it is advised to analyze the integrand</span>
    1156. 

  1156. <a name="l01092"></a>01092 <span class="comment">!                             in order to determine the integration</span>
    1157. 

  1157. <a name="l01093"></a>01093 <span class="comment">!                             difficulties. if the position of a local</span>
    1158. 

  1158. <a name="l01094"></a>01094 <span class="comment">!                             difficulty can be determined (i.e.</span>
    1159. 

  1159. <a name="l01095"></a>01095 <span class="comment">!                             singularity, discontinuity within the</span>
    1160. 

  1160. <a name="l01096"></a>01096 <span class="comment">!                             interval), it should be supplied to the</span>
    1161. 

  1161. <a name="l01097"></a>01097 <span class="comment">!                             routine as an element of the vector</span>
    1162. 

  1162. <a name="l01098"></a>01098 <span class="comment">!                             points. if necessary, an appropriate</span>
    1163. 

  1163. <a name="l01099"></a>01099 <span class="comment">!                             special-purpose integrator must be used,</span>
    1164. 

  1164. <a name="l01100"></a>01100 <span class="comment">!                             which is designed for handling the type</span>
    1165. 

  1165. <a name="l01101"></a>01101 <span class="comment">!                             of difficulty involved.</span>
    1166. 

  1166. <a name="l01102"></a>01102 <span class="comment">!                         = 2 the occurrence of roundoff error is</span>
    1167. 

  1167. <a name="l01103"></a>01103 <span class="comment">!                             detected, which prevents the requested</span>
    1168. 

  1168. <a name="l01104"></a>01104 <span class="comment">!                             tolerance from being achieved.</span>
    1169. 

  1169. <a name="l01105"></a>01105 <span class="comment">!                             the error may be under-estimated.</span>
    1170. 

  1170. <a name="l01106"></a>01106 <span class="comment">!                         = 3 extremely bad integrand behavior occurs</span>
    1171. 

  1171. <a name="l01107"></a>01107 <span class="comment">!                             at some points of the integration</span>
    1172. 

  1172. <a name="l01108"></a>01108 <span class="comment">!                             interval.</span>
    1173. 

  1173. <a name="l01109"></a>01109 <span class="comment">!                         = 4 the algorithm does not converge. roundoff</span>
    1174. 

  1174. <a name="l01110"></a>01110 <span class="comment">!                             error is detected in the extrapolation</span>
    1175. 

  1175. <a name="l01111"></a>01111 <span class="comment">!                             table. it is presumed that the requested</span>
    1176. 

  1176. <a name="l01112"></a>01112 <span class="comment">!                             tolerance cannot be achieved, and that</span>
    1177. 

  1177. <a name="l01113"></a>01113 <span class="comment">!                             the returned result is the best which</span>
    1178. 

  1178. <a name="l01114"></a>01114 <span class="comment">!                             can be obtained.</span>
    1179. 

  1179. <a name="l01115"></a>01115 <span class="comment">!                         = 5 the integral is probably divergent, or</span>
    1180. 

  1180. <a name="l01116"></a>01116 <span class="comment">!                             slowly convergent. it must be noted that</span>
    1181. 

  1181. <a name="l01117"></a>01117 <span class="comment">!                             divergence can occur with any other value</span>
    1182. 

  1182. <a name="l01118"></a>01118 <span class="comment">!                             of ier &gt; 0.</span>
    1183. 

  1183. <a name="l01119"></a>01119 <span class="comment">!                         = 6 the input is invalid because</span>
    1184. 

  1184. <a name="l01120"></a>01120 <span class="comment">!                             npts2 &lt; 2 or</span>
    1185. 

  1185. <a name="l01121"></a>01121 <span class="comment">!                             break points are specified outside</span>
    1186. 

  1186. <a name="l01122"></a>01122 <span class="comment">!                             the integration range or</span>
    1187. 

  1187. <a name="l01123"></a>01123 <span class="comment">!                             epsabs &lt; 0 and epsrel &lt; 0,</span>
    1188. 

  1188. <a name="l01124"></a>01124 <span class="comment">!                             or limit &lt; npts2.</span>
    1189. 

  1189. <a name="l01125"></a>01125 <span class="comment">!                             result, abserr, neval are set to zero.</span>
    1190. 

  1190. <a name="l01126"></a>01126 <span class="comment">!</span>
    1191. 

  1191. <a name="l01127"></a>01127 <span class="comment">!  Local parameters:</span>
    1192. 

  1192. <a name="l01128"></a>01128 <span class="comment">!</span>
    1193. 

  1193. <a name="l01129"></a>01129 <span class="comment">!            the dimension of rlist2 is determined by the value of</span>
    1194. 

  1194. <a name="l01130"></a>01130 <span class="comment">!            limexp in QEXTR (rlist2 should be of dimension</span>
    1195. 

  1195. <a name="l01131"></a>01131 <span class="comment">!            (limexp+2) at least).</span>
    1196. 

  1196. <a name="l01132"></a>01132 <span class="comment">!</span>
    1197. 

  1197. <a name="l01133"></a>01133 <span class="comment">!           alist     - list of left end points of all subintervals</span>
    1198. 

  1198. <a name="l01134"></a>01134 <span class="comment">!                       considered up to now</span>
    1199. 

  1199. <a name="l01135"></a>01135 <span class="comment">!           blist     - list of right end points of all subintervals</span>
    1200. 

  1200. <a name="l01136"></a>01136 <span class="comment">!                       considered up to now</span>
    1201. 

  1201. <a name="l01137"></a>01137 <span class="comment">!           rlist(i)  - approximation to the integral over</span>
    1202. 

  1202. <a name="l01138"></a>01138 <span class="comment">!                       (alist(i),blist(i))</span>
    1203. 

  1203. <a name="l01139"></a>01139 <span class="comment">!           rlist2    - array of dimension at least limexp+2</span>
    1204. 

  1204. <a name="l01140"></a>01140 <span class="comment">!                       containing the part of the epsilon table which</span>
    1205. 

  1205. <a name="l01141"></a>01141 <span class="comment">!                       is still needed for further computations</span>
    1206. 

  1206. <a name="l01142"></a>01142 <span class="comment">!           elist(i)  - error estimate applying to rlist(i)</span>
    1207. 

  1207. <a name="l01143"></a>01143 <span class="comment">!           maxerr    - pointer to the interval with largest error</span>
    1208. 

  1208. <a name="l01144"></a>01144 <span class="comment">!                       estimate</span>
    1209. 

  1209. <a name="l01145"></a>01145 <span class="comment">!           errmax    - elist(maxerr)</span>
    1210. 

  1210. <a name="l01146"></a>01146 <span class="comment">!           erlast    - error on the interval currently subdivided</span>
    1211. 

  1211. <a name="l01147"></a>01147 <span class="comment">!                       (before that subdivision has taken place)</span>
    1212. 

  1212. <a name="l01148"></a>01148 <span class="comment">!           area      - sum of the integrals over the subintervals</span>
    1213. 

  1213. <a name="l01149"></a>01149 <span class="comment">!           errsum    - sum of the errors over the subintervals</span>
    1214. 

  1214. <a name="l01150"></a>01150 <span class="comment">!           errbnd    - requested accuracy max(epsabs,epsrel*</span>
    1215. 

  1215. <a name="l01151"></a>01151 <span class="comment">!                       abs(result))</span>
    1216. 

  1216. <a name="l01152"></a>01152 <span class="comment">!           *****1    - variable for the left subinterval</span>
    1217. 

  1217. <a name="l01153"></a>01153 <span class="comment">!           *****2    - variable for the right subinterval</span>
    1218. 

  1218. <a name="l01154"></a>01154 <span class="comment">!           last      - index for subdivision</span>
    1219. 

  1219. <a name="l01155"></a>01155 <span class="comment">!           nres      - number of calls to the extrapolation routine</span>
    1220. 

  1220. <a name="l01156"></a>01156 <span class="comment">!           numrl2    - number of elements in rlist2. if an appropriate</span>
    1221. 

  1221. <a name="l01157"></a>01157 <span class="comment">!                       approximation to the compounded integral has</span>
    1222. 

  1222. <a name="l01158"></a>01158 <span class="comment">!                       obtained, it is put in rlist2(numrl2) after</span>
    1223. 

  1223. <a name="l01159"></a>01159 <span class="comment">!                       numrl2 has been increased by one.</span>
    1224. 

  1224. <a name="l01160"></a>01160 <span class="comment">!           erlarg    - sum of the errors over the intervals larger</span>
    1225. 

  1225. <a name="l01161"></a>01161 <span class="comment">!                       than the smallest interval considered up to now</span>
    1226. 

  1226. <a name="l01162"></a>01162 <span class="comment">!           extrap    - logical variable denoting that the routine</span>
    1227. 

  1227. <a name="l01163"></a>01163 <span class="comment">!                       is attempting to perform extrapolation. i.e.</span>
    1228. 

  1228. <a name="l01164"></a>01164 <span class="comment">!                       before subdividing the smallest interval we</span>
    1229. 

  1229. <a name="l01165"></a>01165 <span class="comment">!                       try to decrease the value of erlarg.</span>
    1230. 

  1230. <a name="l01166"></a>01166 <span class="comment">!           noext     - logical variable denoting that extrapolation is</span>
    1231. 

  1231. <a name="l01167"></a>01167 <span class="comment">!                       no longer allowed (true-value)</span>
    1232. 

  1232. <a name="l01168"></a>01168 <span class="comment">!</span>
    1233. 

  1233. <a name="l01169"></a>01169   <span class="keyword">implicit none</span>
    1234. 

  1234. <a name="l01170"></a>01170 
    1235. 

  1235. <a name="l01171"></a>01171   <span class="keywordtype">integer</span>, <span class="keywordtype">parameter</span> :: limit = 500
    1236. 

  1236. <a name="l01172"></a>01172 
    1237. 

  1237. <a name="l01173"></a>01173   <span class="keywordtype">real</span> a
    1238. 

  1238. <a name="l01174"></a>01174   <span class="keywordtype">real</span> abseps
    1239. 

  1239. <a name="l01175"></a>01175   <span class="keywordtype">real</span> abserr
    1240. 

  1240. <a name="l01176"></a>01176   <span class="keywordtype">real</span> alist(limit)
    1241. 

  1241. <a name="l01177"></a>01177   <span class="keywordtype">real</span> area
    1242. 

  1242. <a name="l01178"></a>01178   <span class="keywordtype">real</span> area1
    1243. 

  1243. <a name="l01179"></a>01179   <span class="keywordtype">real</span> area12
    1244. 

  1244. <a name="l01180"></a>01180   <span class="keywordtype">real</span> area2
    1245. 

  1245. <a name="l01181"></a>01181   <span class="keywordtype">real</span> a1
    1246. 

  1246. <a name="l01182"></a>01182   <span class="keywordtype">real</span> a2
    1247. 

  1247. <a name="l01183"></a>01183   <span class="keywordtype">real</span> b
    1248. 

  1248. <a name="l01184"></a>01184   <span class="keywordtype">real</span> blist(limit)
    1249. 

  1249. <a name="l01185"></a>01185   <span class="keywordtype">real</span> b1
    1250. 

  1250. <a name="l01186"></a>01186   <span class="keywordtype">real</span> b2
    1251. 

  1251. <a name="l01187"></a>01187   <span class="keywordtype">real</span> correc
    1252. 

  1252. <a name="l01188"></a>01188   <span class="keywordtype">real</span> defabs
    1253. 

  1253. <a name="l01189"></a>01189   <span class="keywordtype">real</span> defab1
    1254. 

  1254. <a name="l01190"></a>01190   <span class="keywordtype">real</span> defab2
    1255. 

  1255. <a name="l01191"></a>01191   <span class="keywordtype">real</span> dres
    1256. 

  1256. <a name="l01192"></a>01192   <span class="keywordtype">real</span> elist(limit)
    1257. 

  1257. <a name="l01193"></a>01193   <span class="keywordtype">real</span> epsabs
    1258. 

  1258. <a name="l01194"></a>01194   <span class="keywordtype">real</span> epsrel
    1259. 

  1259. <a name="l01195"></a>01195   <span class="keywordtype">real</span> erlarg
    1260. 

  1260. <a name="l01196"></a>01196   <span class="keywordtype">real</span> erlast
    1261. 

  1261. <a name="l01197"></a>01197   <span class="keywordtype">real</span> errbnd
    1262. 

  1262. <a name="l01198"></a>01198   <span class="keywordtype">real</span> errmax
    1263. 

  1263. <a name="l01199"></a>01199   <span class="keywordtype">real</span> error1
    1264. 

  1264. <a name="l01200"></a>01200   <span class="keywordtype">real</span> erro12
    1265. 

  1265. <a name="l01201"></a>01201   <span class="keywordtype">real</span> error2
    1266. 

  1266. <a name="l01202"></a>01202   <span class="keywordtype">real</span> errsum
    1267. 

  1267. <a name="l01203"></a>01203   <span class="keywordtype">real</span> ertest
    1268. 

  1268. <a name="l01204"></a>01204   <span class="keywordtype">logical</span> extrap
    1269. 

  1269. <a name="l01205"></a>01205   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    1270. 

  1270. <a name="l01206"></a>01206   <span class="keywordtype">integer</span> i
    1271. 

  1271. <a name="l01207"></a>01207   <span class="keywordtype">integer</span> id
    1272. 

  1272. <a name="l01208"></a>01208   <span class="keywordtype">integer</span> ier
    1273. 

  1273. <a name="l01209"></a>01209   <span class="keywordtype">integer</span> ierro
    1274. 

  1274. <a name="l01210"></a>01210   <span class="keywordtype">integer</span> ind1
    1275. 

  1275. <a name="l01211"></a>01211   <span class="keywordtype">integer</span> ind2
    1276. 

  1276. <a name="l01212"></a>01212   <span class="keywordtype">integer</span> iord(limit)
    1277. 

  1277. <a name="l01213"></a>01213   <span class="keywordtype">integer</span> iroff1
    1278. 

  1278. <a name="l01214"></a>01214   <span class="keywordtype">integer</span> iroff2
    1279. 

  1279. <a name="l01215"></a>01215   <span class="keywordtype">integer</span> iroff3
    1280. 

  1280. <a name="l01216"></a>01216   <span class="keywordtype">integer</span> j
    1281. 

  1281. <a name="l01217"></a>01217   <span class="keywordtype">integer</span> jlow
    1282. 

  1282. <a name="l01218"></a>01218   <span class="keywordtype">integer</span> jupbnd
    1283. 

  1283. <a name="l01219"></a>01219   <span class="keywordtype">integer</span> k
    1284. 

  1284. <a name="l01220"></a>01220   <span class="keywordtype">integer</span> ksgn
    1285. 

  1285. <a name="l01221"></a>01221   <span class="keywordtype">integer</span> ktmin
    1286. 

  1286. <a name="l01222"></a>01222   <span class="keywordtype">integer</span> last
    1287. 

  1287. <a name="l01223"></a>01223   <span class="keywordtype">integer</span> levcur
    1288. 

  1288. <a name="l01224"></a>01224   <span class="keywordtype">integer</span> level(limit)
    1289. 

  1289. <a name="l01225"></a>01225   <span class="keywordtype">integer</span> levmax
    1290. 

  1290. <a name="l01226"></a>01226   <span class="keywordtype">integer</span> maxerr
    1291. 

  1291. <a name="l01227"></a>01227   <span class="keywordtype">integer</span> ndin(40)
    1292. 

  1292. <a name="l01228"></a>01228   <span class="keywordtype">integer</span> neval
    1293. 

  1293. <a name="l01229"></a>01229   <span class="keywordtype">integer</span> nint
    1294. 

  1294. <a name="l01230"></a>01230   <span class="keywordtype">logical</span> noext
    1295. 

  1295. <a name="l01231"></a>01231   <span class="keywordtype">integer</span> npts
    1296. 

  1296. <a name="l01232"></a>01232   <span class="keywordtype">integer</span> npts2
    1297. 

  1297. <a name="l01233"></a>01233   <span class="keywordtype">integer</span> nres
    1298. 

  1298. <a name="l01234"></a>01234   <span class="keywordtype">integer</span> nrmax
    1299. 

  1299. <a name="l01235"></a>01235   <span class="keywordtype">integer</span> numrl2
    1300. 

  1300. <a name="l01236"></a>01236   <span class="keywordtype">real</span> points(40)
    1301. 

  1301. <a name="l01237"></a>01237   <span class="keywordtype">real</span> pts(40)
    1302. 

  1302. <a name="l01238"></a>01238   <span class="keywordtype">real</span> resa
    1303. 

  1303. <a name="l01239"></a>01239   <span class="keywordtype">real</span> resabs
    1304. 

  1304. <a name="l01240"></a>01240   <span class="keywordtype">real</span> reseps
    1305. 

  1305. <a name="l01241"></a>01241   <span class="keywordtype">real</span> result
    1306. 

  1306. <a name="l01242"></a>01242   <span class="keywordtype">real</span> res3la(3)
    1307. 

  1307. <a name="l01243"></a>01243   <span class="keywordtype">real</span> rlist(limit)
    1308. 

  1308. <a name="l01244"></a>01244   <span class="keywordtype">real</span> rlist2(52)
    1309. 

  1309. <a name="l01245"></a>01245   <span class="keywordtype">real</span> sign
    1310. 

  1310. <a name="l01246"></a>01246   <span class="keywordtype">real</span> temp
    1311. 

  1311. <a name="l01247"></a>01247 <span class="comment">!</span>
    1312. 

  1312. <a name="l01248"></a>01248 <span class="comment">!  Test on validity of parameters.</span>
    1313. 

  1313. <a name="l01249"></a>01249 <span class="comment">!</span>
    1314. 

  1314. <a name="l01250"></a>01250   ier = 0
    1315. 

  1315. <a name="l01251"></a>01251   neval = 0
    1316. 

  1316. <a name="l01252"></a>01252   last = 0
    1317. 

  1317. <a name="l01253"></a>01253   result = 0.0e+00
    1318. 

  1318. <a name="l01254"></a>01254   abserr = 0.0e+00
    1319. 

  1319. <a name="l01255"></a>01255   alist(1) = a
    1320. 

  1320. <a name="l01256"></a>01256   blist(1) = b
    1321. 

  1321. <a name="l01257"></a>01257   rlist(1) = 0.0e+00
    1322. 

  1322. <a name="l01258"></a>01258   elist(1) = 0.0e+00
    1323. 

  1323. <a name="l01259"></a>01259   iord(1) = 0
    1324. 

  1324. <a name="l01260"></a>01260   level(1) = 0
    1325. 

  1325. <a name="l01261"></a>01261   npts = npts2 - 2
    1326. 

  1326. <a name="l01262"></a>01262 
    1327. 

  1327. <a name="l01263"></a>01263   <span class="keyword">if</span> ( npts2 &lt; 2 ) <span class="keyword">then</span>
    1328. 

  1328. <a name="l01264"></a>01264     ier = 6
    1329. 

  1329. <a name="l01265"></a>01265     return
    1330. 

  1330. <a name="l01266"></a>01266   <span class="keyword">else</span> <span class="keyword">if</span> ( limit &lt;= npts .or. ( epsabs &lt; 0.0e+00 .and. &amp;
    1331. 

  1331. <a name="l01267"></a>01267     epsrel &lt; 0.0e+00) ) <span class="keyword">then</span>
    1332. 

  1332. <a name="l01268"></a>01268     ier = 6
    1333. 

  1333. <a name="l01269"></a>01269     return
    1334. 

  1334. <a name="l01270"></a>01270   <span class="keyword">end if</span>
    1335. 

  1335. <a name="l01271"></a>01271 <span class="comment">!</span>
    1336. 

  1336. <a name="l01272"></a>01272 <span class="comment">!  If any break points are provided, sort them into an</span>
    1337. 

  1337. <a name="l01273"></a>01273 <span class="comment">!  ascending sequence.</span>
    1338. 

  1338. <a name="l01274"></a>01274 <span class="comment">!</span>
    1339. 

  1339. <a name="l01275"></a>01275   <span class="keyword">if</span> ( b &lt; a ) <span class="keyword">then</span>
    1340. 

  1340. <a name="l01276"></a>01276     sign = -1.0e+00
    1341. 

  1341. <a name="l01277"></a>01277   <span class="keyword">else</span>
    1342. 

  1342. <a name="l01278"></a>01278     sign = +1.0E+00
    1343. 

  1343. <a name="l01279"></a>01279   <span class="keyword">end if</span>
    1344. 

  1344. <a name="l01280"></a>01280 
    1345. 

  1345. <a name="l01281"></a>01281   pts(1) = min ( a, b )
    1346. 

  1346. <a name="l01282"></a>01282 
    1347. 

  1347. <a name="l01283"></a>01283   <span class="keyword">do</span> i = 1, npts
    1348. 

  1348. <a name="l01284"></a>01284     pts(i+1) = points(i)
    1349. 

  1349. <a name="l01285"></a>01285   <span class="keyword">end do</span>
    1350. 

  1350. <a name="l01286"></a>01286 
    1351. 

  1351. <a name="l01287"></a>01287   pts(npts+2) = max ( a, b )
    1352. 

  1352. <a name="l01288"></a>01288   nint = npts+1
    1353. 

  1353. <a name="l01289"></a>01289   a1 = pts(1)
    1354. 

  1354. <a name="l01290"></a>01290 
    1355. 

  1355. <a name="l01291"></a>01291   <span class="keyword">if</span> ( npts /= 0 ) <span class="keyword">then</span>
    1356. 

  1356. <a name="l01292"></a>01292 
    1357. 

  1357. <a name="l01293"></a>01293     <span class="keyword">do</span> i = 1, nint
    1358. 

  1358. <a name="l01294"></a>01294       <span class="keyword">do</span> j = i+1, nint+1
    1359. 

  1359. <a name="l01295"></a>01295         <span class="keyword">if</span> ( pts(j) &lt; pts(i) ) <span class="keyword">then</span>
    1360. 

  1360. <a name="l01296"></a>01296           temp   = pts(i)
    1361. 

  1361. <a name="l01297"></a>01297           pts(i) = pts(j)
    1362. 

  1362. <a name="l01298"></a>01298           pts(j) = temp
    1363. 

  1363. <a name="l01299"></a>01299         <span class="keyword">end if</span>
    1364. 

  1364. <a name="l01300"></a>01300       <span class="keyword">end do</span>
    1365. 

  1365. <a name="l01301"></a>01301     <span class="keyword">end do</span>
    1366. 

  1366. <a name="l01302"></a>01302 
    1367. 

  1367. <a name="l01303"></a>01303     <span class="keyword">if</span> ( pts(1) /= min ( a, b ) .or. pts(nint+1) /= max ( a, b ) ) <span class="keyword">then</span>
    1368. 

  1368. <a name="l01304"></a>01304       ier = 6
    1369. 

  1369. <a name="l01305"></a>01305       return
    1370. 

  1370. <a name="l01306"></a>01306     <span class="keyword">end if</span>
    1371. 

  1371. <a name="l01307"></a>01307 
    1372. 

  1372. <a name="l01308"></a>01308   <span class="keyword">end if</span>
    1373. 

  1373. <a name="l01309"></a>01309 <span class="comment">!</span>
    1374. 

  1374. <a name="l01310"></a>01310 <span class="comment">!  Compute first integral and error approximations.</span>
    1375. 

  1375. <a name="l01311"></a>01311 <span class="comment">!</span>
    1376. 

  1376. <a name="l01312"></a>01312   resabs = 0.0e+00
    1377. 

  1377. <a name="l01313"></a>01313 
    1378. 

  1378. <a name="l01314"></a>01314   <span class="keyword">do</span> i = 1, nint
    1379. 

  1379. <a name="l01315"></a>01315 
    1380. 

  1380. <a name="l01316"></a>01316     b1 = pts(i+1)
    1381. 

  1381. <a name="l01317"></a>01317     call <a class="code" href="quadpack_8f90.html#a27241a527b249e9de59a5ed6bee5f805">qk21 </a>( f, a1, b1, area1, error1, defabs, resa )
    1382. 

  1382. <a name="l01318"></a>01318     abserr = abserr + error1
    1383. 

  1383. <a name="l01319"></a>01319     result = result + area1
    1384. 

  1384. <a name="l01320"></a>01320     ndin(i) = 0
    1385. 

  1385. <a name="l01321"></a>01321 
    1386. 

  1386. <a name="l01322"></a>01322     <span class="keyword">if</span> ( error1 == resa .and. error1 /= 0.0e+00 ) <span class="keyword">then</span>
    1387. 

  1387. <a name="l01323"></a>01323       ndin(i) = 1
    1388. 

  1388. <a name="l01324"></a>01324     <span class="keyword">end if</span>
    1389. 

  1389. <a name="l01325"></a>01325 
    1390. 

  1390. <a name="l01326"></a>01326     resabs = resabs + defabs
    1391. 

  1391. <a name="l01327"></a>01327     level(i) = 0
    1392. 

  1392. <a name="l01328"></a>01328     elist(i) = error1
    1393. 

  1393. <a name="l01329"></a>01329     alist(i) = a1
    1394. 

  1394. <a name="l01330"></a>01330     blist(i) = b1
    1395. 

  1395. <a name="l01331"></a>01331     rlist(i) = area1
    1396. 

  1396. <a name="l01332"></a>01332     iord(i) = i
    1397. 

  1397. <a name="l01333"></a>01333     a1 = b1
    1398. 

  1398. <a name="l01334"></a>01334 
    1399. 

  1399. <a name="l01335"></a>01335   <span class="keyword">end do</span>
    1400. 

  1400. <a name="l01336"></a>01336 
    1401. 

  1401. <a name="l01337"></a>01337   errsum = 0.0e+00
    1402. 

  1402. <a name="l01338"></a>01338 
    1403. 

  1403. <a name="l01339"></a>01339   <span class="keyword">do</span> i = 1, nint
    1404. 

  1404. <a name="l01340"></a>01340     <span class="keyword">if</span> ( ndin(i) == 1 ) <span class="keyword">then</span>
    1405. 

  1405. <a name="l01341"></a>01341       elist(i) = abserr
    1406. 

  1406. <a name="l01342"></a>01342     <span class="keyword">end if</span>
    1407. 

  1407. <a name="l01343"></a>01343     errsum = errsum + elist(i)
    1408. 

  1408. <a name="l01344"></a>01344   <span class="keyword">end do</span>
    1409. 

  1409. <a name="l01345"></a>01345 <span class="comment">!</span>
    1410. 

  1410. <a name="l01346"></a>01346 <span class="comment">!  Test on accuracy.</span>
    1411. 

  1411. <a name="l01347"></a>01347 <span class="comment">!</span>
    1412. 

  1412. <a name="l01348"></a>01348   last = nint
    1413. 

  1413. <a name="l01349"></a>01349   neval = 21 * nint
    1414. 

  1414. <a name="l01350"></a>01350   dres = abs ( result )
    1415. 

  1415. <a name="l01351"></a>01351   errbnd = max ( epsabs, epsrel * dres )
    1416. 

  1416. <a name="l01352"></a>01352 
    1417. 

  1417. <a name="l01353"></a>01353   <span class="keyword">if</span> ( abserr &lt;= 1.0e+02 * epsilon ( resabs ) * resabs .and. &amp;
    1418. 

  1418. <a name="l01354"></a>01354     abserr &gt; errbnd ) <span class="keyword">then</span>
    1419. 

  1419. <a name="l01355"></a>01355     ier = 2
    1420. 

  1420. <a name="l01356"></a>01356   <span class="keyword">end if</span>
    1421. 

  1421. <a name="l01357"></a>01357 
    1422. 

  1422. <a name="l01358"></a>01358   <span class="keyword">if</span> ( nint /= 1 ) <span class="keyword">then</span>
    1423. 

  1423. <a name="l01359"></a>01359 
    1424. 

  1424. <a name="l01360"></a>01360     <span class="keyword">do</span> i = 1, npts
    1425. 

  1425. <a name="l01361"></a>01361 
    1426. 

  1426. <a name="l01362"></a>01362       jlow = i+1
    1427. 

  1427. <a name="l01363"></a>01363       ind1 = iord(i)
    1428. 

  1428. <a name="l01364"></a>01364 
    1429. 

  1429. <a name="l01365"></a>01365       <span class="keyword">do</span> j = jlow, nint
    1430. 

  1430. <a name="l01366"></a>01366         ind2 = iord(j)
    1431. 

  1431. <a name="l01367"></a>01367         <span class="keyword">if</span> ( elist(ind1) &lt;= elist(ind2) ) <span class="keyword">then</span>
    1432. 

  1432. <a name="l01368"></a>01368           ind1 = ind2
    1433. 

  1433. <a name="l01369"></a>01369           k = j
    1434. 

  1434. <a name="l01370"></a>01370         <span class="keyword">end if</span>
    1435. 

  1435. <a name="l01371"></a>01371       <span class="keyword">end do</span>
    1436. 

  1436. <a name="l01372"></a>01372 
    1437. 

  1437. <a name="l01373"></a>01373       <span class="keyword">if</span> ( ind1 /= iord(i) ) <span class="keyword">then</span>
    1438. 

  1438. <a name="l01374"></a>01374         iord(k) = iord(i)
    1439. 

  1439. <a name="l01375"></a>01375         iord(i) = ind1
    1440. 

  1440. <a name="l01376"></a>01376       <span class="keyword">end if</span>
    1441. 

  1441. <a name="l01377"></a>01377 
    1442. 

  1442. <a name="l01378"></a>01378     <span class="keyword">end do</span>
    1443. 

  1443. <a name="l01379"></a>01379 
    1444. 

  1444. <a name="l01380"></a>01380     <span class="keyword">if</span> ( limit &lt; npts2 ) <span class="keyword">then</span>
    1445. 

  1445. <a name="l01381"></a>01381       ier = 1
    1446. 

  1446. <a name="l01382"></a>01382     <span class="keyword">end if</span>
    1447. 

  1447. <a name="l01383"></a>01383 
    1448. 

  1448. <a name="l01384"></a>01384   <span class="keyword">end if</span>
    1449. 

  1449. <a name="l01385"></a>01385 
    1450. 

  1450. <a name="l01386"></a>01386   <span class="keyword">if</span> ( ier /= 0 .or. abserr &lt;= errbnd ) <span class="keyword">then</span>
    1451. 

  1451. <a name="l01387"></a>01387     return
    1452. 

  1452. <a name="l01388"></a>01388   <span class="keyword">end if</span>
    1453. 

  1453. <a name="l01389"></a>01389 <span class="comment">!</span>
    1454. 

  1454. <a name="l01390"></a>01390 <span class="comment">!  Initialization</span>
    1455. 

  1455. <a name="l01391"></a>01391 <span class="comment">!</span>
    1456. 

  1456. <a name="l01392"></a>01392   rlist2(1) = result
    1457. 

  1457. <a name="l01393"></a>01393   maxerr = iord(1)
    1458. 

  1458. <a name="l01394"></a>01394   errmax = elist(maxerr)
    1459. 

  1459. <a name="l01395"></a>01395   area = result
    1460. 

  1460. <a name="l01396"></a>01396   nrmax = 1
    1461. 

  1461. <a name="l01397"></a>01397   nres = 0
    1462. 

  1462. <a name="l01398"></a>01398   numrl2 = 1
    1463. 

  1463. <a name="l01399"></a>01399   ktmin = 0
    1464. 

  1464. <a name="l01400"></a>01400   extrap = .false.
    1465. 

  1465. <a name="l01401"></a>01401   noext = .false.
    1466. 

  1466. <a name="l01402"></a>01402   erlarg = errsum
    1467. 

  1467. <a name="l01403"></a>01403   ertest = errbnd
    1468. 

  1468. <a name="l01404"></a>01404   levmax = 1
    1469. 

  1469. <a name="l01405"></a>01405   iroff1 = 0
    1470. 

  1470. <a name="l01406"></a>01406   iroff2 = 0
    1471. 

  1471. <a name="l01407"></a>01407   iroff3 = 0
    1472. 

  1472. <a name="l01408"></a>01408   ierro = 0
    1473. 

  1473. <a name="l01409"></a>01409   abserr = huge ( abserr )
    1474. 

  1474. <a name="l01410"></a>01410 
    1475. 

  1475. <a name="l01411"></a>01411   <span class="keyword">if</span> ( dres &gt;= ( 1.0e+00 - 0.5E+00 * epsilon ( resabs ) ) * resabs ) <span class="keyword">then</span>
    1476. 

  1476. <a name="l01412"></a>01412     ksgn = 1
    1477. 

  1477. <a name="l01413"></a>01413   <span class="keyword">else</span>
    1478. 

  1478. <a name="l01414"></a>01414     ksgn = -1
    1479. 

  1479. <a name="l01415"></a>01415   <span class="keyword">end if</span>
    1480. 

  1480. <a name="l01416"></a>01416 
    1481. 

  1481. <a name="l01417"></a>01417   <span class="keyword">do</span> last = npts2, limit
    1482. 

  1482. <a name="l01418"></a>01418 <span class="comment">!</span>
    1483. 

  1483. <a name="l01419"></a>01419 <span class="comment">!  Bisect the subinterval with the NRMAX-th largest error estimate.</span>
    1484. 

  1484. <a name="l01420"></a>01420 <span class="comment">!</span>
    1485. 

  1485. <a name="l01421"></a>01421     levcur = level(maxerr) + 1
    1486. 

  1486. <a name="l01422"></a>01422     a1 = alist(maxerr)
    1487. 

  1487. <a name="l01423"></a>01423     b1 = 0.5E+00 * ( alist(maxerr) + blist(maxerr) )
    1488. 

  1488. <a name="l01424"></a>01424     a2 = b1
    1489. 

  1489. <a name="l01425"></a>01425     b2 = blist(maxerr)
    1490. 

  1490. <a name="l01426"></a>01426     erlast = errmax
    1491. 

  1491. <a name="l01427"></a>01427     call <a class="code" href="quadpack_8f90.html#a27241a527b249e9de59a5ed6bee5f805">qk21 </a>( f, a1, b1, area1, error1, resa, defab1 )
    1492. 

  1492. <a name="l01428"></a>01428     call <a class="code" href="quadpack_8f90.html#a27241a527b249e9de59a5ed6bee5f805">qk21 </a>( f, a2, b2, area2, error2, resa, defab2 )
    1493. 

  1493. <a name="l01429"></a>01429 <span class="comment">!</span>
    1494. 

  1494. <a name="l01430"></a>01430 <span class="comment">!  Improve previous approximations to integral and error</span>
    1495. 

  1495. <a name="l01431"></a>01431 <span class="comment">!  and test for accuracy.</span>
    1496. 

  1496. <a name="l01432"></a>01432 <span class="comment">!</span>
    1497. 

  1497. <a name="l01433"></a>01433     neval = neval + 42
    1498. 

  1498. <a name="l01434"></a>01434     area12 = area1 + area2
    1499. 

  1499. <a name="l01435"></a>01435     erro12 = error1 + error2
    1500. 

  1500. <a name="l01436"></a>01436     errsum = errsum + erro12 -errmax
    1501. 

  1501. <a name="l01437"></a>01437     area = area + area12 - rlist(maxerr)
    1502. 

  1502. <a name="l01438"></a>01438 
    1503. 

  1503. <a name="l01439"></a>01439     <span class="keyword">if</span> ( defab1 /= error1 .and. defab2 /= error2 ) <span class="keyword">then</span>
    1504. 

  1504. <a name="l01440"></a>01440 
    1505. 

  1505. <a name="l01441"></a>01441       <span class="keyword">if</span> ( abs ( rlist ( maxerr ) - area12 ) &lt;= 1.0e-05 * abs(area12) .and. &amp;
    1506. 

  1506. <a name="l01442"></a>01442         erro12 &gt;= 9.9e-01 * errmax ) <span class="keyword">then</span>
    1507. 

  1507. <a name="l01443"></a>01443 
    1508. 

  1508. <a name="l01444"></a>01444         <span class="keyword">if</span> ( extrap ) <span class="keyword">then</span>
    1509. 

  1509. <a name="l01445"></a>01445           iroff2 = iroff2+1
    1510. 

  1510. <a name="l01446"></a>01446         <span class="keyword">else</span>
    1511. 

  1511. <a name="l01447"></a>01447           iroff1 = iroff1+1
    1512. 

  1512. <a name="l01448"></a>01448         <span class="keyword">end if</span>
    1513. 

  1513. <a name="l01449"></a>01449 
    1514. 

  1514. <a name="l01450"></a>01450       <span class="keyword">end if</span>
    1515. 

  1515. <a name="l01451"></a>01451 
    1516. 

  1516. <a name="l01452"></a>01452       <span class="keyword">if</span> ( last &gt; 10 .and. erro12 &gt; errmax ) <span class="keyword">then</span>
    1517. 

  1517. <a name="l01453"></a>01453         iroff3 = iroff3 + 1
    1518. 

  1518. <a name="l01454"></a>01454       <span class="keyword">end if</span>
    1519. 

  1519. <a name="l01455"></a>01455 
    1520. 

  1520. <a name="l01456"></a>01456     <span class="keyword">end if</span>
    1521. 

  1521. <a name="l01457"></a>01457 
    1522. 

  1522. <a name="l01458"></a>01458     level(maxerr) = levcur
    1523. 

  1523. <a name="l01459"></a>01459     level(last) = levcur
    1524. 

  1524. <a name="l01460"></a>01460     rlist(maxerr) = area1
    1525. 

  1525. <a name="l01461"></a>01461     rlist(last) = area2
    1526. 

  1526. <a name="l01462"></a>01462     errbnd = max ( epsabs, epsrel * abs ( area ) )
    1527. 

  1527. <a name="l01463"></a>01463 <span class="comment">!</span>
    1528. 

  1528. <a name="l01464"></a>01464 <span class="comment">!  Test for roundoff error and eventually set error flag.</span>
    1529. 

  1529. <a name="l01465"></a>01465 <span class="comment">!</span>
    1530. 

  1530. <a name="l01466"></a>01466     <span class="keyword">if</span> ( 10 &lt;= iroff1 + iroff2 .or. 20 &lt;= iroff3 ) <span class="keyword">then</span>
    1531. 

  1531. <a name="l01467"></a>01467       ier = 2
    1532. 

  1532. <a name="l01468"></a>01468     <span class="keyword">end if</span>
    1533. 

  1533. <a name="l01469"></a>01469 
    1534. 

  1534. <a name="l01470"></a>01470     <span class="keyword">if</span> ( 5 &lt;= iroff2 ) <span class="keyword">then</span>
    1535. 

  1535. <a name="l01471"></a>01471       ierro = 3
    1536. 

  1536. <a name="l01472"></a>01472     <span class="keyword">end if</span>
    1537. 

  1537. <a name="l01473"></a>01473 <span class="comment">!</span>
    1538. 

  1538. <a name="l01474"></a>01474 <span class="comment">!  Set error flag in the case that the number of subintervals</span>
    1539. 

  1539. <a name="l01475"></a>01475 <span class="comment">!  equals limit.</span>
    1540. 

  1540. <a name="l01476"></a>01476 <span class="comment">!</span>
    1541. 

  1541. <a name="l01477"></a>01477     <span class="keyword">if</span> ( last == limit ) <span class="keyword">then</span>
    1542. 

  1542. <a name="l01478"></a>01478       ier = 1
    1543. 

  1543. <a name="l01479"></a>01479     <span class="keyword">end if</span>
    1544. 

  1544. <a name="l01480"></a>01480 <span class="comment">!</span>
    1545. 

  1545. <a name="l01481"></a>01481 <span class="comment">!  Set error flag in the case of bad integrand behavior</span>
    1546. 

  1546. <a name="l01482"></a>01482 <span class="comment">!  at a point of the integration range</span>
    1547. 

  1547. <a name="l01483"></a>01483 <span class="comment">!</span>
    1548. 

  1548. <a name="l01484"></a>01484     <span class="keyword">if</span> ( max ( abs(a1), abs(b2)) &lt;= ( 1.0e+00 + 1.0e+03 * epsilon ( a1 ) )* &amp;
    1549. 

  1549. <a name="l01485"></a>01485     ( abs ( a2 ) + 1.0e+03 * tiny ( a2 ) ) ) <span class="keyword">then</span>
    1550. 

  1550. <a name="l01486"></a>01486       ier = 4
    1551. 

  1551. <a name="l01487"></a>01487     <span class="keyword">end if</span>
    1552. 

  1552. <a name="l01488"></a>01488 <span class="comment">!</span>
    1553. 

  1553. <a name="l01489"></a>01489 <span class="comment">!  Append the newly-created intervals to the list.</span>
    1554. 

  1554. <a name="l01490"></a>01490 <span class="comment">!</span>
    1555. 

  1555. <a name="l01491"></a>01491     <span class="keyword">if</span> ( error2 &lt;= error1 ) <span class="keyword">then</span>
    1556. 

  1556. <a name="l01492"></a>01492       alist(last) = a2
    1557. 

  1557. <a name="l01493"></a>01493       blist(maxerr) = b1
    1558. 

  1558. <a name="l01494"></a>01494       blist(last) = b2
    1559. 

  1559. <a name="l01495"></a>01495       elist(maxerr) = error1
    1560. 

  1560. <a name="l01496"></a>01496       elist(last) = error2
    1561. 

  1561. <a name="l01497"></a>01497     <span class="keyword">else</span>
    1562. 

  1562. <a name="l01498"></a>01498       alist(maxerr) = a2
    1563. 

  1563. <a name="l01499"></a>01499       alist(last) = a1
    1564. 

  1564. <a name="l01500"></a>01500       blist(last) = b1
    1565. 

  1565. <a name="l01501"></a>01501       rlist(maxerr) = area2
    1566. 

  1566. <a name="l01502"></a>01502       rlist(last) = area1
    1567. 

  1567. <a name="l01503"></a>01503       elist(maxerr) = error2
    1568. 

  1568. <a name="l01504"></a>01504       elist(last) = error1
    1569. 

  1569. <a name="l01505"></a>01505     <span class="keyword">end if</span>
    1570. 

  1570. <a name="l01506"></a>01506 <span class="comment">!</span>
    1571. 

  1571. <a name="l01507"></a>01507 <span class="comment">!  Call QSORT to maintain the descending ordering</span>
    1572. 

  1572. <a name="l01508"></a>01508 <span class="comment">!  in the list of error estimates and select the subinterval</span>
    1573. 

  1573. <a name="l01509"></a>01509 <span class="comment">!  with nrmax-th largest error estimate (to be bisected next).</span>
    1574. 

  1574. <a name="l01510"></a>01510 <span class="comment">!</span>
    1575. 

  1575. <a name="l01511"></a>01511     call <a class="code" href="quadpack_8f90.html#a55e08a684c5a6315fb37dd0fdc66d8e6">qsort </a>( limit, last, maxerr, errmax, elist, iord, nrmax )
    1576. 

  1576. <a name="l01512"></a>01512 
    1577. 

  1577. <a name="l01513"></a>01513     <span class="keyword">if</span> ( errsum &lt;= errbnd ) <span class="keyword">then</span>
    1578. 

  1578. <a name="l01514"></a>01514       go to 190
    1579. 

  1579. <a name="l01515"></a>01515     <span class="keyword">end if</span>
    1580. 

  1580. <a name="l01516"></a>01516 
    1581. 

  1581. <a name="l01517"></a>01517     <span class="keyword">if</span> ( ier /= 0 ) <span class="keyword">then</span>
    1582. 

  1582. <a name="l01518"></a>01518       exit
    1583. 

  1583. <a name="l01519"></a>01519     <span class="keyword">end if</span>
    1584. 

  1584. <a name="l01520"></a>01520 
    1585. 

  1585. <a name="l01521"></a>01521     <span class="keyword">if</span> ( noext ) <span class="keyword">then</span>
    1586. 

  1586. <a name="l01522"></a>01522       cycle
    1587. 

  1587. <a name="l01523"></a>01523     <span class="keyword">end if</span>
    1588. 

  1588. <a name="l01524"></a>01524 
    1589. 

  1589. <a name="l01525"></a>01525     erlarg = erlarg - erlast
    1590. 

  1590. <a name="l01526"></a>01526 
    1591. 

  1591. <a name="l01527"></a>01527     <span class="keyword">if</span> ( levcur+1 &lt;= levmax ) <span class="keyword">then</span>
    1592. 

  1592. <a name="l01528"></a>01528       erlarg = erlarg + erro12
    1593. 

  1593. <a name="l01529"></a>01529     <span class="keyword">end if</span>
    1594. 

  1594. <a name="l01530"></a>01530 <span class="comment">!</span>
    1595. 

  1595. <a name="l01531"></a>01531 <span class="comment">!  Test whether the interval to be bisected next is the</span>
    1596. 

  1596. <a name="l01532"></a>01532 <span class="comment">!  smallest interval.</span>
    1597. 

  1597. <a name="l01533"></a>01533 <span class="comment">!</span>
    1598. 

  1598. <a name="l01534"></a>01534     <span class="keyword">if</span> ( .not. extrap ) <span class="keyword">then</span>
    1599. 

  1599. <a name="l01535"></a>01535 
    1600. 

  1600. <a name="l01536"></a>01536       <span class="keyword">if</span> ( level(maxerr)+1 &lt;= levmax ) <span class="keyword">then</span>
    1601. 

  1601. <a name="l01537"></a>01537         cycle
    1602. 

  1602. <a name="l01538"></a>01538       <span class="keyword">end if</span>
    1603. 

  1603. <a name="l01539"></a>01539 
    1604. 

  1604. <a name="l01540"></a>01540       extrap = .true.
    1605. 

  1605. <a name="l01541"></a>01541       nrmax = 2
    1606. 

  1606. <a name="l01542"></a>01542 
    1607. 

  1607. <a name="l01543"></a>01543     <span class="keyword">end if</span>
    1608. 

  1608. <a name="l01544"></a>01544 <span class="comment">!</span>
    1609. 

  1609. <a name="l01545"></a>01545 <span class="comment">!  The smallest interval has the largest error.</span>
    1610. 

  1610. <a name="l01546"></a>01546 <span class="comment">!  Before bisecting decrease the sum of the errors over the</span>
    1611. 

  1611. <a name="l01547"></a>01547 <span class="comment">!  larger intervals (erlarg) and perform extrapolation.</span>
    1612. 

  1612. <a name="l01548"></a>01548 <span class="comment">!</span>
    1613. 

  1613. <a name="l01549"></a>01549     <span class="keyword">if</span> ( ierro /= 3 .and. erlarg &gt; ertest ) <span class="keyword">then</span>
    1614. 

  1614. <a name="l01550"></a>01550 
    1615. 

  1615. <a name="l01551"></a>01551       id = nrmax
    1616. 

  1616. <a name="l01552"></a>01552       jupbnd = last
    1617. 

  1617. <a name="l01553"></a>01553       <span class="keyword">if</span> ( last &gt; (2+limit/2) ) <span class="keyword">then</span>
    1618. 

  1618. <a name="l01554"></a>01554         jupbnd = limit+3-last
    1619. 

  1619. <a name="l01555"></a>01555       <span class="keyword">end if</span>
    1620. 

  1620. <a name="l01556"></a>01556 
    1621. 

  1621. <a name="l01557"></a>01557       <span class="keyword">do</span> k = id, jupbnd
    1622. 

  1622. <a name="l01558"></a>01558         maxerr = iord(nrmax)
    1623. 

  1623. <a name="l01559"></a>01559         errmax = elist(maxerr)
    1624. 

  1624. <a name="l01560"></a>01560         <span class="keyword">if</span> ( level(maxerr)+1 &lt;= levmax ) <span class="keyword">then</span>
    1625. 

  1625. <a name="l01561"></a>01561           go to 160
    1626. 

  1626. <a name="l01562"></a>01562         <span class="keyword">end if</span>
    1627. 

  1627. <a name="l01563"></a>01563         nrmax = nrmax + 1
    1628. 

  1628. <a name="l01564"></a>01564       <span class="keyword">end do</span>
    1629. 

  1629. <a name="l01565"></a>01565 
    1630. 

  1630. <a name="l01566"></a>01566     <span class="keyword">end if</span>
    1631. 

  1631. <a name="l01567"></a>01567 <span class="comment">!</span>
    1632. 

  1632. <a name="l01568"></a>01568 <span class="comment">!  Perform extrapolation.</span>
    1633. 

  1633. <a name="l01569"></a>01569 <span class="comment">!</span>
    1634. 

  1634. <a name="l01570"></a>01570     numrl2 = numrl2 + 1
    1635. 

  1635. <a name="l01571"></a>01571     rlist2(numrl2) = area
    1636. 

  1636. <a name="l01572"></a>01572 
    1637. 

  1637. <a name="l01573"></a>01573     <span class="keyword">if</span> ( numrl2 &lt;= 2 ) <span class="keyword">then</span>
    1638. 

  1638. <a name="l01574"></a>01574       go to 155
    1639. 

  1639. <a name="l01575"></a>01575     <span class="keyword">end if</span>
    1640. 

  1640. <a name="l01576"></a>01576 
    1641. 

  1641. <a name="l01577"></a>01577     call <a class="code" href="quadpack_8f90.html#a5a75101d080f224c63adde98a0e64386">qextr </a>( numrl2, rlist2, reseps, abseps, res3la, nres )
    1642. 

  1642. <a name="l01578"></a>01578     ktmin = ktmin+1
    1643. 

  1643. <a name="l01579"></a>01579 
    1644. 

  1644. <a name="l01580"></a>01580     <span class="keyword">if</span> ( 5 &lt; ktmin .and. abserr &lt; 1.0e-03 * errsum ) <span class="keyword">then</span>
    1645. 

  1645. <a name="l01581"></a>01581       ier = 5
    1646. 

  1646. <a name="l01582"></a>01582     <span class="keyword">end if</span>
    1647. 

  1647. <a name="l01583"></a>01583 
    1648. 

  1648. <a name="l01584"></a>01584     <span class="keyword">if</span> ( abseps &lt; abserr ) <span class="keyword">then</span>
    1649. 

  1649. <a name="l01585"></a>01585 
    1650. 

  1650. <a name="l01586"></a>01586       ktmin = 0
    1651. 

  1651. <a name="l01587"></a>01587       abserr = abseps
    1652. 

  1652. <a name="l01588"></a>01588       result = reseps
    1653. 

  1653. <a name="l01589"></a>01589       correc = erlarg
    1654. 

  1654. <a name="l01590"></a>01590       ertest = max ( epsabs, epsrel * abs(reseps) )
    1655. 

  1655. <a name="l01591"></a>01591 
    1656. 

  1656. <a name="l01592"></a>01592       <span class="keyword">if</span> ( abserr &lt; ertest ) <span class="keyword">then</span>
    1657. 

  1657. <a name="l01593"></a>01593         exit
    1658. 

  1658. <a name="l01594"></a>01594       <span class="keyword">end if</span>
    1659. 

  1659. <a name="l01595"></a>01595 
    1660. 

  1660. <a name="l01596"></a>01596     <span class="keyword">end if</span>
    1661. 

  1661. <a name="l01597"></a>01597 <span class="comment">!</span>
    1662. 

  1662. <a name="l01598"></a>01598 <span class="comment">!  Prepare bisection of the smallest interval.</span>
    1663. 

  1663. <a name="l01599"></a>01599 <span class="comment">!</span>
    1664. 

  1664. <a name="l01600"></a>01600     <span class="keyword">if</span> ( numrl2 == 1 ) <span class="keyword">then</span>
    1665. 

  1665. <a name="l01601"></a>01601       noext = .true.
    1666. 

  1666. <a name="l01602"></a>01602     <span class="keyword">end if</span>
    1667. 

  1667. <a name="l01603"></a>01603 
    1668. 

  1668. <a name="l01604"></a>01604     <span class="keyword">if</span> ( 5 &lt;= ier ) <span class="keyword">then</span>
    1669. 

  1669. <a name="l01605"></a>01605       exit
    1670. 

  1670. <a name="l01606"></a>01606     <span class="keyword">end if</span>
    1671. 

  1671. <a name="l01607"></a>01607 
    1672. 

  1672. <a name="l01608"></a>01608 155 continue
    1673. 

  1673. <a name="l01609"></a>01609 
    1674. 

  1674. <a name="l01610"></a>01610     maxerr = iord(1)
    1675. 

  1675. <a name="l01611"></a>01611     errmax = elist(maxerr)
    1676. 

  1676. <a name="l01612"></a>01612     nrmax = 1
    1677. 

  1677. <a name="l01613"></a>01613     extrap = .false.
    1678. 

  1678. <a name="l01614"></a>01614     levmax = levmax + 1
    1679. 

  1679. <a name="l01615"></a>01615     erlarg = errsum
    1680. 

  1680. <a name="l01616"></a>01616 
    1681. 

  1681. <a name="l01617"></a>01617 160 continue
    1682. 

  1682. <a name="l01618"></a>01618 
    1683. 

  1683. <a name="l01619"></a>01619   <span class="keyword">end do</span>
    1684. 

  1684. <a name="l01620"></a>01620 <span class="comment">!</span>
    1685. 

  1685. <a name="l01621"></a>01621 <span class="comment">!  Set the final result.</span>
    1686. 

  1686. <a name="l01622"></a>01622 <span class="comment">!</span>
    1687. 

  1687. <a name="l01623"></a>01623   <span class="keyword">if</span> ( abserr == huge ( abserr ) ) <span class="keyword">then</span>
    1688. 

  1688. <a name="l01624"></a>01624     go to 190
    1689. 

  1689. <a name="l01625"></a>01625   <span class="keyword">end if</span>
    1690. 

  1690. <a name="l01626"></a>01626 
    1691. 

  1691. <a name="l01627"></a>01627   <span class="keyword">if</span> ( ( ier + ierro ) == 0 ) <span class="keyword">then</span>
    1692. 

  1692. <a name="l01628"></a>01628     go to 180
    1693. 

  1693. <a name="l01629"></a>01629   <span class="keyword">end if</span>
    1694. 

  1694. <a name="l01630"></a>01630 
    1695. 

  1695. <a name="l01631"></a>01631   <span class="keyword">if</span> ( ierro == 3 ) <span class="keyword">then</span>
    1696. 

  1696. <a name="l01632"></a>01632     abserr = abserr + correc
    1697. 

  1697. <a name="l01633"></a>01633   <span class="keyword">end if</span>
    1698. 

  1698. <a name="l01634"></a>01634 
    1699. 

  1699. <a name="l01635"></a>01635   <span class="keyword">if</span> ( ier == 0 ) <span class="keyword">then</span>
    1700. 

  1700. <a name="l01636"></a>01636     ier = 3
    1701. 

  1701. <a name="l01637"></a>01637   <span class="keyword">end if</span>
    1702. 

  1702. <a name="l01638"></a>01638 
    1703. 

  1703. <a name="l01639"></a>01639   <span class="keyword">if</span> ( result /= 0.0e+00 .and. area /= 0.0e+00 ) <span class="keyword">then</span>
    1704. 

  1704. <a name="l01640"></a>01640     go to 175
    1705. 

  1705. <a name="l01641"></a>01641   <span class="keyword">end if</span>
    1706. 

  1706. <a name="l01642"></a>01642 
    1707. 

  1707. <a name="l01643"></a>01643   <span class="keyword">if</span> ( errsum &lt; abserr ) <span class="keyword">then</span>
    1708. 

  1708. <a name="l01644"></a>01644     go to 190
    1709. 

  1709. <a name="l01645"></a>01645   <span class="keyword">end if</span>
    1710. 

  1710. <a name="l01646"></a>01646 
    1711. 

  1711. <a name="l01647"></a>01647   <span class="keyword">if</span> ( area == 0.0e+00 ) <span class="keyword">then</span>
    1712. 

  1712. <a name="l01648"></a>01648     go to 210
    1713. 

  1713. <a name="l01649"></a>01649   <span class="keyword">end if</span>
    1714. 

  1714. <a name="l01650"></a>01650 
    1715. 

  1715. <a name="l01651"></a>01651   go to 180
    1716. 

  1716. <a name="l01652"></a>01652 
    1717. 

  1717. <a name="l01653"></a>01653 175 continue
    1718. 

  1718. <a name="l01654"></a>01654 
    1719. 

  1719. <a name="l01655"></a>01655   <span class="keyword">if</span> ( abserr / abs(result) &gt; errsum / abs(area) ) <span class="keyword">then</span>
    1720. 

  1720. <a name="l01656"></a>01656     go to 190
    1721. 

  1721. <a name="l01657"></a>01657   <span class="keyword">end if</span>
    1722. 

  1722. <a name="l01658"></a>01658 <span class="comment">!</span>
    1723. 

  1723. <a name="l01659"></a>01659 <span class="comment">!  Test on divergence.</span>
    1724. 

  1724. <a name="l01660"></a>01660 <span class="comment">!</span>
    1725. 

  1725. <a name="l01661"></a>01661   180 continue
    1726. 

  1726. <a name="l01662"></a>01662 
    1727. 

  1727. <a name="l01663"></a>01663   <span class="keyword">if</span> ( ksgn == (-1) .and. max ( abs(result),abs(area)) &lt;=  &amp;
    1728. 

  1728. <a name="l01664"></a>01664     resabs*1.0e-02 ) go to 210
    1729. 

  1729. <a name="l01665"></a>01665 
    1730. 

  1730. <a name="l01666"></a>01666   <span class="keyword">if</span> ( 1.0e-02 &gt; (result/area) .or. (result/area) &gt; 1.0e+02 .or. &amp;
    1731. 

  1731. <a name="l01667"></a>01667     errsum &gt; abs(area) ) <span class="keyword">then</span>
    1732. 

  1732. <a name="l01668"></a>01668     ier = 6
    1733. 

  1733. <a name="l01669"></a>01669   <span class="keyword">end if</span>
    1734. 

  1734. <a name="l01670"></a>01670 
    1735. 

  1735. <a name="l01671"></a>01671   go to 210
    1736. 

  1736. <a name="l01672"></a>01672 <span class="comment">!</span>
    1737. 

  1737. <a name="l01673"></a>01673 <span class="comment">!  Compute global integral sum.</span>
    1738. 

  1738. <a name="l01674"></a>01674 <span class="comment">!</span>
    1739. 

  1739. <a name="l01675"></a>01675 190 continue
    1740. 

  1740. <a name="l01676"></a>01676 
    1741. 

  1741. <a name="l01677"></a>01677   result = sum ( rlist(1:last) )
    1742. 

  1742. <a name="l01678"></a>01678 
    1743. 

  1743. <a name="l01679"></a>01679   abserr = errsum
    1744. 

  1744. <a name="l01680"></a>01680 
    1745. 

  1745. <a name="l01681"></a>01681 210 continue
    1746. 

  1746. <a name="l01682"></a>01682 
    1747. 

  1747. <a name="l01683"></a>01683   <span class="keyword">if</span> ( 2 &lt; ier ) <span class="keyword">then</span>
    1748. 

  1748. <a name="l01684"></a>01684     ier = ier - 1
    1749. 

  1749. <a name="l01685"></a>01685   <span class="keyword">end if</span>
    1750. 

  1750. <a name="l01686"></a>01686 
    1751. 

  1751. <a name="l01687"></a>01687   result = result * sign
    1752. 

  1752. <a name="l01688"></a>01688 
    1753. 

  1753. <a name="l01689"></a>01689   return
    1754. 

  1754. <a name="l01690"></a>01690 <span class="keyword">end</span>
    1755. 

  1755. <a name="l01691"></a><a class="code" href="quadpack_8f90.html#a00a116a91c0699e57d15abc61dcd531b">01691</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a00a116a91c0699e57d15abc61dcd531b">qags</a> ( f, a, b, epsabs, epsrel, result, abserr, neval, ier )
    1756. 

  1756. <a name="l01692"></a>01692 
    1757. 

  1757. <a name="l01693"></a>01693 <span class="comment">!*****************************************************************************80</span>
    1758. 

  1758. <a name="l01694"></a>01694 <span class="comment">!</span>
    1759. 

  1759. <a name="l01695"></a>01695 <span class="comment">!! QAGS estimates the integral of a function.</span>
    1760. 

  1760. <a name="l01696"></a>01696 <span class="comment">!</span>
    1761. 

  1761. <a name="l01697"></a>01697 <span class="comment">!  Discussion:</span>
    1762. 

  1762. <a name="l01698"></a>01698 <span class="comment">!</span>
    1763. 

  1763. <a name="l01699"></a>01699 <span class="comment">!    The routine calculates an approximation RESULT to a definite integral   </span>
    1764. 

  1764. <a name="l01700"></a>01700 <span class="comment">!      I = integral of F over (A,B),</span>
    1765. 

  1765. <a name="l01701"></a>01701 <span class="comment">!    hopefully satisfying</span>
    1766. 

  1766. <a name="l01702"></a>01702 <span class="comment">!      || I - RESULT || &lt;= max ( EPSABS, EPSREL * ||I|| ).</span>
    1767. 

  1767. <a name="l01703"></a>01703 <span class="comment">!</span>
    1768. 

  1768. <a name="l01704"></a>01704 <span class="comment">!  Author:</span>
    1769. 

  1769. <a name="l01705"></a>01705 <span class="comment">!</span>
    1770. 

  1770. <a name="l01706"></a>01706 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    1771. 

  1771. <a name="l01707"></a>01707 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    1772. 

  1772. <a name="l01708"></a>01708 <span class="comment">!</span>
    1773. 

  1773. <a name="l01709"></a>01709 <span class="comment">!  Reference:</span>
    1774. 

  1774. <a name="l01710"></a>01710 <span class="comment">!</span>
    1775. 

  1775. <a name="l01711"></a>01711 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    1776. 

  1776. <a name="l01712"></a>01712 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    1777. 

  1777. <a name="l01713"></a>01713 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    1778. 

  1778. <a name="l01714"></a>01714 <span class="comment">!    Springer Verlag, 1983</span>
    1779. 

  1779. <a name="l01715"></a>01715 <span class="comment">!</span>
    1780. 

  1780. <a name="l01716"></a>01716 <span class="comment">!  Parameters:</span>
    1781. 

  1781. <a name="l01717"></a>01717 <span class="comment">!</span>
    1782. 

  1782. <a name="l01718"></a>01718 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    1783. 

  1783. <a name="l01719"></a>01719 <span class="comment">!      function f ( x )</span>
    1784. 

  1784. <a name="l01720"></a>01720 <span class="comment">!      real f</span>
    1785. 

  1785. <a name="l01721"></a>01721 <span class="comment">!      real x</span>
    1786. 

  1786. <a name="l01722"></a>01722 <span class="comment">!    which evaluates the integrand function.</span>
    1787. 

  1787. <a name="l01723"></a>01723 <span class="comment">!</span>
    1788. 

  1788. <a name="l01724"></a>01724 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    1789. 

  1789. <a name="l01725"></a>01725 <span class="comment">!</span>
    1790. 

  1790. <a name="l01726"></a>01726 <span class="comment">!    Input, real EPSABS, EPSREL, the absolute and relative accuracy requested.</span>
    1791. 

  1791. <a name="l01727"></a>01727 <span class="comment">!</span>
    1792. 

  1792. <a name="l01728"></a>01728 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    1793. 

  1793. <a name="l01729"></a>01729 <span class="comment">!</span>
    1794. 

  1794. <a name="l01730"></a>01730 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    1795. 

  1795. <a name="l01731"></a>01731 <span class="comment">!</span>
    1796. 

  1796. <a name="l01732"></a>01732 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    1797. 

  1797. <a name="l01733"></a>01733 <span class="comment">!</span>
    1798. 

  1798. <a name="l01734"></a>01734 <span class="comment">!    Output, integer IER, error flag.</span>
    1799. 

  1799. <a name="l01735"></a>01735 <span class="comment">!                     ier = 0 normal and reliable termination of the</span>
    1800. 

  1800. <a name="l01736"></a>01736 <span class="comment">!                             routine. it is assumed that the requested</span>
    1801. 

  1801. <a name="l01737"></a>01737 <span class="comment">!                             accuracy has been achieved.</span>
    1802. 

  1802. <a name="l01738"></a>01738 <span class="comment">!                     ier &gt; 0 abnormal termination of the routine</span>
    1803. 

  1803. <a name="l01739"></a>01739 <span class="comment">!                             the estimates for integral and error are</span>
    1804. 

  1804. <a name="l01740"></a>01740 <span class="comment">!                             less reliable. it is assumed that the</span>
    1805. 

  1805. <a name="l01741"></a>01741 <span class="comment">!                             requested accuracy has not been achieved.</span>
    1806. 

  1806. <a name="l01742"></a>01742 <span class="comment">!                         = 1 maximum number of subdivisions allowed</span>
    1807. 

  1807. <a name="l01743"></a>01743 <span class="comment">!                             has been achieved. one can allow more sub-</span>
    1808. 

  1808. <a name="l01744"></a>01744 <span class="comment">!                             divisions by increasing the data value of</span>
    1809. 

  1809. <a name="l01745"></a>01745 <span class="comment">!                             limit in qags (and taking the according</span>
    1810. 

  1810. <a name="l01746"></a>01746 <span class="comment">!                             dimension adjustments into account).</span>
    1811. 

  1811. <a name="l01747"></a>01747 <span class="comment">!                             however, if this yields no improvement</span>
    1812. 

  1812. <a name="l01748"></a>01748 <span class="comment">!                             it is advised to analyze the integrand</span>
    1813. 

  1813. <a name="l01749"></a>01749 <span class="comment">!                             in order to determine the integration</span>
    1814. 

  1814. <a name="l01750"></a>01750 <span class="comment">!                             difficulties. if the position of a</span>
    1815. 

  1815. <a name="l01751"></a>01751 <span class="comment">!                             local difficulty can be determined (e.g.</span>
    1816. 

  1816. <a name="l01752"></a>01752 <span class="comment">!                             singularity, discontinuity within the</span>
    1817. 

  1817. <a name="l01753"></a>01753 <span class="comment">!                             interval) one will probably gain from</span>
    1818. 

  1818. <a name="l01754"></a>01754 <span class="comment">!                             splitting up the interval at this point</span>
    1819. 

  1819. <a name="l01755"></a>01755 <span class="comment">!                             and calling the integrator on the sub-</span>
    1820. 

  1820. <a name="l01756"></a>01756 <span class="comment">!                             ranges. if possible, an appropriate</span>
    1821. 

  1821. <a name="l01757"></a>01757 <span class="comment">!                             special-purpose integrator should be used,</span>
    1822. 

  1822. <a name="l01758"></a>01758 <span class="comment">!                             which is designed for handling the type</span>
    1823. 

  1823. <a name="l01759"></a>01759 <span class="comment">!                             of difficulty involved.</span>
    1824. 

  1824. <a name="l01760"></a>01760 <span class="comment">!                         = 2 the occurrence of roundoff error is detec-</span>
    1825. 

  1825. <a name="l01761"></a>01761 <span class="comment">!                             ted, which prevents the requested</span>
    1826. 

  1826. <a name="l01762"></a>01762 <span class="comment">!                             tolerance from being achieved.</span>
    1827. 

  1827. <a name="l01763"></a>01763 <span class="comment">!                             the error may be under-estimated.</span>
    1828. 

  1828. <a name="l01764"></a>01764 <span class="comment">!                         = 3 extremely bad integrand behavior occurs</span>
    1829. 

  1829. <a name="l01765"></a>01765 <span class="comment">!                             at some  points of the integration</span>
    1830. 

  1830. <a name="l01766"></a>01766 <span class="comment">!                             interval.</span>
    1831. 

  1831. <a name="l01767"></a>01767 <span class="comment">!                         = 4 the algorithm does not converge. roundoff</span>
    1832. 

  1832. <a name="l01768"></a>01768 <span class="comment">!                             error is detected in the extrapolation</span>
    1833. 

  1833. <a name="l01769"></a>01769 <span class="comment">!                             table. it is presumed that the requested</span>
    1834. 

  1834. <a name="l01770"></a>01770 <span class="comment">!                             tolerance cannot be achieved, and that the</span>
    1835. 

  1835. <a name="l01771"></a>01771 <span class="comment">!                             returned result is the best which can be</span>
    1836. 

  1836. <a name="l01772"></a>01772 <span class="comment">!                             obtained.</span>
    1837. 

  1837. <a name="l01773"></a>01773 <span class="comment">!                         = 5 the integral is probably divergent, or</span>
    1838. 

  1838. <a name="l01774"></a>01774 <span class="comment">!                             slowly convergent. it must be noted that</span>
    1839. 

  1839. <a name="l01775"></a>01775 <span class="comment">!                             divergence can occur with any other value</span>
    1840. 

  1840. <a name="l01776"></a>01776 <span class="comment">!                             of ier.</span>
    1841. 

  1841. <a name="l01777"></a>01777 <span class="comment">!                         = 6 the input is invalid, because</span>
    1842. 

  1842. <a name="l01778"></a>01778 <span class="comment">!                             epsabs &lt; 0 and epsrel &lt; 0,</span>
    1843. 

  1843. <a name="l01779"></a>01779 <span class="comment">!                             result, abserr and neval are set to zero.</span>
    1844. 

  1844. <a name="l01780"></a>01780 <span class="comment">!</span>
    1845. 

  1845. <a name="l01781"></a>01781 <span class="comment">!  Local Parameters:</span>
    1846. 

  1846. <a name="l01782"></a>01782 <span class="comment">!</span>
    1847. 

  1847. <a name="l01783"></a>01783 <span class="comment">!           alist     - list of left end points of all subintervals</span>
    1848. 

  1848. <a name="l01784"></a>01784 <span class="comment">!                       considered up to now</span>
    1849. 

  1849. <a name="l01785"></a>01785 <span class="comment">!           blist     - list of right end points of all subintervals</span>
    1850. 

  1850. <a name="l01786"></a>01786 <span class="comment">!                       considered up to now</span>
    1851. 

  1851. <a name="l01787"></a>01787 <span class="comment">!           rlist(i)  - approximation to the integral over</span>
    1852. 

  1852. <a name="l01788"></a>01788 <span class="comment">!                       (alist(i),blist(i))</span>
    1853. 

  1853. <a name="l01789"></a>01789 <span class="comment">!           rlist2    - array of dimension at least limexp+2 containing</span>
    1854. 

  1854. <a name="l01790"></a>01790 <span class="comment">!                       the part of the epsilon table which is still</span>
    1855. 

  1855. <a name="l01791"></a>01791 <span class="comment">!                       needed for further computations</span>
    1856. 

  1856. <a name="l01792"></a>01792 <span class="comment">!           elist(i)  - error estimate applying to rlist(i)</span>
    1857. 

  1857. <a name="l01793"></a>01793 <span class="comment">!           maxerr    - pointer to the interval with largest error</span>
    1858. 

  1858. <a name="l01794"></a>01794 <span class="comment">!                       estimate</span>
    1859. 

  1859. <a name="l01795"></a>01795 <span class="comment">!           errmax    - elist(maxerr)</span>
    1860. 

  1860. <a name="l01796"></a>01796 <span class="comment">!           erlast    - error on the interval currently subdivided</span>
    1861. 

  1861. <a name="l01797"></a>01797 <span class="comment">!                       (before that subdivision has taken place)</span>
    1862. 

  1862. <a name="l01798"></a>01798 <span class="comment">!           area      - sum of the integrals over the subintervals</span>
    1863. 

  1863. <a name="l01799"></a>01799 <span class="comment">!           errsum    - sum of the errors over the subintervals</span>
    1864. 

  1864. <a name="l01800"></a>01800 <span class="comment">!           errbnd    - requested accuracy max(epsabs,epsrel*</span>
    1865. 

  1865. <a name="l01801"></a>01801 <span class="comment">!                       abs(result))</span>
    1866. 

  1866. <a name="l01802"></a>01802 <span class="comment">!           *****1    - variable for the left interval</span>
    1867. 

  1867. <a name="l01803"></a>01803 <span class="comment">!           *****2    - variable for the right interval</span>
    1868. 

  1868. <a name="l01804"></a>01804 <span class="comment">!           last      - index for subdivision</span>
    1869. 

  1869. <a name="l01805"></a>01805 <span class="comment">!           nres      - number of calls to the extrapolation routine</span>
    1870. 

  1870. <a name="l01806"></a>01806 <span class="comment">!           numrl2    - number of elements currently in rlist2. if an</span>
    1871. 

  1871. <a name="l01807"></a>01807 <span class="comment">!                       appropriate approximation to the compounded</span>
    1872. 

  1872. <a name="l01808"></a>01808 <span class="comment">!                       integral has been obtained it is put in</span>
    1873. 

  1873. <a name="l01809"></a>01809 <span class="comment">!                       rlist2(numrl2) after numrl2 has been increased</span>
    1874. 

  1874. <a name="l01810"></a>01810 <span class="comment">!                       by one.</span>
    1875. 

  1875. <a name="l01811"></a>01811 <span class="comment">!           small     - length of the smallest interval considered</span>
    1876. 

  1876. <a name="l01812"></a>01812 <span class="comment">!                       up to now, multiplied by 1.5</span>
    1877. 

  1877. <a name="l01813"></a>01813 <span class="comment">!           erlarg    - sum of the errors over the intervals larger</span>
    1878. 

  1878. <a name="l01814"></a>01814 <span class="comment">!                       than the smallest interval considered up to now</span>
    1879. 

  1879. <a name="l01815"></a>01815 <span class="comment">!           extrap    - logical variable denoting that the routine is</span>
    1880. 

  1880. <a name="l01816"></a>01816 <span class="comment">!                       attempting to perform extrapolation i.e. before</span>
    1881. 

  1881. <a name="l01817"></a>01817 <span class="comment">!                       subdividing the smallest interval we try to</span>
    1882. 

  1882. <a name="l01818"></a>01818 <span class="comment">!                       decrease the value of erlarg.</span>
    1883. 

  1883. <a name="l01819"></a>01819 <span class="comment">!           noext     - logical variable denoting that extrapolation</span>
    1884. 

  1884. <a name="l01820"></a>01820 <span class="comment">!                       is no longer allowed (true value)</span>
    1885. 

  1885. <a name="l01821"></a>01821 <span class="comment">!</span>
    1886. 

  1886. <a name="l01822"></a>01822   <span class="keyword">implicit none</span>
    1887. 

  1887. <a name="l01823"></a>01823 
    1888. 

  1888. <a name="l01824"></a>01824   <span class="keywordtype">integer</span>, <span class="keywordtype">parameter</span> :: limit = 500
    1889. 

  1889. <a name="l01825"></a>01825 
    1890. 

  1890. <a name="l01826"></a>01826   <span class="keywordtype">real</span> a
    1891. 

  1891. <a name="l01827"></a>01827   <span class="keywordtype">real</span> abseps
    1892. 

  1892. <a name="l01828"></a>01828   <span class="keywordtype">real</span> abserr
    1893. 

  1893. <a name="l01829"></a>01829   <span class="keywordtype">real</span> alist(limit)
    1894. 

  1894. <a name="l01830"></a>01830   <span class="keywordtype">real</span> area
    1895. 

  1895. <a name="l01831"></a>01831   <span class="keywordtype">real</span> area1
    1896. 

  1896. <a name="l01832"></a>01832   <span class="keywordtype">real</span> area12
    1897. 

  1897. <a name="l01833"></a>01833   <span class="keywordtype">real</span> area2
    1898. 

  1898. <a name="l01834"></a>01834   <span class="keywordtype">real</span> a1
    1899. 

  1899. <a name="l01835"></a>01835   <span class="keywordtype">real</span> a2
    1900. 

  1900. <a name="l01836"></a>01836   <span class="keywordtype">real</span> b
    1901. 

  1901. <a name="l01837"></a>01837   <span class="keywordtype">real</span> blist(limit)
    1902. 

  1902. <a name="l01838"></a>01838   <span class="keywordtype">real</span> b1
    1903. 

  1903. <a name="l01839"></a>01839   <span class="keywordtype">real</span> b2
    1904. 

  1904. <a name="l01840"></a>01840   <span class="keywordtype">real</span> correc
    1905. 

  1905. <a name="l01841"></a>01841   <span class="keywordtype">real</span> defabs
    1906. 

  1906. <a name="l01842"></a>01842   <span class="keywordtype">real</span> defab1
    1907. 

  1907. <a name="l01843"></a>01843   <span class="keywordtype">real</span> defab2
    1908. 

  1908. <a name="l01844"></a>01844   <span class="keywordtype">real</span> dres
    1909. 

  1909. <a name="l01845"></a>01845   <span class="keywordtype">real</span> elist(limit)
    1910. 

  1910. <a name="l01846"></a>01846   <span class="keywordtype">real</span> epsabs
    1911. 

  1911. <a name="l01847"></a>01847   <span class="keywordtype">real</span> epsrel
    1912. 

  1912. <a name="l01848"></a>01848   <span class="keywordtype">real</span> erlarg
    1913. 

  1913. <a name="l01849"></a>01849   <span class="keywordtype">real</span> erlast
    1914. 

  1914. <a name="l01850"></a>01850   <span class="keywordtype">real</span> errbnd
    1915. 

  1915. <a name="l01851"></a>01851   <span class="keywordtype">real</span> errmax
    1916. 

  1916. <a name="l01852"></a>01852   <span class="keywordtype">real</span> error1
    1917. 

  1917. <a name="l01853"></a>01853   <span class="keywordtype">real</span> error2
    1918. 

  1918. <a name="l01854"></a>01854   <span class="keywordtype">real</span> erro12
    1919. 

  1919. <a name="l01855"></a>01855   <span class="keywordtype">real</span> errsum
    1920. 

  1920. <a name="l01856"></a>01856   <span class="keywordtype">real</span> ertest
    1921. 

  1921. <a name="l01857"></a>01857   <span class="keywordtype">logical</span> extrap
    1922. 

  1922. <a name="l01858"></a>01858   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    1923. 

  1923. <a name="l01859"></a>01859   <span class="keywordtype">integer</span> id
    1924. 

  1924. <a name="l01860"></a>01860   <span class="keywordtype">integer</span> ier
    1925. 

  1925. <a name="l01861"></a>01861   <span class="keywordtype">integer</span> ierro
    1926. 

  1926. <a name="l01862"></a>01862   <span class="keywordtype">integer</span> iord(limit)
    1927. 

  1927. <a name="l01863"></a>01863   <span class="keywordtype">integer</span> iroff1
    1928. 

  1928. <a name="l01864"></a>01864   <span class="keywordtype">integer</span> iroff2
    1929. 

  1929. <a name="l01865"></a>01865   <span class="keywordtype">integer</span> iroff3
    1930. 

  1930. <a name="l01866"></a>01866   <span class="keywordtype">integer</span> jupbnd
    1931. 

  1931. <a name="l01867"></a>01867   <span class="keywordtype">integer</span> k
    1932. 

  1932. <a name="l01868"></a>01868   <span class="keywordtype">integer</span> ksgn
    1933. 

  1933. <a name="l01869"></a>01869   <span class="keywordtype">integer</span> ktmin
    1934. 

  1934. <a name="l01870"></a>01870   <span class="keywordtype">integer</span> last
    1935. 

  1935. <a name="l01871"></a>01871   <span class="keywordtype">logical</span> noext
    1936. 

  1936. <a name="l01872"></a>01872   <span class="keywordtype">integer</span> maxerr
    1937. 

  1937. <a name="l01873"></a>01873   <span class="keywordtype">integer</span> neval
    1938. 

  1938. <a name="l01874"></a>01874   <span class="keywordtype">integer</span> nres
    1939. 

  1939. <a name="l01875"></a>01875   <span class="keywordtype">integer</span> nrmax
    1940. 

  1940. <a name="l01876"></a>01876   <span class="keywordtype">integer</span> numrl2
    1941. 

  1941. <a name="l01877"></a>01877   <span class="keywordtype">real</span> resabs
    1942. 

  1942. <a name="l01878"></a>01878   <span class="keywordtype">real</span> reseps
    1943. 

  1943. <a name="l01879"></a>01879   <span class="keywordtype">real</span> result
    1944. 

  1944. <a name="l01880"></a>01880   <span class="keywordtype">real</span> res3la(3)
    1945. 

  1945. <a name="l01881"></a>01881   <span class="keywordtype">real</span> rlist(limit)
    1946. 

  1946. <a name="l01882"></a>01882   <span class="keywordtype">real</span> rlist2(52)
    1947. 

  1947. <a name="l01883"></a>01883   <span class="keywordtype">real</span> small
    1948. 

  1948. <a name="l01884"></a>01884 <span class="comment">!</span>
    1949. 

  1949. <a name="l01885"></a>01885 <span class="comment">!  The dimension of rlist2 is determined by the value of</span>
    1950. 

  1950. <a name="l01886"></a>01886 <span class="comment">!  limexp in QEXTR (rlist2 should be of dimension</span>
    1951. 

  1951. <a name="l01887"></a>01887 <span class="comment">!  (limexp+2) at least).</span>
    1952. 

  1952. <a name="l01888"></a>01888 <span class="comment">!</span>
    1953. 

  1953. <a name="l01889"></a>01889 <span class="comment">!  Test on validity of parameters.</span>
    1954. 

  1954. <a name="l01890"></a>01890 <span class="comment">!</span>
    1955. 

  1955. <a name="l01891"></a>01891   ier = 0
    1956. 

  1956. <a name="l01892"></a>01892   neval = 0
    1957. 

  1957. <a name="l01893"></a>01893   last = 0
    1958. 

  1958. <a name="l01894"></a>01894   result = 0.0e+00
    1959. 

  1959. <a name="l01895"></a>01895   abserr = 0.0e+00
    1960. 

  1960. <a name="l01896"></a>01896   alist(1) = a
    1961. 

  1961. <a name="l01897"></a>01897   blist(1) = b
    1962. 

  1962. <a name="l01898"></a>01898   rlist(1) = 0.0e+00
    1963. 

  1963. <a name="l01899"></a>01899   elist(1) = 0.0e+00
    1964. 

  1964. <a name="l01900"></a>01900 
    1965. 

  1965. <a name="l01901"></a>01901   <span class="keyword">if</span> ( epsabs &lt; 0.0e+00 .and. epsrel &lt; 0.0e+00 ) <span class="keyword">then</span>
    1966. 

  1966. <a name="l01902"></a>01902     ier = 6
    1967. 

  1967. <a name="l01903"></a>01903     return
    1968. 

  1968. <a name="l01904"></a>01904   <span class="keyword">end if</span>
    1969. 

  1969. <a name="l01905"></a>01905 <span class="comment">!</span>
    1970. 

  1970. <a name="l01906"></a>01906 <span class="comment">!  First approximation to the integral.</span>
    1971. 

  1971. <a name="l01907"></a>01907 <span class="comment">!</span>
    1972. 

  1972. <a name="l01908"></a>01908   ierro = 0
    1973. 

  1973. <a name="l01909"></a>01909   call <a class="code" href="quadpack_8f90.html#a27241a527b249e9de59a5ed6bee5f805">qk21 </a>( f, a, b, result, abserr, defabs, resabs )
    1974. 

  1974. <a name="l01910"></a>01910 <span class="comment">!</span>
    1975. 

  1975. <a name="l01911"></a>01911 <span class="comment">!  Test on accuracy.</span>
    1976. 

  1976. <a name="l01912"></a>01912 <span class="comment">!</span>
    1977. 

  1977. <a name="l01913"></a>01913   dres = abs ( result )
    1978. 

  1978. <a name="l01914"></a>01914   errbnd = max ( epsabs, epsrel * dres )
    1979. 

  1979. <a name="l01915"></a>01915   last = 1
    1980. 

  1980. <a name="l01916"></a>01916   rlist(1) = result
    1981. 

  1981. <a name="l01917"></a>01917   elist(1) = abserr
    1982. 

  1982. <a name="l01918"></a>01918   iord(1) = 1
    1983. 

  1983. <a name="l01919"></a>01919 
    1984. 

  1984. <a name="l01920"></a>01920   <span class="keyword">if</span> ( abserr &lt;= 1.0e+02 * epsilon ( defabs ) * defabs .and. &amp;
    1985. 

  1985. <a name="l01921"></a>01921     abserr &gt; errbnd ) <span class="keyword">then</span>
    1986. 

  1986. <a name="l01922"></a>01922     ier = 2
    1987. 

  1987. <a name="l01923"></a>01923   <span class="keyword">end if</span>
    1988. 

  1988. <a name="l01924"></a>01924 
    1989. 

  1989. <a name="l01925"></a>01925   <span class="keyword">if</span> ( limit == 1 ) <span class="keyword">then</span>
    1990. 

  1990. <a name="l01926"></a>01926     ier = 1
    1991. 

  1991. <a name="l01927"></a>01927   <span class="keyword">end if</span>
    1992. 

  1992. <a name="l01928"></a>01928 
    1993. 

  1993. <a name="l01929"></a>01929   <span class="keyword">if</span> ( ier /= 0 .or. (abserr &lt;= errbnd .and. abserr /= resabs ) .or. &amp;
    1994. 

  1994. <a name="l01930"></a>01930     abserr == 0.0e+00 ) go to 140
    1995. 

  1995. <a name="l01931"></a>01931 <span class="comment">!</span>
    1996. 

  1996. <a name="l01932"></a>01932 <span class="comment">!  Initialization.</span>
    1997. 

  1997. <a name="l01933"></a>01933 <span class="comment">!</span>
    1998. 

  1998. <a name="l01934"></a>01934   rlist2(1) = result
    1999. 

  1999. <a name="l01935"></a>01935   errmax = abserr
    2000. 

  2000. <a name="l01936"></a>01936   maxerr = 1
    2001. 

  2001. <a name="l01937"></a>01937   area = result
    2002. 

  2002. <a name="l01938"></a>01938   errsum = abserr
    2003. 

  2003. <a name="l01939"></a>01939   abserr = huge ( abserr )
    2004. 

  2004. <a name="l01940"></a>01940   nrmax = 1
    2005. 

  2005. <a name="l01941"></a>01941   nres = 0
    2006. 

  2006. <a name="l01942"></a>01942   numrl2 = 2
    2007. 

  2007. <a name="l01943"></a>01943   ktmin = 0
    2008. 

  2008. <a name="l01944"></a>01944   extrap = .false.
    2009. 

  2009. <a name="l01945"></a>01945   noext = .false.
    2010. 

  2010. <a name="l01946"></a>01946   iroff1 = 0
    2011. 

  2011. <a name="l01947"></a>01947   iroff2 = 0
    2012. 

  2012. <a name="l01948"></a>01948   iroff3 = 0
    2013. 

  2013. <a name="l01949"></a>01949 
    2014. 

  2014. <a name="l01950"></a>01950   <span class="keyword">if</span> ( dres &gt;= (1.0e+00-5.0e+01* epsilon ( defabs ) ) * defabs ) <span class="keyword">then</span>
    2015. 

  2015. <a name="l01951"></a>01951     ksgn = 1
    2016. 

  2016. <a name="l01952"></a>01952   <span class="keyword">else</span>
    2017. 

  2017. <a name="l01953"></a>01953     ksgn = -1
    2018. 

  2018. <a name="l01954"></a>01954   <span class="keyword">end if</span>
    2019. 

  2019. <a name="l01955"></a>01955 
    2020. 

  2020. <a name="l01956"></a>01956   <span class="keyword">do</span> last = 2, limit
    2021. 

  2021. <a name="l01957"></a>01957 <span class="comment">!</span>
    2022. 

  2022. <a name="l01958"></a>01958 <span class="comment">!  Bisect the subinterval with the nrmax-th largest error estimate.</span>
    2023. 

  2023. <a name="l01959"></a>01959 <span class="comment">!</span>
    2024. 

  2024. <a name="l01960"></a>01960     a1 = alist(maxerr)
    2025. 

  2025. <a name="l01961"></a>01961     b1 = 5.0e-01 * ( alist(maxerr) + blist(maxerr) )
    2026. 

  2026. <a name="l01962"></a>01962     a2 = b1
    2027. 

  2027. <a name="l01963"></a>01963     b2 = blist(maxerr)
    2028. 

  2028. <a name="l01964"></a>01964     erlast = errmax
    2029. 

  2029. <a name="l01965"></a>01965     call <a class="code" href="quadpack_8f90.html#a27241a527b249e9de59a5ed6bee5f805">qk21 </a>( f, a1, b1, area1, error1, resabs, defab1 )
    2030. 

  2030. <a name="l01966"></a>01966     call <a class="code" href="quadpack_8f90.html#a27241a527b249e9de59a5ed6bee5f805">qk21 </a>( f, a2, b2, area2, error2, resabs, defab2 )
    2031. 

  2031. <a name="l01967"></a>01967 <span class="comment">!</span>
    2032. 

  2032. <a name="l01968"></a>01968 <span class="comment">!  Improve previous approximations to integral and error</span>
    2033. 

  2033. <a name="l01969"></a>01969 <span class="comment">!  and test for accuracy.</span>
    2034. 

  2034. <a name="l01970"></a>01970 <span class="comment">!</span>
    2035. 

  2035. <a name="l01971"></a>01971     area12 = area1+area2
    2036. 

  2036. <a name="l01972"></a>01972     erro12 = error1+error2
    2037. 

  2037. <a name="l01973"></a>01973     errsum = errsum+erro12-errmax
    2038. 

  2038. <a name="l01974"></a>01974     area = area+area12-rlist(maxerr)
    2039. 

  2039. <a name="l01975"></a>01975 
    2040. 

  2040. <a name="l01976"></a>01976     <span class="keyword">if</span> ( defab1 == error1 .or. defab2 == error2 ) go to 15
    2041. 

  2041. <a name="l01977"></a>01977 
    2042. 

  2042. <a name="l01978"></a>01978     <span class="keyword">if</span> ( abs ( rlist(maxerr) - area12) &gt; 1.0e-05 * abs(area12) &amp;
    2043. 

  2043. <a name="l01979"></a>01979       .or. erro12 &lt; 9.9e-01 * errmax ) go to 10
    2044. 

  2044. <a name="l01980"></a>01980 
    2045. 

  2045. <a name="l01981"></a>01981     <span class="keyword">if</span> ( extrap ) <span class="keyword">then</span>
    2046. 

  2046. <a name="l01982"></a>01982       iroff2 = iroff2+1
    2047. 

  2047. <a name="l01983"></a>01983     <span class="keyword">else</span>
    2048. 

  2048. <a name="l01984"></a>01984       iroff1 = iroff1+1
    2049. 

  2049. <a name="l01985"></a>01985     <span class="keyword">end if</span>
    2050. 

  2050. <a name="l01986"></a>01986 
    2051. 

  2051. <a name="l01987"></a>01987 10  continue
    2052. 

  2052. <a name="l01988"></a>01988 
    2053. 

  2053. <a name="l01989"></a>01989     <span class="keyword">if</span> ( last &gt; 10 .and. erro12 &gt; errmax ) <span class="keyword">then</span>
    2054. 

  2054. <a name="l01990"></a>01990       iroff3 = iroff3+1
    2055. 

  2055. <a name="l01991"></a>01991     <span class="keyword">end if</span>
    2056. 

  2056. <a name="l01992"></a>01992 
    2057. 

  2057. <a name="l01993"></a>01993 15  continue
    2058. 

  2058. <a name="l01994"></a>01994 
    2059. 

  2059. <a name="l01995"></a>01995     rlist(maxerr) = area1
    2060. 

  2060. <a name="l01996"></a>01996     rlist(last) = area2
    2061. 

  2061. <a name="l01997"></a>01997     errbnd = max ( epsabs, epsrel*abs(area) )
    2062. 

  2062. <a name="l01998"></a>01998 <span class="comment">!</span>
    2063. 

  2063. <a name="l01999"></a>01999 <span class="comment">!  Test for roundoff error and eventually set error flag.</span>
    2064. 

  2064. <a name="l02000"></a>02000 <span class="comment">!</span>
    2065. 

  2065. <a name="l02001"></a>02001     <span class="keyword">if</span> ( iroff1+iroff2 &gt;= 10 .or. iroff3 &gt;= 20 ) <span class="keyword">then</span>
    2066. 

  2066. <a name="l02002"></a>02002       ier = 2
    2067. 

  2067. <a name="l02003"></a>02003     <span class="keyword">end if</span>
    2068. 

  2068. <a name="l02004"></a>02004 
    2069. 

  2069. <a name="l02005"></a>02005     <span class="keyword">if</span> ( iroff2 &gt;= 5 ) <span class="keyword">then</span>
    2070. 

  2070. <a name="l02006"></a>02006       ierro = 3
    2071. 

  2071. <a name="l02007"></a>02007     <span class="keyword">end if</span>
    2072. 

  2072. <a name="l02008"></a>02008 <span class="comment">!</span>
    2073. 

  2073. <a name="l02009"></a>02009 <span class="comment">!  Set error flag in the case that the number of subintervals</span>
    2074. 

  2074. <a name="l02010"></a>02010 <span class="comment">!  equals limit.</span>
    2075. 

  2075. <a name="l02011"></a>02011 <span class="comment">!</span>
    2076. 

  2076. <a name="l02012"></a>02012     <span class="keyword">if</span> ( last == limit ) <span class="keyword">then</span>
    2077. 

  2077. <a name="l02013"></a>02013       ier = 1
    2078. 

  2078. <a name="l02014"></a>02014     <span class="keyword">end if</span>
    2079. 

  2079. <a name="l02015"></a>02015 <span class="comment">!</span>
    2080. 

  2080. <a name="l02016"></a>02016 <span class="comment">!  Set error flag in the case of bad integrand behavior</span>
    2081. 

  2081. <a name="l02017"></a>02017 <span class="comment">!  at a point of the integration range.</span>
    2082. 

  2082. <a name="l02018"></a>02018 <span class="comment">!</span>
    2083. 

  2083. <a name="l02019"></a>02019     <span class="keyword">if</span> ( max ( abs(a1),abs(b2)) &lt;= (1.0e+00+1.0e+03* epsilon ( a1 ) )* &amp;
    2084. 

  2084. <a name="l02020"></a>02020       (abs(a2)+1.0e+03* tiny ( a2 ) ) ) <span class="keyword">then</span>
    2085. 

  2085. <a name="l02021"></a>02021       ier = 4
    2086. 

  2086. <a name="l02022"></a>02022     <span class="keyword">end if</span>
    2087. 

  2087. <a name="l02023"></a>02023 <span class="comment">!</span>
    2088. 

  2088. <a name="l02024"></a>02024 <span class="comment">!  Append the newly-created intervals to the list.</span>
    2089. 

  2089. <a name="l02025"></a>02025 <span class="comment">!</span>
    2090. 

  2090. <a name="l02026"></a>02026     <span class="keyword">if</span> ( error2 &lt;= error1 ) <span class="keyword">then</span>
    2091. 

  2091. <a name="l02027"></a>02027       alist(last) = a2
    2092. 

  2092. <a name="l02028"></a>02028       blist(maxerr) = b1
    2093. 

  2093. <a name="l02029"></a>02029       blist(last) = b2
    2094. 

  2094. <a name="l02030"></a>02030       elist(maxerr) = error1
    2095. 

  2095. <a name="l02031"></a>02031       elist(last) = error2
    2096. 

  2096. <a name="l02032"></a>02032     <span class="keyword">else</span>
    2097. 

  2097. <a name="l02033"></a>02033       alist(maxerr) = a2
    2098. 

  2098. <a name="l02034"></a>02034       alist(last) = a1
    2099. 

  2099. <a name="l02035"></a>02035       blist(last) = b1
    2100. 

  2100. <a name="l02036"></a>02036       rlist(maxerr) = area2
    2101. 

  2101. <a name="l02037"></a>02037       rlist(last) = area1
    2102. 

  2102. <a name="l02038"></a>02038       elist(maxerr) = error2
    2103. 

  2103. <a name="l02039"></a>02039       elist(last) = error1
    2104. 

  2104. <a name="l02040"></a>02040     <span class="keyword">end if</span>
    2105. 

  2105. <a name="l02041"></a>02041 <span class="comment">!</span>
    2106. 

  2106. <a name="l02042"></a>02042 <span class="comment">!  Call QSORT to maintain the descending ordering</span>
    2107. 

  2107. <a name="l02043"></a>02043 <span class="comment">!  in the list of error estimates and select the subinterval</span>
    2108. 

  2108. <a name="l02044"></a>02044 <span class="comment">!  with nrmax-th largest error estimate (to be bisected next).</span>
    2109. 

  2109. <a name="l02045"></a>02045 <span class="comment">!</span>
    2110. 

  2110. <a name="l02046"></a>02046     call <a class="code" href="quadpack_8f90.html#a55e08a684c5a6315fb37dd0fdc66d8e6">qsort </a>( limit, last, maxerr, errmax, elist, iord, nrmax )
    2111. 

  2111. <a name="l02047"></a>02047 
    2112. 

  2112. <a name="l02048"></a>02048     <span class="keyword">if</span> ( errsum &lt;= errbnd ) go to 115
    2113. 

  2113. <a name="l02049"></a>02049 
    2114. 

  2114. <a name="l02050"></a>02050     <span class="keyword">if</span> ( ier /= 0 ) <span class="keyword">then</span>
    2115. 

  2115. <a name="l02051"></a>02051       exit
    2116. 

  2116. <a name="l02052"></a>02052     <span class="keyword">end if</span>
    2117. 

  2117. <a name="l02053"></a>02053 
    2118. 

  2118. <a name="l02054"></a>02054     <span class="keyword">if</span> ( last == 2 ) go to 80
    2119. 

  2119. <a name="l02055"></a>02055     <span class="keyword">if</span> ( noext ) go to 90
    2120. 

  2120. <a name="l02056"></a>02056 
    2121. 

  2121. <a name="l02057"></a>02057     erlarg = erlarg-erlast
    2122. 

  2122. <a name="l02058"></a>02058 
    2123. 

  2123. <a name="l02059"></a>02059     <span class="keyword">if</span> ( abs(b1-a1) &gt; small ) <span class="keyword">then</span>
    2124. 

  2124. <a name="l02060"></a>02060       erlarg = erlarg+erro12
    2125. 

  2125. <a name="l02061"></a>02061     <span class="keyword">end if</span>
    2126. 

  2126. <a name="l02062"></a>02062 <span class="comment">!</span>
    2127. 

  2127. <a name="l02063"></a>02063 <span class="comment">!  Test whether the interval to be bisected next is the</span>
    2128. 

  2128. <a name="l02064"></a>02064 <span class="comment">!  smallest interval.</span>
    2129. 

  2129. <a name="l02065"></a>02065 <span class="comment">!</span>
    2130. 

  2130. <a name="l02066"></a>02066     <span class="keyword">if</span> ( .not. extrap ) <span class="keyword">then</span>
    2131. 

  2131. <a name="l02067"></a>02067       <span class="keyword">if</span> ( abs(blist(maxerr)-alist(maxerr)) &gt; small ) go to 90
    2132. 

  2132. <a name="l02068"></a>02068       extrap = .true.
    2133. 

  2133. <a name="l02069"></a>02069       nrmax = 2
    2134. 

  2134. <a name="l02070"></a>02070     <span class="keyword">end if</span>
    2135. 

  2135. <a name="l02071"></a>02071 
    2136. 

  2136. <a name="l02072"></a>02072 <span class="comment">!40  continue</span>
    2137. 

  2137. <a name="l02073"></a>02073 <span class="comment">!</span>
    2138. 

  2138. <a name="l02074"></a>02074 <span class="comment">!  The smallest interval has the largest error.</span>
    2139. 

  2139. <a name="l02075"></a>02075 <span class="comment">!  Before bisecting decrease the sum of the errors over the</span>
    2140. 

  2140. <a name="l02076"></a>02076 <span class="comment">!  larger intervals (erlarg) and perform extrapolation.</span>
    2141. 

  2141. <a name="l02077"></a>02077 <span class="comment">!</span>
    2142. 

  2142. <a name="l02078"></a>02078     <span class="keyword">if</span> ( ierro /= 3 .and. erlarg &gt; ertest ) <span class="keyword">then</span>
    2143. 

  2143. <a name="l02079"></a>02079 
    2144. 

  2144. <a name="l02080"></a>02080       id = nrmax
    2145. 

  2145. <a name="l02081"></a>02081       jupbnd = last
    2146. 

  2146. <a name="l02082"></a>02082 
    2147. 

  2147. <a name="l02083"></a>02083       <span class="keyword">if</span> ( last &gt; (2+limit/2) ) <span class="keyword">then</span>
    2148. 

  2148. <a name="l02084"></a>02084         jupbnd = limit+3-last
    2149. 

  2149. <a name="l02085"></a>02085       <span class="keyword">end if</span>
    2150. 

  2150. <a name="l02086"></a>02086 
    2151. 

  2151. <a name="l02087"></a>02087       <span class="keyword">do</span> k = id, jupbnd
    2152. 

  2152. <a name="l02088"></a>02088         maxerr = iord(nrmax)
    2153. 

  2153. <a name="l02089"></a>02089         errmax = elist(maxerr)
    2154. 

  2154. <a name="l02090"></a>02090         <span class="keyword">if</span> ( abs(blist(maxerr)-alist(maxerr)) &gt; small ) <span class="keyword">then</span>
    2155. 

  2155. <a name="l02091"></a>02091           go to 90
    2156. 

  2156. <a name="l02092"></a>02092         <span class="keyword">end if</span>
    2157. 

  2157. <a name="l02093"></a>02093         nrmax = nrmax+1
    2158. 

  2158. <a name="l02094"></a>02094       <span class="keyword">end do</span>
    2159. 

  2159. <a name="l02095"></a>02095 
    2160. 

  2160. <a name="l02096"></a>02096     <span class="keyword">end if</span>
    2161. 

  2161. <a name="l02097"></a>02097 <span class="comment">!</span>
    2162. 

  2162. <a name="l02098"></a>02098 <span class="comment">!  Perform extrapolation.</span>
    2163. 

  2163. <a name="l02099"></a>02099 <span class="comment">!</span>
    2164. 

  2164. <a name="l02100"></a>02100 <span class="comment">!60  continue</span>
    2165. 

  2165. <a name="l02101"></a>02101 
    2166. 

  2166. <a name="l02102"></a>02102     numrl2 = numrl2+1
    2167. 

  2167. <a name="l02103"></a>02103     rlist2(numrl2) = area
    2168. 

  2168. <a name="l02104"></a>02104     call <a class="code" href="quadpack_8f90.html#a5a75101d080f224c63adde98a0e64386">qextr </a>( numrl2, rlist2, reseps, abseps, res3la, nres )
    2169. 

  2169. <a name="l02105"></a>02105     ktmin = ktmin+1
    2170. 

  2170. <a name="l02106"></a>02106 
    2171. 

  2171. <a name="l02107"></a>02107     <span class="keyword">if</span> ( ktmin &gt; 5 .and. abserr &lt; 1.0e-03 * errsum ) <span class="keyword">then</span>
    2172. 

  2172. <a name="l02108"></a>02108       ier = 5
    2173. 

  2173. <a name="l02109"></a>02109     <span class="keyword">end if</span>
    2174. 

  2174. <a name="l02110"></a>02110 
    2175. 

  2175. <a name="l02111"></a>02111     <span class="keyword">if</span> ( abseps &lt; abserr ) <span class="keyword">then</span>
    2176. 

  2176. <a name="l02112"></a>02112 
    2177. 

  2177. <a name="l02113"></a>02113       ktmin = 0
    2178. 

  2178. <a name="l02114"></a>02114       abserr = abseps
    2179. 

  2179. <a name="l02115"></a>02115       result = reseps
    2180. 

  2180. <a name="l02116"></a>02116       correc = erlarg
    2181. 

  2181. <a name="l02117"></a>02117       ertest = max ( epsabs,epsrel*abs(reseps))
    2182. 

  2182. <a name="l02118"></a>02118 
    2183. 

  2183. <a name="l02119"></a>02119       <span class="keyword">if</span> ( abserr &lt;= ertest ) <span class="keyword">then</span>
    2184. 

  2184. <a name="l02120"></a>02120         exit
    2185. 

  2185. <a name="l02121"></a>02121       <span class="keyword">end if</span>
    2186. 

  2186. <a name="l02122"></a>02122 
    2187. 

  2187. <a name="l02123"></a>02123     <span class="keyword">end if</span>
    2188. 

  2188. <a name="l02124"></a>02124 <span class="comment">!</span>
    2189. 

  2189. <a name="l02125"></a>02125 <span class="comment">!  Prepare bisection of the smallest interval.</span>
    2190. 

  2190. <a name="l02126"></a>02126 <span class="comment">!</span>
    2191. 

  2191. <a name="l02127"></a>02127     <span class="keyword">if</span> ( numrl2 == 1 ) <span class="keyword">then</span>
    2192. 

  2192. <a name="l02128"></a>02128       noext = .true.
    2193. 

  2193. <a name="l02129"></a>02129     <span class="keyword">end if</span>
    2194. 

  2194. <a name="l02130"></a>02130 
    2195. 

  2195. <a name="l02131"></a>02131     <span class="keyword">if</span> ( ier == 5 ) <span class="keyword">then</span>
    2196. 

  2196. <a name="l02132"></a>02132       exit
    2197. 

  2197. <a name="l02133"></a>02133     <span class="keyword">end if</span>
    2198. 

  2198. <a name="l02134"></a>02134 
    2199. 

  2199. <a name="l02135"></a>02135     maxerr = iord(1)
    2200. 

  2200. <a name="l02136"></a>02136     errmax = elist(maxerr)
    2201. 

  2201. <a name="l02137"></a>02137     nrmax = 1
    2202. 

  2202. <a name="l02138"></a>02138     extrap = .false.
    2203. 

  2203. <a name="l02139"></a>02139     small = small * 5.0e-01
    2204. 

  2204. <a name="l02140"></a>02140     erlarg = errsum
    2205. 

  2205. <a name="l02141"></a>02141     go to 90
    2206. 

  2206. <a name="l02142"></a>02142 
    2207. 

  2207. <a name="l02143"></a>02143 80  continue
    2208. 

  2208. <a name="l02144"></a>02144 
    2209. 

  2209. <a name="l02145"></a>02145     small = abs ( b - a ) * 3.75e-01
    2210. 

  2210. <a name="l02146"></a>02146     erlarg = errsum
    2211. 

  2211. <a name="l02147"></a>02147     ertest = errbnd
    2212. 

  2212. <a name="l02148"></a>02148     rlist2(2) = area
    2213. 

  2213. <a name="l02149"></a>02149 
    2214. 

  2214. <a name="l02150"></a>02150 90  continue
    2215. 

  2215. <a name="l02151"></a>02151 
    2216. 

  2216. <a name="l02152"></a>02152   <span class="keyword">end do</span>
    2217. 

  2217. <a name="l02153"></a>02153 <span class="comment">!</span>
    2218. 

  2218. <a name="l02154"></a>02154 <span class="comment">!  Set final result and error estimate.</span>
    2219. 

  2219. <a name="l02155"></a>02155 <span class="comment">!</span>
    2220. 

  2220. <a name="l02156"></a>02156   <span class="keyword">if</span> ( abserr == huge ( abserr ) ) <span class="keyword">then</span>
    2221. 

  2221. <a name="l02157"></a>02157     go to 115
    2222. 

  2222. <a name="l02158"></a>02158   <span class="keyword">end if</span>
    2223. 

  2223. <a name="l02159"></a>02159 
    2224. 

  2224. <a name="l02160"></a>02160   <span class="keyword">if</span> ( ier + ierro == 0 ) <span class="keyword">then</span>
    2225. 

  2225. <a name="l02161"></a>02161     go to 110
    2226. 

  2226. <a name="l02162"></a>02162   <span class="keyword">end if</span>
    2227. 

  2227. <a name="l02163"></a>02163 
    2228. 

  2228. <a name="l02164"></a>02164   <span class="keyword">if</span> ( ierro == 3 ) <span class="keyword">then</span>
    2229. 

  2229. <a name="l02165"></a>02165     abserr = abserr + correc
    2230. 

  2230. <a name="l02166"></a>02166   <span class="keyword">end if</span>
    2231. 

  2231. <a name="l02167"></a>02167 
    2232. 

  2232. <a name="l02168"></a>02168   <span class="keyword">if</span> ( ier == 0 ) <span class="keyword">then</span>
    2233. 

  2233. <a name="l02169"></a>02169     ier = 3
    2234. 

  2234. <a name="l02170"></a>02170   <span class="keyword">end if</span>
    2235. 

  2235. <a name="l02171"></a>02171 
    2236. 

  2236. <a name="l02172"></a>02172   <span class="keyword">if</span> ( result /= 0.0e+00.and.area /= 0.0e+00 ) <span class="keyword">then</span>
    2237. 

  2237. <a name="l02173"></a>02173     go to 105
    2238. 

  2238. <a name="l02174"></a>02174   <span class="keyword">end if</span>
    2239. 

  2239. <a name="l02175"></a>02175 
    2240. 

  2240. <a name="l02176"></a>02176   <span class="keyword">if</span> ( abserr &gt; errsum ) go to 115
    2241. 

  2241. <a name="l02177"></a>02177   <span class="keyword">if</span> ( area == 0.0e+00 ) go to 130
    2242. 

  2242. <a name="l02178"></a>02178   go to 110
    2243. 

  2243. <a name="l02179"></a>02179 
    2244. 

  2244. <a name="l02180"></a>02180 105 continue
    2245. 

  2245. <a name="l02181"></a>02181 
    2246. 

  2246. <a name="l02182"></a>02182   <span class="keyword">if</span> ( abserr/abs(result) &gt; errsum/abs(area) ) go to 115
    2247. 

  2247. <a name="l02183"></a>02183 <span class="comment">!</span>
    2248. 

  2248. <a name="l02184"></a>02184 <span class="comment">!  Test on divergence.</span>
    2249. 

  2249. <a name="l02185"></a>02185 <span class="comment">!</span>
    2250. 

  2250. <a name="l02186"></a>02186 110 continue
    2251. 

  2251. <a name="l02187"></a>02187 
    2252. 

  2252. <a name="l02188"></a>02188   <span class="keyword">if</span> ( ksgn == (-1).and.max ( abs(result),abs(area)) &lt;=  &amp;
    2253. 

  2253. <a name="l02189"></a>02189    defabs*1.0e-02 ) go to 130
    2254. 

  2254. <a name="l02190"></a>02190 
    2255. 

  2255. <a name="l02191"></a>02191   <span class="keyword">if</span> ( 1.0e-02 &gt; (result/area) .or. (result/area) &gt; 1.0e+02 &amp;
    2256. 

  2256. <a name="l02192"></a>02192    .or. errsum &gt; abs(area) ) <span class="keyword">then</span>
    2257. 

  2257. <a name="l02193"></a>02193     ier = 6
    2258. 

  2258. <a name="l02194"></a>02194   <span class="keyword">end if</span>
    2259. 

  2259. <a name="l02195"></a>02195 
    2260. 

  2260. <a name="l02196"></a>02196   go to 130
    2261. 

  2261. <a name="l02197"></a>02197 <span class="comment">!</span>
    2262. 

  2262. <a name="l02198"></a>02198 <span class="comment">!  Compute global integral sum.</span>
    2263. 

  2263. <a name="l02199"></a>02199 <span class="comment">!</span>
    2264. 

  2264. <a name="l02200"></a>02200 115 continue
    2265. 

  2265. <a name="l02201"></a>02201 
    2266. 

  2266. <a name="l02202"></a>02202   result = sum ( rlist(1:last) )
    2267. 

  2267. <a name="l02203"></a>02203 
    2268. 

  2268. <a name="l02204"></a>02204   abserr = errsum
    2269. 

  2269. <a name="l02205"></a>02205 
    2270. 

  2270. <a name="l02206"></a>02206 130 continue
    2271. 

  2271. <a name="l02207"></a>02207  
    2272. 

  2272. <a name="l02208"></a>02208   <span class="keyword">if</span> ( 2 &lt; ier ) <span class="keyword">then</span>
    2273. 

  2273. <a name="l02209"></a>02209     ier = ier - 1
    2274. 

  2274. <a name="l02210"></a>02210   <span class="keyword">end if</span>
    2275. 

  2275. <a name="l02211"></a>02211 
    2276. 

  2276. <a name="l02212"></a>02212 140 continue
    2277. 

  2277. <a name="l02213"></a>02213 
    2278. 

  2278. <a name="l02214"></a>02214   neval = 42*last-21
    2279. 

  2279. <a name="l02215"></a>02215 
    2280. 

  2280. <a name="l02216"></a>02216   return
    2281. 

  2281. <a name="l02217"></a>02217 <span class="keyword">end</span>
    2282. 

  2282. <a name="l02218"></a><a class="code" href="quadpack_8f90.html#a4cea9ad83248026209e702bb01abb7da">02218</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a4cea9ad83248026209e702bb01abb7da">qawc</a> ( f, a, b, c, epsabs, epsrel, result, abserr, neval, ier )
    2283. 

  2283. <a name="l02219"></a>02219 
    2284. 

  2284. <a name="l02220"></a>02220 <span class="comment">!*****************************************************************************80</span>
    2285. 

  2285. <a name="l02221"></a>02221 <span class="comment">!</span>
    2286. 

  2286. <a name="l02222"></a>02222 <span class="comment">!! QAWC computes a Cauchy principal value.</span>
    2287. 

  2287. <a name="l02223"></a>02223 <span class="comment">!</span>
    2288. 

  2288. <a name="l02224"></a>02224 <span class="comment">!  Discussion:</span>
    2289. 

  2289. <a name="l02225"></a>02225 <span class="comment">!</span>
    2290. 

  2290. <a name="l02226"></a>02226 <span class="comment">!    The routine calculates an approximation RESULT to a Cauchy principal</span>
    2291. 

  2291. <a name="l02227"></a>02227 <span class="comment">!    value </span>
    2292. 

  2292. <a name="l02228"></a>02228 <span class="comment">!      I = integral of F*W over (A,B),</span>
    2293. 

  2293. <a name="l02229"></a>02229 <span class="comment">!    with</span>
    2294. 

  2294. <a name="l02230"></a>02230 <span class="comment">!      W(X) = 1 / (X-C),</span>
    2295. 

  2295. <a name="l02231"></a>02231 <span class="comment">!    with C distinct from A and B, hopefully satisfying</span>
    2296. 

  2296. <a name="l02232"></a>02232 <span class="comment">!      || I - RESULT || &lt;= max ( EPSABS, EPSREL * ||I|| ).</span>
    2297. 

  2297. <a name="l02233"></a>02233 <span class="comment">!</span>
    2298. 

  2298. <a name="l02234"></a>02234 <span class="comment">!  Author:</span>
    2299. 

  2299. <a name="l02235"></a>02235 <span class="comment">!</span>
    2300. 

  2300. <a name="l02236"></a>02236 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    2301. 

  2301. <a name="l02237"></a>02237 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    2302. 

  2302. <a name="l02238"></a>02238 <span class="comment">!</span>
    2303. 

  2303. <a name="l02239"></a>02239 <span class="comment">!  Reference:</span>
    2304. 

  2304. <a name="l02240"></a>02240 <span class="comment">!</span>
    2305. 

  2305. <a name="l02241"></a>02241 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    2306. 

  2306. <a name="l02242"></a>02242 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    2307. 

  2307. <a name="l02243"></a>02243 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    2308. 

  2308. <a name="l02244"></a>02244 <span class="comment">!    Springer Verlag, 1983</span>
    2309. 

  2309. <a name="l02245"></a>02245 <span class="comment">!</span>
    2310. 

  2310. <a name="l02246"></a>02246 <span class="comment">!  Parameters:</span>
    2311. 

  2311. <a name="l02247"></a>02247 <span class="comment">!</span>
    2312. 

  2312. <a name="l02248"></a>02248 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    2313. 

  2313. <a name="l02249"></a>02249 <span class="comment">!      function f ( x )</span>
    2314. 

  2314. <a name="l02250"></a>02250 <span class="comment">!      real f</span>
    2315. 

  2315. <a name="l02251"></a>02251 <span class="comment">!      real x</span>
    2316. 

  2316. <a name="l02252"></a>02252 <span class="comment">!    which evaluates the integrand function.</span>
    2317. 

  2317. <a name="l02253"></a>02253 <span class="comment">!</span>
    2318. 

  2318. <a name="l02254"></a>02254 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    2319. 

  2319. <a name="l02255"></a>02255 <span class="comment">!</span>
    2320. 

  2320. <a name="l02256"></a>02256 <span class="comment">!    Input, real C, a parameter in the weight function, which must</span>
    2321. 

  2321. <a name="l02257"></a>02257 <span class="comment">!    not be equal to A or B.</span>
    2322. 

  2322. <a name="l02258"></a>02258 <span class="comment">!</span>
    2323. 

  2323. <a name="l02259"></a>02259 <span class="comment">!    Input, real EPSABS, EPSREL, the absolute and relative accuracy requested.</span>
    2324. 

  2324. <a name="l02260"></a>02260 <span class="comment">!</span>
    2325. 

  2325. <a name="l02261"></a>02261 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    2326. 

  2326. <a name="l02262"></a>02262 <span class="comment">!</span>
    2327. 

  2327. <a name="l02263"></a>02263 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    2328. 

  2328. <a name="l02264"></a>02264 <span class="comment">!</span>
    2329. 

  2329. <a name="l02265"></a>02265 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    2330. 

  2330. <a name="l02266"></a>02266 <span class="comment">!</span>
    2331. 

  2331. <a name="l02267"></a>02267 <span class="comment">!            ier    - integer</span>
    2332. 

  2332. <a name="l02268"></a>02268 <span class="comment">!                     ier = 0 normal and reliable termination of the</span>
    2333. 

  2333. <a name="l02269"></a>02269 <span class="comment">!                             routine. it is assumed that the requested</span>
    2334. 

  2334. <a name="l02270"></a>02270 <span class="comment">!                             accuracy has been achieved.</span>
    2335. 

  2335. <a name="l02271"></a>02271 <span class="comment">!                     ier &gt; 0 abnormal termination of the routine</span>
    2336. 

  2336. <a name="l02272"></a>02272 <span class="comment">!                             the estimates for integral and error are</span>
    2337. 

  2337. <a name="l02273"></a>02273 <span class="comment">!                             less reliable. it is assumed that the</span>
    2338. 

  2338. <a name="l02274"></a>02274 <span class="comment">!                             requested accuracy has not been achieved.</span>
    2339. 

  2339. <a name="l02275"></a>02275 <span class="comment">!                     ier = 1 maximum number of subdivisions allowed</span>
    2340. 

  2340. <a name="l02276"></a>02276 <span class="comment">!                             has been achieved. one can allow more sub-</span>
    2341. 

  2341. <a name="l02277"></a>02277 <span class="comment">!                             divisions by increasing the data value of</span>
    2342. 

  2342. <a name="l02278"></a>02278 <span class="comment">!                             limit in qawc (and taking the according</span>
    2343. 

  2343. <a name="l02279"></a>02279 <span class="comment">!                             dimension adjustments into account).</span>
    2344. 

  2344. <a name="l02280"></a>02280 <span class="comment">!                             however, if this yields no improvement it</span>
    2345. 

  2345. <a name="l02281"></a>02281 <span class="comment">!                             is advised to analyze the integrand in</span>
    2346. 

  2346. <a name="l02282"></a>02282 <span class="comment">!                             order to determine the integration</span>
    2347. 

  2347. <a name="l02283"></a>02283 <span class="comment">!                             difficulties. if the position of a local</span>
    2348. 

  2348. <a name="l02284"></a>02284 <span class="comment">!                             difficulty can be determined (e.g.</span>
    2349. 

  2349. <a name="l02285"></a>02285 <span class="comment">!                             singularity, discontinuity within the</span>
    2350. 

  2350. <a name="l02286"></a>02286 <span class="comment">!                             interval one will probably gain from</span>
    2351. 

  2351. <a name="l02287"></a>02287 <span class="comment">!                             splitting up the interval at this point</span>
    2352. 

  2352. <a name="l02288"></a>02288 <span class="comment">!                             and calling appropriate integrators on the</span>
    2353. 

  2353. <a name="l02289"></a>02289 <span class="comment">!                             subranges.</span>
    2354. 

  2354. <a name="l02290"></a>02290 <span class="comment">!                         = 2 the occurrence of roundoff error is detec-</span>
    2355. 

  2355. <a name="l02291"></a>02291 <span class="comment">!                             ted, which prevents the requested</span>
    2356. 

  2356. <a name="l02292"></a>02292 <span class="comment">!                             tolerance from being achieved.</span>
    2357. 

  2357. <a name="l02293"></a>02293 <span class="comment">!                         = 3 extremely bad integrand behavior occurs</span>
    2358. 

  2358. <a name="l02294"></a>02294 <span class="comment">!                             at some points of the integration</span>
    2359. 

  2359. <a name="l02295"></a>02295 <span class="comment">!                             interval.</span>
    2360. 

  2360. <a name="l02296"></a>02296 <span class="comment">!                         = 6 the input is invalid, because</span>
    2361. 

  2361. <a name="l02297"></a>02297 <span class="comment">!                             c = a or c = b or</span>
    2362. 

  2362. <a name="l02298"></a>02298 <span class="comment">!                             epsabs &lt; 0 and epsrel &lt; 0,</span>
    2363. 

  2363. <a name="l02299"></a>02299 <span class="comment">!                             result, abserr, neval are set to zero.</span>
    2364. 

  2364. <a name="l02300"></a>02300 <span class="comment">!</span>
    2365. 

  2365. <a name="l02301"></a>02301 <span class="comment">!  Local parameters:</span>
    2366. 

  2366. <a name="l02302"></a>02302 <span class="comment">!</span>
    2367. 

  2367. <a name="l02303"></a>02303 <span class="comment">!    LIMIT is the maximum number of subintervals allowed in the</span>
    2368. 

  2368. <a name="l02304"></a>02304 <span class="comment">!    subdivision process of qawce. take care that limit &gt;= 1.</span>
    2369. 

  2369. <a name="l02305"></a>02305 <span class="comment">!</span>
    2370. 

  2370. <a name="l02306"></a>02306   <span class="keyword">implicit none</span>
    2371. 

  2371. <a name="l02307"></a>02307 
    2372. 

  2372. <a name="l02308"></a>02308   <span class="keywordtype">integer</span>, <span class="keywordtype">parameter</span> :: limit = 500
    2373. 

  2373. <a name="l02309"></a>02309 
    2374. 

  2374. <a name="l02310"></a>02310   <span class="keywordtype">real</span> a
    2375. 

  2375. <a name="l02311"></a>02311   <span class="keywordtype">real</span> abserr
    2376. 

  2376. <a name="l02312"></a>02312   <span class="keywordtype">real</span> alist(limit)
    2377. 

  2377. <a name="l02313"></a>02313   <span class="keywordtype">real</span> b
    2378. 

  2378. <a name="l02314"></a>02314   <span class="keywordtype">real</span> blist(limit)
    2379. 

  2379. <a name="l02315"></a>02315   <span class="keywordtype">real</span> elist(limit)
    2380. 

  2380. <a name="l02316"></a>02316   <span class="keywordtype">real</span> c
    2381. 

  2381. <a name="l02317"></a>02317   <span class="keywordtype">real</span> epsabs
    2382. 

  2382. <a name="l02318"></a>02318   <span class="keywordtype">real</span> epsrel
    2383. 

  2383. <a name="l02319"></a>02319   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    2384. 

  2384. <a name="l02320"></a>02320   <span class="keywordtype">integer</span> ier
    2385. 

  2385. <a name="l02321"></a>02321   <span class="keywordtype">integer</span> iord(limit)
    2386. 

  2386. <a name="l02322"></a>02322   <span class="keywordtype">integer</span> last
    2387. 

  2387. <a name="l02323"></a>02323   <span class="keywordtype">integer</span> neval
    2388. 

  2388. <a name="l02324"></a>02324   <span class="keywordtype">real</span> result
    2389. 

  2389. <a name="l02325"></a>02325   <span class="keywordtype">real</span> rlist(limit)
    2390. 

  2390. <a name="l02326"></a>02326 
    2391. 

  2391. <a name="l02327"></a>02327   call <a class="code" href="quadpack_8f90.html#a51d7f754a9214f7490c035740fc0aef7">qawce </a>( f, a, b, c, epsabs, epsrel, limit, result, abserr, neval, ier, &amp;
    2392. 

  2392. <a name="l02328"></a>02328     alist, blist, rlist, elist, iord, last )
    2393. 

  2393. <a name="l02329"></a>02329 
    2394. 

  2394. <a name="l02330"></a>02330   return
    2395. 

  2395. <a name="l02331"></a>02331 <span class="keyword">end</span>
    2396. 

  2396. <a name="l02332"></a><a class="code" href="quadpack_8f90.html#a51d7f754a9214f7490c035740fc0aef7">02332</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a51d7f754a9214f7490c035740fc0aef7">qawce</a> ( f, a, b, c, epsabs, epsrel, limit, result, abserr, neval, &amp;
    2397. 

  2397. <a name="l02333"></a>02333   ier, alist, blist, rlist, elist, iord, last )
    2398. 

  2398. <a name="l02334"></a>02334 
    2399. 

  2399. <a name="l02335"></a>02335 <span class="comment">!*****************************************************************************80</span>
    2400. 

  2400. <a name="l02336"></a>02336 <span class="comment">!</span>
    2401. 

  2401. <a name="l02337"></a>02337 <span class="comment">!! QAWCE computes a Cauchy principal value.</span>
    2402. 

  2402. <a name="l02338"></a>02338 <span class="comment">!</span>
    2403. 

  2403. <a name="l02339"></a>02339 <span class="comment">!  Discussion:</span>
    2404. 

  2404. <a name="l02340"></a>02340 <span class="comment">!</span>
    2405. 

  2405. <a name="l02341"></a>02341 <span class="comment">!    The routine calculates an approximation RESULT to a Cauchy principal</span>
    2406. 

  2406. <a name="l02342"></a>02342 <span class="comment">!    value   </span>
    2407. 

  2407. <a name="l02343"></a>02343 <span class="comment">!      I = integral of F*W over (A,B),</span>
    2408. 

  2408. <a name="l02344"></a>02344 <span class="comment">!    with</span>
    2409. 

  2409. <a name="l02345"></a>02345 <span class="comment">!      W(X) = 1 / ( X - C ),</span>
    2410. 

  2410. <a name="l02346"></a>02346 <span class="comment">!    with C distinct from A and B, hopefully satisfying</span>
    2411. 

  2411. <a name="l02347"></a>02347 <span class="comment">!      | I - RESULT | &lt;= max ( EPSABS, EPSREL * |I| ).</span>
    2412. 

  2412. <a name="l02348"></a>02348 <span class="comment">!</span>
    2413. 

  2413. <a name="l02349"></a>02349 <span class="comment">!  Author:</span>
    2414. 

  2414. <a name="l02350"></a>02350 <span class="comment">!</span>
    2415. 

  2415. <a name="l02351"></a>02351 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    2416. 

  2416. <a name="l02352"></a>02352 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    2417. 

  2417. <a name="l02353"></a>02353 <span class="comment">!</span>
    2418. 

  2418. <a name="l02354"></a>02354 <span class="comment">!  Reference:</span>
    2419. 

  2419. <a name="l02355"></a>02355 <span class="comment">!</span>
    2420. 

  2420. <a name="l02356"></a>02356 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    2421. 

  2421. <a name="l02357"></a>02357 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    2422. 

  2422. <a name="l02358"></a>02358 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    2423. 

  2423. <a name="l02359"></a>02359 <span class="comment">!    Springer Verlag, 1983</span>
    2424. 

  2424. <a name="l02360"></a>02360 <span class="comment">!</span>
    2425. 

  2425. <a name="l02361"></a>02361 <span class="comment">!  Parameters:</span>
    2426. 

  2426. <a name="l02362"></a>02362 <span class="comment">!</span>
    2427. 

  2427. <a name="l02363"></a>02363 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    2428. 

  2428. <a name="l02364"></a>02364 <span class="comment">!      function f ( x )</span>
    2429. 

  2429. <a name="l02365"></a>02365 <span class="comment">!      real f</span>
    2430. 

  2430. <a name="l02366"></a>02366 <span class="comment">!      real x</span>
    2431. 

  2431. <a name="l02367"></a>02367 <span class="comment">!    which evaluates the integrand function.</span>
    2432. 

  2432. <a name="l02368"></a>02368 <span class="comment">!</span>
    2433. 

  2433. <a name="l02369"></a>02369 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    2434. 

  2434. <a name="l02370"></a>02370 <span class="comment">!</span>
    2435. 

  2435. <a name="l02371"></a>02371 <span class="comment">!    Input, real C, a parameter in the weight function, which cannot be</span>
    2436. 

  2436. <a name="l02372"></a>02372 <span class="comment">!    equal to A or B.</span>
    2437. 

  2437. <a name="l02373"></a>02373 <span class="comment">!</span>
    2438. 

  2438. <a name="l02374"></a>02374 <span class="comment">!    Input, real EPSABS, EPSREL, the absolute and relative accuracy requested.</span>
    2439. 

  2439. <a name="l02375"></a>02375 <span class="comment">!</span>
    2440. 

  2440. <a name="l02376"></a>02376 <span class="comment">!    Input, integer LIMIT, the upper bound on the number of subintervals that</span>
    2441. 

  2441. <a name="l02377"></a>02377 <span class="comment">!    will be used in the partition of [A,B].  LIMIT is typically 500.</span>
    2442. 

  2442. <a name="l02378"></a>02378 <span class="comment">!</span>
    2443. 

  2443. <a name="l02379"></a>02379 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    2444. 

  2444. <a name="l02380"></a>02380 <span class="comment">!</span>
    2445. 

  2445. <a name="l02381"></a>02381 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    2446. 

  2446. <a name="l02382"></a>02382 <span class="comment">!</span>
    2447. 

  2447. <a name="l02383"></a>02383 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    2448. 

  2448. <a name="l02384"></a>02384 <span class="comment">!</span>
    2449. 

  2449. <a name="l02385"></a>02385 <span class="comment">!            ier    - integer</span>
    2450. 

  2450. <a name="l02386"></a>02386 <span class="comment">!                     ier = 0 normal and reliable termination of the</span>
    2451. 

  2451. <a name="l02387"></a>02387 <span class="comment">!                             routine. it is assumed that the requested</span>
    2452. 

  2452. <a name="l02388"></a>02388 <span class="comment">!                             accuracy has been achieved.</span>
    2453. 

  2453. <a name="l02389"></a>02389 <span class="comment">!                     ier &gt; 0 abnormal termination of the routine</span>
    2454. 

  2454. <a name="l02390"></a>02390 <span class="comment">!                             the estimates for integral and error are</span>
    2455. 

  2455. <a name="l02391"></a>02391 <span class="comment">!                             less reliable. it is assumed that the</span>
    2456. 

  2456. <a name="l02392"></a>02392 <span class="comment">!                             requested accuracy has not been achieved.</span>
    2457. 

  2457. <a name="l02393"></a>02393 <span class="comment">!                     ier = 1 maximum number of subdivisions allowed</span>
    2458. 

  2458. <a name="l02394"></a>02394 <span class="comment">!                             has been achieved. one can allow more sub-</span>
    2459. 

  2459. <a name="l02395"></a>02395 <span class="comment">!                             divisions by increasing the value of</span>
    2460. 

  2460. <a name="l02396"></a>02396 <span class="comment">!                             limit. however, if this yields no</span>
    2461. 

  2461. <a name="l02397"></a>02397 <span class="comment">!                             improvement it is advised to analyze the</span>
    2462. 

  2462. <a name="l02398"></a>02398 <span class="comment">!                             integrand, in order to determine the</span>
    2463. 

  2463. <a name="l02399"></a>02399 <span class="comment">!                             integration difficulties.  if the position</span>
    2464. 

  2464. <a name="l02400"></a>02400 <span class="comment">!                             of a local difficulty can be determined</span>
    2465. 

  2465. <a name="l02401"></a>02401 <span class="comment">!                             (e.g. singularity, discontinuity within</span>
    2466. 

  2466. <a name="l02402"></a>02402 <span class="comment">!                             the interval) one will probably gain</span>
    2467. 

  2467. <a name="l02403"></a>02403 <span class="comment">!                             from splitting up the interval at this</span>
    2468. 

  2468. <a name="l02404"></a>02404 <span class="comment">!                             point and calling appropriate integrators</span>
    2469. 

  2469. <a name="l02405"></a>02405 <span class="comment">!                             on the subranges.</span>
    2470. 

  2470. <a name="l02406"></a>02406 <span class="comment">!                         = 2 the occurrence of roundoff error is detec-</span>
    2471. 

  2471. <a name="l02407"></a>02407 <span class="comment">!                             ted, which prevents the requested</span>
    2472. 

  2472. <a name="l02408"></a>02408 <span class="comment">!                             tolerance from being achieved.</span>
    2473. 

  2473. <a name="l02409"></a>02409 <span class="comment">!                         = 3 extremely bad integrand behavior occurs</span>
    2474. 

  2474. <a name="l02410"></a>02410 <span class="comment">!                             at some interior points of the integration</span>
    2475. 

  2475. <a name="l02411"></a>02411 <span class="comment">!                             interval.</span>
    2476. 

  2476. <a name="l02412"></a>02412 <span class="comment">!                         = 6 the input is invalid, because</span>
    2477. 

  2477. <a name="l02413"></a>02413 <span class="comment">!                             c = a or c = b or</span>
    2478. 

  2478. <a name="l02414"></a>02414 <span class="comment">!                             epsabs &lt; 0 and epsrel &lt; 0,</span>
    2479. 

  2479. <a name="l02415"></a>02415 <span class="comment">!                             or limit &lt; 1.</span>
    2480. 

  2480. <a name="l02416"></a>02416 <span class="comment">!                             result, abserr, neval, rlist(1), elist(1),</span>
    2481. 

  2481. <a name="l02417"></a>02417 <span class="comment">!                             iord(1) and last are set to zero.</span>
    2482. 

  2482. <a name="l02418"></a>02418 <span class="comment">!                             alist(1) and blist(1) are set to a and b</span>
    2483. 

  2483. <a name="l02419"></a>02419 <span class="comment">!                             respectively.</span>
    2484. 

  2484. <a name="l02420"></a>02420 <span class="comment">!</span>
    2485. 

  2485. <a name="l02421"></a>02421 <span class="comment">!    Workspace, real ALIST(LIMIT), BLIST(LIMIT), contains in entries 1 </span>
    2486. 

  2486. <a name="l02422"></a>02422 <span class="comment">!    through LAST the left and right ends of the partition subintervals.</span>
    2487. 

  2487. <a name="l02423"></a>02423 <span class="comment">!</span>
    2488. 

  2488. <a name="l02424"></a>02424 <span class="comment">!    Workspace, real RLIST(LIMIT), contains in entries 1 through LAST</span>
    2489. 

  2489. <a name="l02425"></a>02425 <span class="comment">!    the integral approximations on the subintervals.</span>
    2490. 

  2490. <a name="l02426"></a>02426 <span class="comment">!</span>
    2491. 

  2491. <a name="l02427"></a>02427 <span class="comment">!    Workspace, real ELIST(LIMIT), contains in entries 1 through LAST</span>
    2492. 

  2492. <a name="l02428"></a>02428 <span class="comment">!    the absolute error estimates on the subintervals.</span>
    2493. 

  2493. <a name="l02429"></a>02429 <span class="comment">!</span>
    2494. 

  2494. <a name="l02430"></a>02430 <span class="comment">!            iord    - integer</span>
    2495. 

  2495. <a name="l02431"></a>02431 <span class="comment">!                      vector of dimension at least limit, the first k</span>
    2496. 

  2496. <a name="l02432"></a>02432 <span class="comment">!                      elements of which are pointers to the error</span>
    2497. 

  2497. <a name="l02433"></a>02433 <span class="comment">!                      estimates over the subintervals, so that</span>
    2498. 

  2498. <a name="l02434"></a>02434 <span class="comment">!                      elist(iord(1)), ...,  elist(iord(k)) with</span>
    2499. 

  2499. <a name="l02435"></a>02435 <span class="comment">!                      k = last if last &lt;= (limit/2+2), and</span>
    2500. 

  2500. <a name="l02436"></a>02436 <span class="comment">!                      k = limit+1-last otherwise, form a decreasing</span>
    2501. 

  2501. <a name="l02437"></a>02437 <span class="comment">!                      sequence.</span>
    2502. 

  2502. <a name="l02438"></a>02438 <span class="comment">!</span>
    2503. 

  2503. <a name="l02439"></a>02439 <span class="comment">!            last    - integer</span>
    2504. 

  2504. <a name="l02440"></a>02440 <span class="comment">!                      number of subintervals actually produced in</span>
    2505. 

  2505. <a name="l02441"></a>02441 <span class="comment">!                      the subdivision process</span>
    2506. 

  2506. <a name="l02442"></a>02442 <span class="comment">!</span>
    2507. 

  2507. <a name="l02443"></a>02443 <span class="comment">!  Local parameters:</span>
    2508. 

  2508. <a name="l02444"></a>02444 <span class="comment">!</span>
    2509. 

  2509. <a name="l02445"></a>02445 <span class="comment">!           alist     - list of left end points of all subintervals</span>
    2510. 

  2510. <a name="l02446"></a>02446 <span class="comment">!                       considered up to now</span>
    2511. 

  2511. <a name="l02447"></a>02447 <span class="comment">!           blist     - list of right end points of all subintervals</span>
    2512. 

  2512. <a name="l02448"></a>02448 <span class="comment">!                       considered up to now</span>
    2513. 

  2513. <a name="l02449"></a>02449 <span class="comment">!           rlist(i)  - approximation to the integral over</span>
    2514. 

  2514. <a name="l02450"></a>02450 <span class="comment">!                       (alist(i),blist(i))</span>
    2515. 

  2515. <a name="l02451"></a>02451 <span class="comment">!           elist(i)  - error estimate applying to rlist(i)</span>
    2516. 

  2516. <a name="l02452"></a>02452 <span class="comment">!           maxerr    - pointer to the interval with largest error</span>
    2517. 

  2517. <a name="l02453"></a>02453 <span class="comment">!                       estimate</span>
    2518. 

  2518. <a name="l02454"></a>02454 <span class="comment">!           errmax    - elist(maxerr)</span>
    2519. 

  2519. <a name="l02455"></a>02455 <span class="comment">!           area      - sum of the integrals over the subintervals</span>
    2520. 

  2520. <a name="l02456"></a>02456 <span class="comment">!           errsum    - sum of the errors over the subintervals</span>
    2521. 

  2521. <a name="l02457"></a>02457 <span class="comment">!           errbnd    - requested accuracy max(epsabs,epsrel*</span>
    2522. 

  2522. <a name="l02458"></a>02458 <span class="comment">!                       abs(result))</span>
    2523. 

  2523. <a name="l02459"></a>02459 <span class="comment">!           *****1    - variable for the left subinterval</span>
    2524. 

  2524. <a name="l02460"></a>02460 <span class="comment">!           *****2    - variable for the right subinterval</span>
    2525. 

  2525. <a name="l02461"></a>02461 <span class="comment">!           last      - index for subdivision</span>
    2526. 

  2526. <a name="l02462"></a>02462 <span class="comment">!</span>
    2527. 

  2527. <a name="l02463"></a>02463   <span class="keyword">implicit none</span>
    2528. 

  2528. <a name="l02464"></a>02464 
    2529. 

  2529. <a name="l02465"></a>02465   <span class="keywordtype">integer</span> limit
    2530. 

  2530. <a name="l02466"></a>02466 
    2531. 

  2531. <a name="l02467"></a>02467   <span class="keywordtype">real</span> a
    2532. 

  2532. <a name="l02468"></a>02468   <span class="keywordtype">real</span> aa
    2533. 

  2533. <a name="l02469"></a>02469   <span class="keywordtype">real</span> abserr
    2534. 

  2534. <a name="l02470"></a>02470   <span class="keywordtype">real</span> alist(limit)
    2535. 

  2535. <a name="l02471"></a>02471   <span class="keywordtype">real</span> area
    2536. 

  2536. <a name="l02472"></a>02472   <span class="keywordtype">real</span> area1
    2537. 

  2537. <a name="l02473"></a>02473   <span class="keywordtype">real</span> area12
    2538. 

  2538. <a name="l02474"></a>02474   <span class="keywordtype">real</span> area2
    2539. 

  2539. <a name="l02475"></a>02475   <span class="keywordtype">real</span> a1
    2540. 

  2540. <a name="l02476"></a>02476   <span class="keywordtype">real</span> a2
    2541. 

  2541. <a name="l02477"></a>02477   <span class="keywordtype">real</span> b
    2542. 

  2542. <a name="l02478"></a>02478   <span class="keywordtype">real</span> bb
    2543. 

  2543. <a name="l02479"></a>02479   <span class="keywordtype">real</span> blist(limit)
    2544. 

  2544. <a name="l02480"></a>02480   <span class="keywordtype">real</span> b1
    2545. 

  2545. <a name="l02481"></a>02481   <span class="keywordtype">real</span> b2
    2546. 

  2546. <a name="l02482"></a>02482   <span class="keywordtype">real</span> c
    2547. 

  2547. <a name="l02483"></a>02483   <span class="keywordtype">real</span> elist(limit)
    2548. 

  2548. <a name="l02484"></a>02484   <span class="keywordtype">real</span> epsabs
    2549. 

  2549. <a name="l02485"></a>02485   <span class="keywordtype">real</span> epsrel
    2550. 

  2550. <a name="l02486"></a>02486   <span class="keywordtype">real</span> errbnd
    2551. 

  2551. <a name="l02487"></a>02487   <span class="keywordtype">real</span> errmax
    2552. 

  2552. <a name="l02488"></a>02488   <span class="keywordtype">real</span> error1
    2553. 

  2553. <a name="l02489"></a>02489   <span class="keywordtype">real</span> error2
    2554. 

  2554. <a name="l02490"></a>02490   <span class="keywordtype">real</span> erro12
    2555. 

  2555. <a name="l02491"></a>02491   <span class="keywordtype">real</span> errsum
    2556. 

  2556. <a name="l02492"></a>02492   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    2557. 

  2557. <a name="l02493"></a>02493   <span class="keywordtype">integer</span> ier
    2558. 

  2558. <a name="l02494"></a>02494   <span class="keywordtype">integer</span> iord(limit)
    2559. 

  2559. <a name="l02495"></a>02495   <span class="keywordtype">integer</span> iroff1
    2560. 

  2560. <a name="l02496"></a>02496   <span class="keywordtype">integer</span> iroff2
    2561. 

  2561. <a name="l02497"></a>02497   <span class="keywordtype">integer</span> krule
    2562. 

  2562. <a name="l02498"></a>02498   <span class="keywordtype">integer</span> last
    2563. 

  2563. <a name="l02499"></a>02499   <span class="keywordtype">integer</span> maxerr
    2564. 

  2564. <a name="l02500"></a>02500   <span class="keywordtype">integer</span> nev
    2565. 

  2565. <a name="l02501"></a>02501   <span class="keywordtype">integer</span> neval
    2566. 

  2566. <a name="l02502"></a>02502   <span class="keywordtype">integer</span> nrmax
    2567. 

  2567. <a name="l02503"></a>02503   <span class="keywordtype">real</span> result
    2568. 

  2568. <a name="l02504"></a>02504   <span class="keywordtype">real</span> rlist(limit)
    2569. 

  2569. <a name="l02505"></a>02505 <span class="comment">!</span>
    2570. 

  2570. <a name="l02506"></a>02506 <span class="comment">!  Test on validity of parameters.</span>
    2571. 

  2571. <a name="l02507"></a>02507 <span class="comment">!</span>
    2572. 

  2572. <a name="l02508"></a>02508   ier = 0
    2573. 

  2573. <a name="l02509"></a>02509   neval = 0
    2574. 

  2574. <a name="l02510"></a>02510   last = 0
    2575. 

  2575. <a name="l02511"></a>02511   alist(1) = a
    2576. 

  2576. <a name="l02512"></a>02512   blist(1) = b
    2577. 

  2577. <a name="l02513"></a>02513   rlist(1) = 0.0e+00
    2578. 

  2578. <a name="l02514"></a>02514   elist(1) = 0.0e+00
    2579. 

  2579. <a name="l02515"></a>02515   iord(1) = 0
    2580. 

  2580. <a name="l02516"></a>02516   result = 0.0e+00
    2581. 

  2581. <a name="l02517"></a>02517   abserr = 0.0e+00
    2582. 

  2582. <a name="l02518"></a>02518 
    2583. 

  2583. <a name="l02519"></a>02519   <span class="keyword">if</span> ( c == a  ) <span class="keyword">then</span>
    2584. 

  2584. <a name="l02520"></a>02520     ier = 6
    2585. 

  2585. <a name="l02521"></a>02521     return
    2586. 

  2586. <a name="l02522"></a>02522   <span class="keyword">else</span> <span class="keyword">if</span> ( c == b ) <span class="keyword">then</span>
    2587. 

  2587. <a name="l02523"></a>02523     ier = 6
    2588. 

  2588. <a name="l02524"></a>02524     return
    2589. 

  2589. <a name="l02525"></a>02525   <span class="keyword">else</span> <span class="keyword">if</span> ( epsabs &lt; 0.0e+00 .and. epsrel &lt; 0.0e+00 ) <span class="keyword">then</span>
    2590. 

  2590. <a name="l02526"></a>02526     ier = 6
    2591. 

  2591. <a name="l02527"></a>02527     return
    2592. 

  2592. <a name="l02528"></a>02528   <span class="keyword">end if</span>
    2593. 

  2593. <a name="l02529"></a>02529 <span class="comment">!</span>
    2594. 

  2594. <a name="l02530"></a>02530 <span class="comment">!  First approximation to the integral.</span>
    2595. 

  2595. <a name="l02531"></a>02531 <span class="comment">!</span>
    2596. 

  2596. <a name="l02532"></a>02532   <span class="keyword">if</span> ( a &lt;= b ) <span class="keyword">then</span>
    2597. 

  2597. <a name="l02533"></a>02533     aa = a
    2598. 

  2598. <a name="l02534"></a>02534     bb = b
    2599. 

  2599. <a name="l02535"></a>02535   <span class="keyword">else</span>
    2600. 

  2600. <a name="l02536"></a>02536     aa = b
    2601. 

  2601. <a name="l02537"></a>02537     bb = a
    2602. 

  2602. <a name="l02538"></a>02538   <span class="keyword">end if</span>
    2603. 

  2603. <a name="l02539"></a>02539 
    2604. 

  2604. <a name="l02540"></a>02540   krule = 1
    2605. 

  2605. <a name="l02541"></a>02541   call <a class="code" href="quadpack_8f90.html#af8148c1623b7cf59159c491cfb1856f4">qc25c </a>( f, aa, bb, c, result, abserr, krule, neval )
    2606. 

  2606. <a name="l02542"></a>02542   last = 1
    2607. 

  2607. <a name="l02543"></a>02543   rlist(1) = result
    2608. 

  2608. <a name="l02544"></a>02544   elist(1) = abserr
    2609. 

  2609. <a name="l02545"></a>02545   iord(1) = 1
    2610. 

  2610. <a name="l02546"></a>02546   alist(1) = a
    2611. 

  2611. <a name="l02547"></a>02547   blist(1) = b
    2612. 

  2612. <a name="l02548"></a>02548 <span class="comment">!</span>
    2613. 

  2613. <a name="l02549"></a>02549 <span class="comment">!  Test on accuracy.</span>
    2614. 

  2614. <a name="l02550"></a>02550 <span class="comment">!</span>
    2615. 

  2615. <a name="l02551"></a>02551   errbnd = max ( epsabs, epsrel * abs(result) )
    2616. 

  2616. <a name="l02552"></a>02552 
    2617. 

  2617. <a name="l02553"></a>02553   <span class="keyword">if</span> ( limit == 1 ) <span class="keyword">then</span>
    2618. 

  2618. <a name="l02554"></a>02554     ier = 1
    2619. 

  2619. <a name="l02555"></a>02555     go to 70
    2620. 

  2620. <a name="l02556"></a>02556   <span class="keyword">end if</span>
    2621. 

  2621. <a name="l02557"></a>02557 
    2622. 

  2622. <a name="l02558"></a>02558   <span class="keyword">if</span> ( abserr &lt; min ( 1.0e-02 * abs(result), errbnd)  ) <span class="keyword">then</span>
    2623. 

  2623. <a name="l02559"></a>02559     go to 70
    2624. 

  2624. <a name="l02560"></a>02560   <span class="keyword">end if</span>
    2625. 

  2625. <a name="l02561"></a>02561 <span class="comment">!</span>
    2626. 

  2626. <a name="l02562"></a>02562 <span class="comment">!  Initialization</span>
    2627. 

  2627. <a name="l02563"></a>02563 <span class="comment">!</span>
    2628. 

  2628. <a name="l02564"></a>02564   alist(1) = aa
    2629. 

  2629. <a name="l02565"></a>02565   blist(1) = bb
    2630. 

  2630. <a name="l02566"></a>02566   rlist(1) = result
    2631. 

  2631. <a name="l02567"></a>02567   errmax = abserr
    2632. 

  2632. <a name="l02568"></a>02568   maxerr = 1
    2633. 

  2633. <a name="l02569"></a>02569   area = result
    2634. 

  2634. <a name="l02570"></a>02570   errsum = abserr
    2635. 

  2635. <a name="l02571"></a>02571   nrmax = 1
    2636. 

  2636. <a name="l02572"></a>02572   iroff1 = 0
    2637. 

  2637. <a name="l02573"></a>02573   iroff2 = 0
    2638. 

  2638. <a name="l02574"></a>02574 
    2639. 

  2639. <a name="l02575"></a>02575   <span class="keyword">do</span> last = 2, limit
    2640. 

  2640. <a name="l02576"></a>02576 <span class="comment">!</span>
    2641. 

  2641. <a name="l02577"></a>02577 <span class="comment">!  Bisect the subinterval with nrmax-th largest error estimate.</span>
    2642. 

  2642. <a name="l02578"></a>02578 <span class="comment">!</span>
    2643. 

  2643. <a name="l02579"></a>02579     a1 = alist(maxerr)
    2644. 

  2644. <a name="l02580"></a>02580     b1 = 5.0e-01*(alist(maxerr)+blist(maxerr))
    2645. 

  2645. <a name="l02581"></a>02581     b2 = blist(maxerr)
    2646. 

  2646. <a name="l02582"></a>02582 
    2647. 

  2647. <a name="l02583"></a>02583     <span class="keyword">if</span> ( c &lt;= b1 .and. a1 &lt; c ) <span class="keyword">then</span>
    2648. 

  2648. <a name="l02584"></a>02584       b1 = 5.0e-01*(c+b2)
    2649. 

  2649. <a name="l02585"></a>02585     <span class="keyword">end if</span>
    2650. 

  2650. <a name="l02586"></a>02586 
    2651. 

  2651. <a name="l02587"></a>02587     <span class="keyword">if</span> ( b1 &lt; c .and. c &lt; b2 ) <span class="keyword">then</span>
    2652. 

  2652. <a name="l02588"></a>02588       b1 = 5.0e-01 * ( a1 + c )
    2653. 

  2653. <a name="l02589"></a>02589     <span class="keyword">end if</span>
    2654. 

  2654. <a name="l02590"></a>02590 
    2655. 

  2655. <a name="l02591"></a>02591     a2 = b1
    2656. 

  2656. <a name="l02592"></a>02592     krule = 2
    2657. 

  2657. <a name="l02593"></a>02593 
    2658. 

  2658. <a name="l02594"></a>02594     call <a class="code" href="quadpack_8f90.html#af8148c1623b7cf59159c491cfb1856f4">qc25c </a>( f, a1, b1, c, area1, error1, krule, nev )
    2659. 

  2659. <a name="l02595"></a>02595     neval = neval+nev
    2660. 

  2660. <a name="l02596"></a>02596 
    2661. 

  2661. <a name="l02597"></a>02597     call <a class="code" href="quadpack_8f90.html#af8148c1623b7cf59159c491cfb1856f4">qc25c </a>( f, a2, b2, c, area2, error2, krule, nev )
    2662. 

  2662. <a name="l02598"></a>02598     neval = neval+nev
    2663. 

  2663. <a name="l02599"></a>02599 <span class="comment">!</span>
    2664. 

  2664. <a name="l02600"></a>02600 <span class="comment">!  Improve previous approximations to integral and error</span>
    2665. 

  2665. <a name="l02601"></a>02601 <span class="comment">!  and test for accuracy.</span>
    2666. 

  2666. <a name="l02602"></a>02602 <span class="comment">!</span>
    2667. 

  2667. <a name="l02603"></a>02603     area12 = area1 + area2
    2668. 

  2668. <a name="l02604"></a>02604     erro12 = error1 + error2
    2669. 

  2669. <a name="l02605"></a>02605     errsum = errsum + erro12 - errmax
    2670. 

  2670. <a name="l02606"></a>02606     area = area + area12 - rlist(maxerr)
    2671. 

  2671. <a name="l02607"></a>02607 
    2672. 

  2672. <a name="l02608"></a>02608     <span class="keyword">if</span> ( abs ( rlist(maxerr)-area12) &lt; 1.0e-05 * abs(area12) &amp;
    2673. 

  2673. <a name="l02609"></a>02609       .and. erro12 &gt;= 9.9e-01 * errmax .and. krule == 0 ) &amp;
    2674. 

  2674. <a name="l02610"></a>02610       iroff1 = iroff1+1
    2675. 

  2675. <a name="l02611"></a>02611 
    2676. 

  2676. <a name="l02612"></a>02612     <span class="keyword">if</span> ( last &gt; 10.and.erro12 &gt; errmax .and. krule == 0 ) <span class="keyword">then</span>
    2677. 

  2677. <a name="l02613"></a>02613       iroff2 = iroff2+1
    2678. 

  2678. <a name="l02614"></a>02614     <span class="keyword">end if</span>
    2679. 

  2679. <a name="l02615"></a>02615 
    2680. 

  2680. <a name="l02616"></a>02616     rlist(maxerr) = area1
    2681. 

  2681. <a name="l02617"></a>02617     rlist(last) = area2
    2682. 

  2682. <a name="l02618"></a>02618     errbnd = max ( epsabs, epsrel * abs(area) )
    2683. 

  2683. <a name="l02619"></a>02619 
    2684. 

  2684. <a name="l02620"></a>02620     <span class="keyword">if</span> ( errsum &gt; errbnd ) <span class="keyword">then</span>
    2685. 

  2685. <a name="l02621"></a>02621 <span class="comment">!</span>
    2686. 

  2686. <a name="l02622"></a>02622 <span class="comment">!  Test for roundoff error and eventually set error flag.</span>
    2687. 

  2687. <a name="l02623"></a>02623 <span class="comment">!</span>
    2688. 

  2688. <a name="l02624"></a>02624       <span class="keyword">if</span> ( iroff1 &gt;= 6 .and. iroff2 &gt; 20 ) <span class="keyword">then</span>
    2689. 

  2689. <a name="l02625"></a>02625         ier = 2
    2690. 

  2690. <a name="l02626"></a>02626       <span class="keyword">end if</span>
    2691. 

  2691. <a name="l02627"></a>02627 <span class="comment">!</span>
    2692. 

  2692. <a name="l02628"></a>02628 <span class="comment">!  Set error flag in the case that number of interval</span>
    2693. 

  2693. <a name="l02629"></a>02629 <span class="comment">!  bisections exceeds limit.</span>
    2694. 

  2694. <a name="l02630"></a>02630 <span class="comment">!</span>
    2695. 

  2695. <a name="l02631"></a>02631       <span class="keyword">if</span> ( last == limit ) <span class="keyword">then</span>
    2696. 

  2696. <a name="l02632"></a>02632         ier = 1
    2697. 

  2697. <a name="l02633"></a>02633       <span class="keyword">end if</span>
    2698. 

  2698. <a name="l02634"></a>02634 <span class="comment">!</span>
    2699. 

  2699. <a name="l02635"></a>02635 <span class="comment">!  Set error flag in the case of bad integrand behavior at</span>
    2700. 

  2700. <a name="l02636"></a>02636 <span class="comment">!  a point of the integration range.</span>
    2701. 

  2701. <a name="l02637"></a>02637 <span class="comment">!</span>
    2702. 

  2702. <a name="l02638"></a>02638       <span class="keyword">if</span> ( max ( abs(a1), abs(b2) ) &lt;= ( 1.0e+00 + 1.0e+03 * epsilon ( a1 ) ) &amp;
    2703. 

  2703. <a name="l02639"></a>02639         *( abs(a2)+1.0e+03* tiny ( a2 ) )) <span class="keyword">then</span>
    2704. 

  2704. <a name="l02640"></a>02640         ier = 3
    2705. 

  2705. <a name="l02641"></a>02641       <span class="keyword">end if</span>
    2706. 

  2706. <a name="l02642"></a>02642 
    2707. 

  2707. <a name="l02643"></a>02643     <span class="keyword">end if</span>
    2708. 

  2708. <a name="l02644"></a>02644 <span class="comment">!</span>
    2709. 

  2709. <a name="l02645"></a>02645 <span class="comment">!  Append the newly-created intervals to the list.</span>
    2710. 

  2710. <a name="l02646"></a>02646 <span class="comment">!</span>
    2711. 

  2711. <a name="l02647"></a>02647     <span class="keyword">if</span> ( error2 &lt;= error1 ) <span class="keyword">then</span>
    2712. 

  2712. <a name="l02648"></a>02648       alist(last) = a2
    2713. 

  2713. <a name="l02649"></a>02649       blist(maxerr) = b1
    2714. 

  2714. <a name="l02650"></a>02650       blist(last) = b2
    2715. 

  2715. <a name="l02651"></a>02651       elist(maxerr) = error1
    2716. 

  2716. <a name="l02652"></a>02652       elist(last) = error2
    2717. 

  2717. <a name="l02653"></a>02653     <span class="keyword">else</span>
    2718. 

  2718. <a name="l02654"></a>02654       alist(maxerr) = a2
    2719. 

  2719. <a name="l02655"></a>02655       alist(last) = a1
    2720. 

  2720. <a name="l02656"></a>02656       blist(last) = b1
    2721. 

  2721. <a name="l02657"></a>02657       rlist(maxerr) = area2
    2722. 

  2722. <a name="l02658"></a>02658       rlist(last) = area1
    2723. 

  2723. <a name="l02659"></a>02659       elist(maxerr) = error2
    2724. 

  2724. <a name="l02660"></a>02660       elist(last) = error1
    2725. 

  2725. <a name="l02661"></a>02661     <span class="keyword">end if</span>
    2726. 

  2726. <a name="l02662"></a>02662 <span class="comment">!</span>
    2727. 

  2727. <a name="l02663"></a>02663 <span class="comment">!  Call QSORT to maintain the descending ordering</span>
    2728. 

  2728. <a name="l02664"></a>02664 <span class="comment">!  in the list of error estimates and select the subinterval</span>
    2729. 

  2729. <a name="l02665"></a>02665 <span class="comment">!  with NRMAX-th largest error estimate (to be bisected next).</span>
    2730. 

  2730. <a name="l02666"></a>02666 <span class="comment">!</span>
    2731. 

  2731. <a name="l02667"></a>02667     call <a class="code" href="quadpack_8f90.html#a55e08a684c5a6315fb37dd0fdc66d8e6">qsort </a>( limit, last, maxerr, errmax, elist, iord, nrmax )
    2732. 

  2732. <a name="l02668"></a>02668 
    2733. 

  2733. <a name="l02669"></a>02669     <span class="keyword">if</span> ( ier /= 0 .or. errsum &lt;= errbnd ) <span class="keyword">then</span>
    2734. 

  2734. <a name="l02670"></a>02670       exit
    2735. 

  2735. <a name="l02671"></a>02671     <span class="keyword">end if</span>
    2736. 

  2736. <a name="l02672"></a>02672 
    2737. 

  2737. <a name="l02673"></a>02673   <span class="keyword">end do</span>
    2738. 

  2738. <a name="l02674"></a>02674 <span class="comment">!</span>
    2739. 

  2739. <a name="l02675"></a>02675 <span class="comment">!  Compute final result.</span>
    2740. 

  2740. <a name="l02676"></a>02676 <span class="comment">!</span>
    2741. 

  2741. <a name="l02677"></a>02677   result = sum ( rlist(1:last) )
    2742. 

  2742. <a name="l02678"></a>02678 
    2743. 

  2743. <a name="l02679"></a>02679   abserr = errsum
    2744. 

  2744. <a name="l02680"></a>02680 
    2745. 

  2745. <a name="l02681"></a>02681 70 continue 
    2746. 

  2746. <a name="l02682"></a>02682 
    2747. 

  2747. <a name="l02683"></a>02683   <span class="keyword">if</span> ( aa == b ) <span class="keyword">then</span>
    2748. 

  2748. <a name="l02684"></a>02684     result = - result
    2749. 

  2749. <a name="l02685"></a>02685   <span class="keyword">end if</span>
    2750. 

  2750. <a name="l02686"></a>02686 
    2751. 

  2751. <a name="l02687"></a>02687   return
    2752. 

  2752. <a name="l02688"></a>02688 <span class="keyword">end</span>
    2753. 

  2753. <a name="l02689"></a><a class="code" href="quadpack_8f90.html#aefd54eff8d0418a0f533f571d80ec5e5">02689</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#aefd54eff8d0418a0f533f571d80ec5e5">qawf</a> ( f, a, omega, integr, epsabs, result, abserr, neval, ier )
    2754. 

  2754. <a name="l02690"></a>02690 
    2755. 

  2755. <a name="l02691"></a>02691 <span class="comment">!*****************************************************************************80</span>
    2756. 

  2756. <a name="l02692"></a>02692 <span class="comment">!</span>
    2757. 

  2757. <a name="l02693"></a>02693 <span class="comment">!! QAWF computes Fourier integrals over the interval [ A, +Infinity ).</span>
    2758. 

  2758. <a name="l02694"></a>02694 <span class="comment">!</span>
    2759. 

  2759. <a name="l02695"></a>02695 <span class="comment">!  Discussion:</span>
    2760. 

  2760. <a name="l02696"></a>02696 <span class="comment">!</span>
    2761. 

  2761. <a name="l02697"></a>02697 <span class="comment">!    The routine calculates an approximation RESULT to a definite integral  </span>
    2762. 

  2762. <a name="l02698"></a>02698 <span class="comment">! </span>
    2763. 

  2763. <a name="l02699"></a>02699 <span class="comment">!      I = integral of F*COS(OMEGA*X) </span>
    2764. 

  2764. <a name="l02700"></a>02700 <span class="comment">!    or </span>
    2765. 

  2765. <a name="l02701"></a>02701 <span class="comment">!      I = integral of F*SIN(OMEGA*X) </span>
    2766. 

  2766. <a name="l02702"></a>02702 <span class="comment">!</span>
    2767. 

  2767. <a name="l02703"></a>02703 <span class="comment">!    over the interval [A,+Infinity), hopefully satisfying</span>
    2768. 

  2768. <a name="l02704"></a>02704 <span class="comment">!</span>
    2769. 

  2769. <a name="l02705"></a>02705 <span class="comment">!      || I - RESULT || &lt;= EPSABS.</span>
    2770. 

  2770. <a name="l02706"></a>02706 <span class="comment">!</span>
    2771. 

  2771. <a name="l02707"></a>02707 <span class="comment">!    If OMEGA = 0 and INTEGR = 1, the integral is calculated by means </span>
    2772. 

  2772. <a name="l02708"></a>02708 <span class="comment">!    of QAGI, and IER has the meaning as described in the comments of QAGI.</span>
    2773. 

  2773. <a name="l02709"></a>02709 <span class="comment">!</span>
    2774. 

  2774. <a name="l02710"></a>02710 <span class="comment">!  Author:</span>
    2775. 

  2775. <a name="l02711"></a>02711 <span class="comment">!</span>
    2776. 

  2776. <a name="l02712"></a>02712 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    2777. 

  2777. <a name="l02713"></a>02713 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    2778. 

  2778. <a name="l02714"></a>02714 <span class="comment">!</span>
    2779. 

  2779. <a name="l02715"></a>02715 <span class="comment">!  Reference:</span>
    2780. 

  2780. <a name="l02716"></a>02716 <span class="comment">!</span>
    2781. 

  2781. <a name="l02717"></a>02717 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    2782. 

  2782. <a name="l02718"></a>02718 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    2783. 

  2783. <a name="l02719"></a>02719 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    2784. 

  2784. <a name="l02720"></a>02720 <span class="comment">!    Springer Verlag, 1983</span>
    2785. 

  2785. <a name="l02721"></a>02721 <span class="comment">!</span>
    2786. 

  2786. <a name="l02722"></a>02722 <span class="comment">!  Parameters:</span>
    2787. 

  2787. <a name="l02723"></a>02723 <span class="comment">!</span>
    2788. 

  2788. <a name="l02724"></a>02724 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    2789. 

  2789. <a name="l02725"></a>02725 <span class="comment">!      function f ( x )</span>
    2790. 

  2790. <a name="l02726"></a>02726 <span class="comment">!      real f</span>
    2791. 

  2791. <a name="l02727"></a>02727 <span class="comment">!      real x</span>
    2792. 

  2792. <a name="l02728"></a>02728 <span class="comment">!    which evaluates the integrand function.</span>
    2793. 

  2793. <a name="l02729"></a>02729 <span class="comment">!</span>
    2794. 

  2794. <a name="l02730"></a>02730 <span class="comment">!    Input, real A, the lower limit of integration.</span>
    2795. 

  2795. <a name="l02731"></a>02731 <span class="comment">!</span>
    2796. 

  2796. <a name="l02732"></a>02732 <span class="comment">!    Input, real OMEGA, the parameter in the weight function.</span>
    2797. 

  2797. <a name="l02733"></a>02733 <span class="comment">!</span>
    2798. 

  2798. <a name="l02734"></a>02734 <span class="comment">!    Input, integer INTEGR, indicates which weight functions is used</span>
    2799. 

  2799. <a name="l02735"></a>02735 <span class="comment">!    = 1, w(x) = cos(omega*x)</span>
    2800. 

  2800. <a name="l02736"></a>02736 <span class="comment">!    = 2, w(x) = sin(omega*x)</span>
    2801. 

  2801. <a name="l02737"></a>02737 <span class="comment">!</span>
    2802. 

  2802. <a name="l02738"></a>02738 <span class="comment">!    Input, real EPSABS, the absolute accuracy requested.</span>
    2803. 

  2803. <a name="l02739"></a>02739 <span class="comment">!</span>
    2804. 

  2804. <a name="l02740"></a>02740 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    2805. 

  2805. <a name="l02741"></a>02741 <span class="comment">!</span>
    2806. 

  2806. <a name="l02742"></a>02742 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    2807. 

  2807. <a name="l02743"></a>02743 <span class="comment">!</span>
    2808. 

  2808. <a name="l02744"></a>02744 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    2809. 

  2809. <a name="l02745"></a>02745 <span class="comment">!</span>
    2810. 

  2810. <a name="l02746"></a>02746 <span class="comment">!            ier    - integer</span>
    2811. 

  2811. <a name="l02747"></a>02747 <span class="comment">!                     ier = 0 normal and reliable termination of the</span>
    2812. 

  2812. <a name="l02748"></a>02748 <span class="comment">!                             routine. it is assumed that the</span>
    2813. 

  2813. <a name="l02749"></a>02749 <span class="comment">!                             requested accuracy has been achieved.</span>
    2814. 

  2814. <a name="l02750"></a>02750 <span class="comment">!                     ier &gt; 0 abnormal termination of the routine.</span>
    2815. 

  2815. <a name="l02751"></a>02751 <span class="comment">!                             the estimates for integral and error are</span>
    2816. 

  2816. <a name="l02752"></a>02752 <span class="comment">!                             less reliable. it is assumed that the</span>
    2817. 

  2817. <a name="l02753"></a>02753 <span class="comment">!                             requested accuracy has not been achieved.</span>
    2818. 

  2818. <a name="l02754"></a>02754 <span class="comment">!                    if omega /= 0</span>
    2819. 

  2819. <a name="l02755"></a>02755 <span class="comment">!                     ier = 6 the input is invalid because</span>
    2820. 

  2820. <a name="l02756"></a>02756 <span class="comment">!                             (integr /= 1 and integr /= 2) or</span>
    2821. 

  2821. <a name="l02757"></a>02757 <span class="comment">!                              epsabs &lt;= 0</span>
    2822. 

  2822. <a name="l02758"></a>02758 <span class="comment">!                              result, abserr, neval, lst are set to</span>
    2823. 

  2823. <a name="l02759"></a>02759 <span class="comment">!                              zero.</span>
    2824. 

  2824. <a name="l02760"></a>02760 <span class="comment">!                         = 7 abnormal termination of the computation</span>
    2825. 

  2825. <a name="l02761"></a>02761 <span class="comment">!                             of one or more subintegrals</span>
    2826. 

  2826. <a name="l02762"></a>02762 <span class="comment">!                         = 8 maximum number of cycles allowed</span>
    2827. 

  2827. <a name="l02763"></a>02763 <span class="comment">!                             has been achieved, i.e. of subintervals</span>
    2828. 

  2828. <a name="l02764"></a>02764 <span class="comment">!                             (a+(k-1)c,a+kc) where</span>
    2829. 

  2829. <a name="l02765"></a>02765 <span class="comment">!                             c = (2*int(abs(omega))+1)*pi/abs(omega),</span>
    2830. 

  2830. <a name="l02766"></a>02766 <span class="comment">!                             for k = 1, 2, ...</span>
    2831. 

  2831. <a name="l02767"></a>02767 <span class="comment">!                         = 9 the extrapolation table constructed for</span>
    2832. 

  2832. <a name="l02768"></a>02768 <span class="comment">!                             convergence acceleration of the series</span>
    2833. 

  2833. <a name="l02769"></a>02769 <span class="comment">!                             formed by the integral contributions</span>
    2834. 

  2834. <a name="l02770"></a>02770 <span class="comment">!                             over the cycles, does not converge to</span>
    2835. 

  2835. <a name="l02771"></a>02771 <span class="comment">!                             within the requested accuracy.</span>
    2836. 

  2836. <a name="l02772"></a>02772 <span class="comment">!</span>
    2837. 

  2837. <a name="l02773"></a>02773 <span class="comment">!  Local parameters:</span>
    2838. 

  2838. <a name="l02774"></a>02774 <span class="comment">!</span>
    2839. 

  2839. <a name="l02775"></a>02775 <span class="comment">!    Integer LIMLST, gives an upper bound on the number of cycles, LIMLST &gt;= 3.</span>
    2840. 

  2840. <a name="l02776"></a>02776 <span class="comment">!    if limlst &lt; 3, the routine will end with ier = 6.</span>
    2841. 

  2841. <a name="l02777"></a>02777 <span class="comment">!</span>
    2842. 

  2842. <a name="l02778"></a>02778 <span class="comment">!    Integer MAXP1, an upper bound on the number of Chebyshev moments which </span>
    2843. 

  2843. <a name="l02779"></a>02779 <span class="comment">!    can be stored, i.e. for the intervals of lengths abs(b-a)*2**(-l), </span>
    2844. 

  2844. <a name="l02780"></a>02780 <span class="comment">!    l = 0,1, ..., maxp1-2, maxp1 &gt;= 1.  if maxp1 &lt; 1, the routine will end</span>
    2845. 

  2845. <a name="l02781"></a>02781 <span class="comment">!    with ier = 6.</span>
    2846. 

  2846. <a name="l02782"></a>02782 <span class="comment">!</span>
    2847. 

  2847. <a name="l02783"></a>02783   <span class="keyword">implicit none</span>
    2848. 

  2848. <a name="l02784"></a>02784 
    2849. 

  2849. <a name="l02785"></a>02785   <span class="keywordtype">integer</span>, <span class="keywordtype">parameter</span> :: limit = 500
    2850. 

  2850. <a name="l02786"></a>02786   <span class="keywordtype">integer</span>, <span class="keywordtype">parameter</span> :: limlst = 50
    2851. 

  2851. <a name="l02787"></a>02787   <span class="keywordtype">integer</span>, <span class="keywordtype">parameter</span> :: maxp1 = 21
    2852. 

  2852. <a name="l02788"></a>02788 
    2853. 

  2853. <a name="l02789"></a>02789   <span class="keywordtype">real</span> a
    2854. 

  2854. <a name="l02790"></a>02790   <span class="keywordtype">real</span> abserr
    2855. 

  2855. <a name="l02791"></a>02791   <span class="keywordtype">real</span> alist(limit)
    2856. 

  2856. <a name="l02792"></a>02792   <span class="keywordtype">real</span> blist(limit)
    2857. 

  2857. <a name="l02793"></a>02793   <span class="keywordtype">real</span> chebmo(maxp1,25)
    2858. 

  2858. <a name="l02794"></a>02794   <span class="keywordtype">real</span> elist(limit)
    2859. 

  2859. <a name="l02795"></a>02795   <span class="keywordtype">real</span> epsabs
    2860. 

  2860. <a name="l02796"></a>02796   <span class="keywordtype">real</span> erlst(limlst)
    2861. 

  2861. <a name="l02797"></a>02797   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    2862. 

  2862. <a name="l02798"></a>02798   <span class="keywordtype">integer</span> ier
    2863. 

  2863. <a name="l02799"></a>02799   <span class="keywordtype">integer</span> integr
    2864. 

  2864. <a name="l02800"></a>02800   <span class="keywordtype">integer</span> iord(limit)
    2865. 

  2865. <a name="l02801"></a>02801   <span class="keywordtype">integer</span> ierlst(limlst)
    2866. 

  2866. <a name="l02802"></a>02802   <span class="keywordtype">integer</span> lst
    2867. 

  2867. <a name="l02803"></a>02803   <span class="keywordtype">integer</span> neval
    2868. 

  2868. <a name="l02804"></a>02804   <span class="keywordtype">integer</span> nnlog(limit)
    2869. 

  2869. <a name="l02805"></a>02805   <span class="keywordtype">real</span> omega
    2870. 

  2870. <a name="l02806"></a>02806   <span class="keywordtype">real</span> result
    2871. 

  2871. <a name="l02807"></a>02807   <span class="keywordtype">real</span> rlist(limit)
    2872. 

  2872. <a name="l02808"></a>02808   <span class="keywordtype">real</span> rslst(limlst)
    2873. 

  2873. <a name="l02809"></a>02809 
    2874. 

  2874. <a name="l02810"></a>02810   ier = 6
    2875. 

  2875. <a name="l02811"></a>02811   neval = 0
    2876. 

  2876. <a name="l02812"></a>02812   result = 0.0e+00
    2877. 

  2877. <a name="l02813"></a>02813   abserr = 0.0e+00
    2878. 

  2878. <a name="l02814"></a>02814 
    2879. 

  2879. <a name="l02815"></a>02815   <span class="keyword">if</span> ( limlst &lt; 3 .or. maxp1 &lt; 1 ) <span class="keyword">then</span>
    2880. 

  2880. <a name="l02816"></a>02816     return
    2881. 

  2881. <a name="l02817"></a>02817   <span class="keyword">end if</span>
    2882. 

  2882. <a name="l02818"></a>02818 
    2883. 

  2883. <a name="l02819"></a>02819   call <a class="code" href="quadpack_8f90.html#abe17af7f3ad5cf264791d326bbd15192">qawfe </a>( f, a, omega, integr, epsabs, limlst, limit, maxp1, &amp;
    2884. 

  2884. <a name="l02820"></a>02820     result, abserr, neval, ier, rslst, erlst, ierlst, lst, alist, blist, &amp;
    2885. 

  2885. <a name="l02821"></a>02821     rlist, elist, iord, nnlog, chebmo )
    2886. 

  2886. <a name="l02822"></a>02822 
    2887. 

  2887. <a name="l02823"></a>02823   return
    2888. 

  2888. <a name="l02824"></a>02824 <span class="keyword">end</span>
    2889. 

  2889. <a name="l02825"></a><a class="code" href="quadpack_8f90.html#abe17af7f3ad5cf264791d326bbd15192">02825</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#abe17af7f3ad5cf264791d326bbd15192">qawfe</a> ( f, a, omega, integr, epsabs, limlst, limit, maxp1, &amp;
    2890. 

  2890. <a name="l02826"></a>02826   result, abserr, neval, ier, rslst, erlst, ierlst, lst, alist, blist, &amp;
    2891. 

  2891. <a name="l02827"></a>02827   rlist, elist, iord, nnlog, chebmo )
    2892. 

  2892. <a name="l02828"></a>02828 
    2893. 

  2893. <a name="l02829"></a>02829 <span class="comment">!*****************************************************************************80</span>
    2894. 

  2894. <a name="l02830"></a>02830 <span class="comment">!</span>
    2895. 

  2895. <a name="l02831"></a>02831 <span class="comment">!! QAWFE computes Fourier integrals.</span>
    2896. 

  2896. <a name="l02832"></a>02832 <span class="comment">!</span>
    2897. 

  2897. <a name="l02833"></a>02833 <span class="comment">!  Discussion:</span>
    2898. 

  2898. <a name="l02834"></a>02834 <span class="comment">!</span>
    2899. 

  2899. <a name="l02835"></a>02835 <span class="comment">!    The routine calculates an approximation RESULT to a definite integral   </span>
    2900. 

  2900. <a name="l02836"></a>02836 <span class="comment">!      I = integral of F*COS(OMEGA*X) or F*SIN(OMEGA*X) over (A,+Infinity),</span>
    2901. 

  2901. <a name="l02837"></a>02837 <span class="comment">!    hopefully satisfying</span>
    2902. 

  2902. <a name="l02838"></a>02838 <span class="comment">!      || I - RESULT || &lt;= EPSABS.</span>
    2903. 

  2903. <a name="l02839"></a>02839 <span class="comment">!</span>
    2904. 

  2904. <a name="l02840"></a>02840 <span class="comment">!  Author:</span>
    2905. 

  2905. <a name="l02841"></a>02841 <span class="comment">!</span>
    2906. 

  2906. <a name="l02842"></a>02842 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    2907. 

  2907. <a name="l02843"></a>02843 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    2908. 

  2908. <a name="l02844"></a>02844 <span class="comment">!</span>
    2909. 

  2909. <a name="l02845"></a>02845 <span class="comment">!  Reference:</span>
    2910. 

  2910. <a name="l02846"></a>02846 <span class="comment">!</span>
    2911. 

  2911. <a name="l02847"></a>02847 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    2912. 

  2912. <a name="l02848"></a>02848 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    2913. 

  2913. <a name="l02849"></a>02849 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    2914. 

  2914. <a name="l02850"></a>02850 <span class="comment">!    Springer Verlag, 1983</span>
    2915. 

  2915. <a name="l02851"></a>02851 <span class="comment">!</span>
    2916. 

  2916. <a name="l02852"></a>02852 <span class="comment">!  Parameters:</span>
    2917. 

  2917. <a name="l02853"></a>02853 <span class="comment">!</span>
    2918. 

  2918. <a name="l02854"></a>02854 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    2919. 

  2919. <a name="l02855"></a>02855 <span class="comment">!      function f ( x )</span>
    2920. 

  2920. <a name="l02856"></a>02856 <span class="comment">!      real f</span>
    2921. 

  2921. <a name="l02857"></a>02857 <span class="comment">!      real x</span>
    2922. 

  2922. <a name="l02858"></a>02858 <span class="comment">!    which evaluates the integrand function.</span>
    2923. 

  2923. <a name="l02859"></a>02859 <span class="comment">!</span>
    2924. 

  2924. <a name="l02860"></a>02860 <span class="comment">!    Input, real A, the lower limit of integration.</span>
    2925. 

  2925. <a name="l02861"></a>02861 <span class="comment">!</span>
    2926. 

  2926. <a name="l02862"></a>02862 <span class="comment">!    Input, real OMEGA, the parameter in the weight function.</span>
    2927. 

  2927. <a name="l02863"></a>02863 <span class="comment">!</span>
    2928. 

  2928. <a name="l02864"></a>02864 <span class="comment">!    Input, integer INTEGR, indicates which weight function is used</span>
    2929. 

  2929. <a name="l02865"></a>02865 <span class="comment">!    = 1      w(x) = cos(omega*x)</span>
    2930. 

  2930. <a name="l02866"></a>02866 <span class="comment">!    = 2      w(x) = sin(omega*x)</span>
    2931. 

  2931. <a name="l02867"></a>02867 <span class="comment">!</span>
    2932. 

  2932. <a name="l02868"></a>02868 <span class="comment">!    Input, real EPSABS, the absolute accuracy requested.</span>
    2933. 

  2933. <a name="l02869"></a>02869 <span class="comment">!</span>
    2934. 

  2934. <a name="l02870"></a>02870 <span class="comment">!    Input, integer LIMLST, an upper bound on the number of cycles.</span>
    2935. 

  2935. <a name="l02871"></a>02871 <span class="comment">!    LIMLST must be at least 1.  In fact, if LIMLST &lt; 3, the routine </span>
    2936. 

  2936. <a name="l02872"></a>02872 <span class="comment">!    will end with IER= 6.</span>
    2937. 

  2937. <a name="l02873"></a>02873 <span class="comment">!</span>
    2938. 

  2938. <a name="l02874"></a>02874 <span class="comment">!    Input, integer LIMIT, an upper bound on the number of subintervals </span>
    2939. 

  2939. <a name="l02875"></a>02875 <span class="comment">!    allowed in the partition of each cycle, limit &gt;= 1.</span>
    2940. 

  2940. <a name="l02876"></a>02876 <span class="comment">!</span>
    2941. 

  2941. <a name="l02877"></a>02877 <span class="comment">!            maxp1  - integer</span>
    2942. 

  2942. <a name="l02878"></a>02878 <span class="comment">!                     gives an upper bound on the number of</span>
    2943. 

  2943. <a name="l02879"></a>02879 <span class="comment">!                     Chebyshev moments which can be stored, i.e.</span>
    2944. 

  2944. <a name="l02880"></a>02880 <span class="comment">!                     for the intervals of lengths abs(b-a)*2**(-l),</span>
    2945. 

  2945. <a name="l02881"></a>02881 <span class="comment">!                     l=0,1, ..., maxp1-2, maxp1 &gt;= 1</span>
    2946. 

  2946. <a name="l02882"></a>02882 <span class="comment">!</span>
    2947. 

  2947. <a name="l02883"></a>02883 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    2948. 

  2948. <a name="l02884"></a>02884 <span class="comment">!</span>
    2949. 

  2949. <a name="l02885"></a>02885 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    2950. 

  2950. <a name="l02886"></a>02886 <span class="comment">!</span>
    2951. 

  2951. <a name="l02887"></a>02887 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    2952. 

  2952. <a name="l02888"></a>02888 <span class="comment">!</span>
    2953. 

  2953. <a name="l02889"></a>02889 <span class="comment">!            ier    - ier = 0 normal and reliable termination of</span>
    2954. 

  2954. <a name="l02890"></a>02890 <span class="comment">!                             the routine. it is assumed that the</span>
    2955. 

  2955. <a name="l02891"></a>02891 <span class="comment">!                             requested accuracy has been achieved.</span>
    2956. 

  2956. <a name="l02892"></a>02892 <span class="comment">!                     ier &gt; 0 abnormal termination of the routine</span>
    2957. 

  2957. <a name="l02893"></a>02893 <span class="comment">!                             the estimates for integral and error</span>
    2958. 

  2958. <a name="l02894"></a>02894 <span class="comment">!                             are less reliable. it is assumed that</span>
    2959. 

  2959. <a name="l02895"></a>02895 <span class="comment">!                             the requested accuracy has not been</span>
    2960. 

  2960. <a name="l02896"></a>02896 <span class="comment">!                             achieved.</span>
    2961. 

  2961. <a name="l02897"></a>02897 <span class="comment">!                    if omega /= 0</span>
    2962. 

  2962. <a name="l02898"></a>02898 <span class="comment">!                     ier = 6 the input is invalid because</span>
    2963. 

  2963. <a name="l02899"></a>02899 <span class="comment">!                             (integr /= 1 and integr /= 2) or</span>
    2964. 

  2964. <a name="l02900"></a>02900 <span class="comment">!                              epsabs &lt;= 0 or limlst &lt; 3.</span>
    2965. 

  2965. <a name="l02901"></a>02901 <span class="comment">!                              result, abserr, neval, lst are set</span>
    2966. 

  2966. <a name="l02902"></a>02902 <span class="comment">!                              to zero.</span>
    2967. 

  2967. <a name="l02903"></a>02903 <span class="comment">!                         = 7 bad integrand behavior occurs within</span>
    2968. 

  2968. <a name="l02904"></a>02904 <span class="comment">!                             one or more of the cycles. location</span>
    2969. 

  2969. <a name="l02905"></a>02905 <span class="comment">!                             and type of the difficulty involved</span>
    2970. 

  2970. <a name="l02906"></a>02906 <span class="comment">!                             can be determined from the vector ierlst.</span>
    2971. 

  2971. <a name="l02907"></a>02907 <span class="comment">!                             here lst is the number of cycles actually</span>
    2972. 

  2972. <a name="l02908"></a>02908 <span class="comment">!                             needed (see below).</span>
    2973. 

  2973. <a name="l02909"></a>02909 <span class="comment">!                             ierlst(k) = 1 the maximum number of</span>
    2974. 

  2974. <a name="l02910"></a>02910 <span class="comment">!                                           subdivisions (= limit)</span>
    2975. 

  2975. <a name="l02911"></a>02911 <span class="comment">!                                           has been achieved on the</span>
    2976. 

  2976. <a name="l02912"></a>02912 <span class="comment">!                                           k th cycle.</span>
    2977. 

  2977. <a name="l02913"></a>02913 <span class="comment">!                                       = 2 occurence of roundoff</span>
    2978. 

  2978. <a name="l02914"></a>02914 <span class="comment">!                                           error is detected and</span>
    2979. 

  2979. <a name="l02915"></a>02915 <span class="comment">!                                           prevents the tolerance</span>
    2980. 

  2980. <a name="l02916"></a>02916 <span class="comment">!                                           imposed on the k th cycle</span>
    2981. 

  2981. <a name="l02917"></a>02917 <span class="comment">!                                           from being acheived.</span>
    2982. 

  2982. <a name="l02918"></a>02918 <span class="comment">!                                       = 3 extremely bad integrand</span>
    2983. 

  2983. <a name="l02919"></a>02919 <span class="comment">!                                           behavior occurs at some</span>
    2984. 

  2984. <a name="l02920"></a>02920 <span class="comment">!                                           points of the k th cycle.</span>
    2985. 

  2985. <a name="l02921"></a>02921 <span class="comment">!                                       = 4 the integration procedure</span>
    2986. 

  2986. <a name="l02922"></a>02922 <span class="comment">!                                           over the k th cycle does</span>
    2987. 

  2987. <a name="l02923"></a>02923 <span class="comment">!                                           not converge (to within the</span>
    2988. 

  2988. <a name="l02924"></a>02924 <span class="comment">!                                           required accuracy) due to</span>
    2989. 

  2989. <a name="l02925"></a>02925 <span class="comment">!                                           roundoff in the</span>
    2990. 

  2990. <a name="l02926"></a>02926 <span class="comment">!                                           extrapolation procedure</span>
    2991. 

  2991. <a name="l02927"></a>02927 <span class="comment">!                                           invoked on this cycle. it</span>
    2992. 

  2992. <a name="l02928"></a>02928 <span class="comment">!                                           is assumed that the result</span>
    2993. 

  2993. <a name="l02929"></a>02929 <span class="comment">!                                           on this interval is the</span>
    2994. 

  2994. <a name="l02930"></a>02930 <span class="comment">!                                           best which can be obtained.</span>
    2995. 

  2995. <a name="l02931"></a>02931 <span class="comment">!                                       = 5 the integral over the k th</span>
    2996. 

  2996. <a name="l02932"></a>02932 <span class="comment">!                                           cycle is probably divergent</span>
    2997. 

  2997. <a name="l02933"></a>02933 <span class="comment">!                                           or slowly convergent. it</span>
    2998. 

  2998. <a name="l02934"></a>02934 <span class="comment">!                                           must be noted that</span>
    2999. 

  2999. <a name="l02935"></a>02935 <span class="comment">!                                           divergence can occur with</span>
    3000. 

  3000. <a name="l02936"></a>02936 <span class="comment">!                                           any other value of</span>
    3001. 

  3001. <a name="l02937"></a>02937 <span class="comment">!                                           ierlst(k).</span>
    3002. 

  3002. <a name="l02938"></a>02938 <span class="comment">!                         = 8 maximum number of  cycles  allowed</span>
    3003. 

  3003. <a name="l02939"></a>02939 <span class="comment">!                             has been achieved, i.e. of subintervals</span>
    3004. 

  3004. <a name="l02940"></a>02940 <span class="comment">!                             (a+(k-1)c,a+kc) where</span>
    3005. 

  3005. <a name="l02941"></a>02941 <span class="comment">!                             c = (2*int(abs(omega))+1)*pi/abs(omega),</span>
    3006. 

  3006. <a name="l02942"></a>02942 <span class="comment">!                             for k = 1, 2, ..., lst.</span>
    3007. 

  3007. <a name="l02943"></a>02943 <span class="comment">!                             one can allow more cycles by increasing</span>
    3008. 

  3008. <a name="l02944"></a>02944 <span class="comment">!                             the value of limlst (and taking the</span>
    3009. 

  3009. <a name="l02945"></a>02945 <span class="comment">!                             according dimension adjustments into</span>
    3010. 

  3010. <a name="l02946"></a>02946 <span class="comment">!                             account).</span>
    3011. 

  3011. <a name="l02947"></a>02947 <span class="comment">!                             examine the array iwork which contains</span>
    3012. 

  3012. <a name="l02948"></a>02948 <span class="comment">!                             the error flags over the cycles, in order</span>
    3013. 

  3013. <a name="l02949"></a>02949 <span class="comment">!                             to eventual look for local integration</span>
    3014. 

  3014. <a name="l02950"></a>02950 <span class="comment">!                             difficulties.</span>
    3015. 

  3015. <a name="l02951"></a>02951 <span class="comment">!                             if the position of a local difficulty can</span>
    3016. 

  3016. <a name="l02952"></a>02952 <span class="comment">!                             be determined (e.g. singularity,</span>
    3017. 

  3017. <a name="l02953"></a>02953 <span class="comment">!                             discontinuity within the interval)</span>
    3018. 

  3018. <a name="l02954"></a>02954 <span class="comment">!                             one will probably gain from splitting</span>
    3019. 

  3019. <a name="l02955"></a>02955 <span class="comment">!                             up the interval at this point and</span>
    3020. 

  3020. <a name="l02956"></a>02956 <span class="comment">!                             calling appopriate integrators on the</span>
    3021. 

  3021. <a name="l02957"></a>02957 <span class="comment">!                             subranges.</span>
    3022. 

  3022. <a name="l02958"></a>02958 <span class="comment">!                         = 9 the extrapolation table constructed for</span>
    3023. 

  3023. <a name="l02959"></a>02959 <span class="comment">!                             convergence acceleration of the series</span>
    3024. 

  3024. <a name="l02960"></a>02960 <span class="comment">!                             formed by the integral contributions</span>
    3025. 

  3025. <a name="l02961"></a>02961 <span class="comment">!                             over the cycles, does not converge to</span>
    3026. 

  3026. <a name="l02962"></a>02962 <span class="comment">!                             within the required accuracy.</span>
    3027. 

  3027. <a name="l02963"></a>02963 <span class="comment">!                             as in the case of ier = 8, it is advised</span>
    3028. 

  3028. <a name="l02964"></a>02964 <span class="comment">!                             to examine the array iwork which contains</span>
    3029. 

  3029. <a name="l02965"></a>02965 <span class="comment">!                             the error flags on the cycles.</span>
    3030. 

  3030. <a name="l02966"></a>02966 <span class="comment">!                    if omega = 0 and integr = 1,</span>
    3031. 

  3031. <a name="l02967"></a>02967 <span class="comment">!                    the integral is calculated by means of qagi</span>
    3032. 

  3032. <a name="l02968"></a>02968 <span class="comment">!                    and ier = ierlst(1) (with meaning as described</span>
    3033. 

  3033. <a name="l02969"></a>02969 <span class="comment">!                    for ierlst(k), k = 1).</span>
    3034. 

  3034. <a name="l02970"></a>02970 <span class="comment">!</span>
    3035. 

  3035. <a name="l02971"></a>02971 <span class="comment">!            rslst  - real</span>
    3036. 

  3036. <a name="l02972"></a>02972 <span class="comment">!                     vector of dimension at least limlst</span>
    3037. 

  3037. <a name="l02973"></a>02973 <span class="comment">!                     rslst(k) contains the integral contribution</span>
    3038. 

  3038. <a name="l02974"></a>02974 <span class="comment">!                     over the interval (a+(k-1)c,a+kc) where</span>
    3039. 

  3039. <a name="l02975"></a>02975 <span class="comment">!                     c = (2*int(abs(omega))+1)*pi/abs(omega),</span>
    3040. 

  3040. <a name="l02976"></a>02976 <span class="comment">!                     k = 1, 2, ..., lst.</span>
    3041. 

  3041. <a name="l02977"></a>02977 <span class="comment">!                     note that, if omega = 0, rslst(1) contains</span>
    3042. 

  3042. <a name="l02978"></a>02978 <span class="comment">!                     the value of the integral over (a,infinity).</span>
    3043. 

  3043. <a name="l02979"></a>02979 <span class="comment">!</span>
    3044. 

  3044. <a name="l02980"></a>02980 <span class="comment">!            erlst  - real</span>
    3045. 

  3045. <a name="l02981"></a>02981 <span class="comment">!                     vector of dimension at least limlst</span>
    3046. 

  3046. <a name="l02982"></a>02982 <span class="comment">!                     erlst(k) contains the error estimate</span>
    3047. 

  3047. <a name="l02983"></a>02983 <span class="comment">!                     corresponding with rslst(k).</span>
    3048. 

  3048. <a name="l02984"></a>02984 <span class="comment">!</span>
    3049. 

  3049. <a name="l02985"></a>02985 <span class="comment">!            ierlst - integer</span>
    3050. 

  3050. <a name="l02986"></a>02986 <span class="comment">!                     vector of dimension at least limlst</span>
    3051. 

  3051. <a name="l02987"></a>02987 <span class="comment">!                     ierlst(k) contains the error flag corresponding</span>
    3052. 

  3052. <a name="l02988"></a>02988 <span class="comment">!                     with rslst(k). for the meaning of the local error</span>
    3053. 

  3053. <a name="l02989"></a>02989 <span class="comment">!                     flags see description of output parameter ier.</span>
    3054. 

  3054. <a name="l02990"></a>02990 <span class="comment">!</span>
    3055. 

  3055. <a name="l02991"></a>02991 <span class="comment">!            lst    - integer</span>
    3056. 

  3056. <a name="l02992"></a>02992 <span class="comment">!                     number of subintervals needed for the integration</span>
    3057. 

  3057. <a name="l02993"></a>02993 <span class="comment">!                     if omega = 0 then lst is set to 1.</span>
    3058. 

  3058. <a name="l02994"></a>02994 <span class="comment">!</span>
    3059. 

  3059. <a name="l02995"></a>02995 <span class="comment">!            alist, blist, rlist, elist - real</span>
    3060. 

  3060. <a name="l02996"></a>02996 <span class="comment">!                     vector of dimension at least limit,</span>
    3061. 

  3061. <a name="l02997"></a>02997 <span class="comment">!</span>
    3062. 

  3062. <a name="l02998"></a>02998 <span class="comment">!            iord, nnlog - integer</span>
    3063. 

  3063. <a name="l02999"></a>02999 <span class="comment">!                     vector of dimension at least limit, providing</span>
    3064. 

  3064. <a name="l03000"></a>03000 <span class="comment">!                     space for the quantities needed in the</span>
    3065. 

  3065. <a name="l03001"></a>03001 <span class="comment">!                     subdivision process of each cycle</span>
    3066. 

  3066. <a name="l03002"></a>03002 <span class="comment">!</span>
    3067. 

  3067. <a name="l03003"></a>03003 <span class="comment">!            chebmo - real</span>
    3068. 

  3068. <a name="l03004"></a>03004 <span class="comment">!                     array of dimension at least (maxp1,25),</span>
    3069. 

  3069. <a name="l03005"></a>03005 <span class="comment">!                     providing space for the Chebyshev moments</span>
    3070. 

  3070. <a name="l03006"></a>03006 <span class="comment">!                     needed within the cycles</span>
    3071. 

  3071. <a name="l03007"></a>03007 <span class="comment">!</span>
    3072. 

  3072. <a name="l03008"></a>03008 <span class="comment">!  Local parameters:</span>
    3073. 

  3073. <a name="l03009"></a>03009 <span class="comment">!</span>
    3074. 

  3074. <a name="l03010"></a>03010 <span class="comment">!           c1, c2    - end points of subinterval (of length</span>
    3075. 

  3075. <a name="l03011"></a>03011 <span class="comment">!                       cycle)</span>
    3076. 

  3076. <a name="l03012"></a>03012 <span class="comment">!           cycle     - (2*int(abs(omega))+1)*pi/abs(omega)</span>
    3077. 

  3077. <a name="l03013"></a>03013 <span class="comment">!           psum      - vector of dimension at least (limexp+2)</span>
    3078. 

  3078. <a name="l03014"></a>03014 <span class="comment">!                       (see routine qextr)</span>
    3079. 

  3079. <a name="l03015"></a>03015 <span class="comment">!                       psum contains the part of the epsilon table</span>
    3080. 

  3080. <a name="l03016"></a>03016 <span class="comment">!                       which is still needed for further computations.</span>
    3081. 

  3081. <a name="l03017"></a>03017 <span class="comment">!                       each element of psum is a partial sum of</span>
    3082. 

  3082. <a name="l03018"></a>03018 <span class="comment">!                       the series which should sum to the value of</span>
    3083. 

  3083. <a name="l03019"></a>03019 <span class="comment">!                       the integral.</span>
    3084. 

  3084. <a name="l03020"></a>03020 <span class="comment">!           errsum    - sum of error estimates over the</span>
    3085. 

  3085. <a name="l03021"></a>03021 <span class="comment">!                       subintervals, calculated cumulatively</span>
    3086. 

  3086. <a name="l03022"></a>03022 <span class="comment">!           epsa      - absolute tolerance requested over current</span>
    3087. 

  3087. <a name="l03023"></a>03023 <span class="comment">!                       subinterval</span>
    3088. 

  3088. <a name="l03024"></a>03024 <span class="comment">!           chebmo    - array containing the modified Chebyshev</span>
    3089. 

  3089. <a name="l03025"></a>03025 <span class="comment">!                       moments (see also routine qc25o)</span>
    3090. 

  3090. <a name="l03026"></a>03026 <span class="comment">!</span>
    3091. 

  3091. <a name="l03027"></a>03027   <span class="keyword">implicit none</span>
    3092. 

  3092. <a name="l03028"></a>03028 
    3093. 

  3093. <a name="l03029"></a>03029   <span class="keywordtype">integer</span> limit
    3094. 

  3094. <a name="l03030"></a>03030   <span class="keywordtype">integer</span> limlst
    3095. 

  3095. <a name="l03031"></a>03031   <span class="keywordtype">integer</span> maxp1
    3096. 

  3096. <a name="l03032"></a>03032 
    3097. 

  3097. <a name="l03033"></a>03033   <span class="keywordtype">real</span> a
    3098. 

  3098. <a name="l03034"></a>03034   <span class="keywordtype">real</span> abseps
    3099. 

  3099. <a name="l03035"></a>03035   <span class="keywordtype">real</span> abserr
    3100. 

  3100. <a name="l03036"></a>03036   <span class="keywordtype">real</span> alist(limit)
    3101. 

  3101. <a name="l03037"></a>03037   <span class="keywordtype">real</span> blist(limit)
    3102. 

  3102. <a name="l03038"></a>03038   <span class="keywordtype">real</span> chebmo(maxp1,25)
    3103. 

  3103. <a name="l03039"></a>03039   <span class="keywordtype">real</span> correc
    3104. 

  3104. <a name="l03040"></a>03040   <span class="keywordtype">real</span> cycle
    3105. 

  3105. <a name="l03041"></a>03041   <span class="keywordtype">real</span> c1
    3106. 

  3106. <a name="l03042"></a>03042   <span class="keywordtype">real</span> c2
    3107. 

  3107. <a name="l03043"></a>03043   <span class="keywordtype">real</span> dl
    3108. 

  3108. <a name="l03044"></a>03044 <span class="comment">! real dla</span>
    3109. 

  3109. <a name="l03045"></a>03045   <span class="keywordtype">real</span> drl
    3110. 

  3110. <a name="l03046"></a>03046   <span class="keywordtype">real</span> elist(limit)
    3111. 

  3111. <a name="l03047"></a>03047   <span class="keywordtype">real</span> ep
    3112. 

  3112. <a name="l03048"></a>03048   <span class="keywordtype">real</span> eps
    3113. 

  3113. <a name="l03049"></a>03049   <span class="keywordtype">real</span> epsa
    3114. 

  3114. <a name="l03050"></a>03050   <span class="keywordtype">real</span> epsabs
    3115. 

  3115. <a name="l03051"></a>03051   <span class="keywordtype">real</span> erlst(limlst)
    3116. 

  3116. <a name="l03052"></a>03052   <span class="keywordtype">real</span> errsum
    3117. 

  3117. <a name="l03053"></a>03053   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    3118. 

  3118. <a name="l03054"></a>03054   <span class="keywordtype">real</span> fact
    3119. 

  3119. <a name="l03055"></a>03055   <span class="keywordtype">integer</span> ier
    3120. 

  3120. <a name="l03056"></a>03056   <span class="keywordtype">integer</span> ierlst(limlst)
    3121. 

  3121. <a name="l03057"></a>03057   <span class="keywordtype">integer</span> integr
    3122. 

  3122. <a name="l03058"></a>03058   <span class="keywordtype">integer</span> iord(limit)
    3123. 

  3123. <a name="l03059"></a>03059   <span class="keywordtype">integer</span> ktmin
    3124. 

  3124. <a name="l03060"></a>03060   <span class="keywordtype">integer</span> l
    3125. 

  3125. <a name="l03061"></a>03061   <span class="keywordtype">integer</span> ll
    3126. 

  3126. <a name="l03062"></a>03062   <span class="keywordtype">integer</span> lst
    3127. 

  3127. <a name="l03063"></a>03063   <span class="keywordtype">integer</span> momcom
    3128. 

  3128. <a name="l03064"></a>03064   <span class="keywordtype">integer</span> nev
    3129. 

  3129. <a name="l03065"></a>03065   <span class="keywordtype">integer</span> neval
    3130. 

  3130. <a name="l03066"></a>03066   <span class="keywordtype">integer</span> nnlog(limit)
    3131. 

  3131. <a name="l03067"></a>03067   <span class="keywordtype">integer</span> nres
    3132. 

  3132. <a name="l03068"></a>03068   <span class="keywordtype">integer</span> numrl2
    3133. 

  3133. <a name="l03069"></a>03069   <span class="keywordtype">real</span> omega
    3134. 

  3134. <a name="l03070"></a>03070   <span class="keywordtype">real</span>, <span class="keywordtype">parameter</span> :: p = 0.9E+00
    3135. 

  3135. <a name="l03071"></a>03071   <span class="keywordtype">real</span>, <span class="keywordtype">parameter</span> :: pi = 3.1415926535897932E+00
    3136. 

  3136. <a name="l03072"></a>03072   <span class="keywordtype">real</span> p1
    3137. 

  3137. <a name="l03073"></a>03073   <span class="keywordtype">real</span> psum(52)
    3138. 

  3138. <a name="l03074"></a>03074   <span class="keywordtype">real</span> reseps
    3139. 

  3139. <a name="l03075"></a>03075   <span class="keywordtype">real</span> result
    3140. 

  3140. <a name="l03076"></a>03076   <span class="keywordtype">real</span> res3la(3)
    3141. 

  3141. <a name="l03077"></a>03077   <span class="keywordtype">real</span> rlist(limit)
    3142. 

  3142. <a name="l03078"></a>03078   <span class="keywordtype">real</span> rslst(limlst)
    3143. 

  3143. <a name="l03079"></a>03079 <span class="comment">!</span>
    3144. 

  3144. <a name="l03080"></a>03080 <span class="comment">!  The dimension of  psum  is determined by the value of</span>
    3145. 

  3145. <a name="l03081"></a>03081 <span class="comment">!  limexp in QEXTR (psum must be</span>
    3146. 

  3146. <a name="l03082"></a>03082 <span class="comment">!  of dimension (limexp+2) at least).</span>
    3147. 

  3147. <a name="l03083"></a>03083 <span class="comment">!</span>
    3148. 

  3148. <a name="l03084"></a>03084 <span class="comment">!  Test on validity of parameters.</span>
    3149. 

  3149. <a name="l03085"></a>03085 <span class="comment">!</span>
    3150. 

  3150. <a name="l03086"></a>03086   result = 0.0e+00
    3151. 

  3151. <a name="l03087"></a>03087   abserr = 0.0e+00
    3152. 

  3152. <a name="l03088"></a>03088   neval = 0
    3153. 

  3153. <a name="l03089"></a>03089   lst = 0
    3154. 

  3154. <a name="l03090"></a>03090   ier = 0
    3155. 

  3155. <a name="l03091"></a>03091 
    3156. 

  3156. <a name="l03092"></a>03092   <span class="keyword">if</span> ( (integr /= 1 .and. integr /= 2 ) .or. &amp;
    3157. 

  3157. <a name="l03093"></a>03093     epsabs &lt;= 0.0e+00 .or. &amp;
    3158. 

  3158. <a name="l03094"></a>03094     limlst &lt; 3 ) <span class="keyword">then</span>
    3159. 

  3159. <a name="l03095"></a>03095     ier = 6
    3160. 

  3160. <a name="l03096"></a>03096     return
    3161. 

  3161. <a name="l03097"></a>03097   <span class="keyword">end if</span>
    3162. 

  3162. <a name="l03098"></a>03098 
    3163. 

  3163. <a name="l03099"></a>03099   <span class="keyword">if</span> ( omega == 0.0e+00 ) <span class="keyword">then</span>
    3164. 

  3164. <a name="l03100"></a>03100 
    3165. 

  3165. <a name="l03101"></a>03101     <span class="keyword">if</span> ( integr == 1 ) <span class="keyword">then</span>
    3166. 

  3166. <a name="l03102"></a>03102       call <a class="code" href="quadpack_8f90.html#ac59eaf7c56c1d421d129425895fa0107">qagi </a>( f, 0.0e+00, 1, epsabs, 0.0e+00, result, abserr, neval, ier )
    3167. 

  3167. <a name="l03103"></a>03103     <span class="keyword">else</span>
    3168. 

  3168. <a name="l03104"></a>03104       result = 0.0E+00
    3169. 

  3169. <a name="l03105"></a>03105       abserr = 0.0E+00
    3170. 

  3170. <a name="l03106"></a>03106       neval = 0
    3171. 

  3171. <a name="l03107"></a>03107       ier = 0
    3172. 

  3172. <a name="l03108"></a>03108     <span class="keyword">end if</span>
    3173. 

  3173. <a name="l03109"></a>03109 
    3174. 

  3174. <a name="l03110"></a>03110     rslst(1) = result
    3175. 

  3175. <a name="l03111"></a>03111     erlst(1) = abserr
    3176. 

  3176. <a name="l03112"></a>03112     ierlst(1) = ier
    3177. 

  3177. <a name="l03113"></a>03113     lst = 1
    3178. 

  3178. <a name="l03114"></a>03114 
    3179. 

  3179. <a name="l03115"></a>03115     return
    3180. 

  3180. <a name="l03116"></a>03116   <span class="keyword">end if</span>
    3181. 

  3181. <a name="l03117"></a>03117 <span class="comment">!</span>
    3182. 

  3182. <a name="l03118"></a>03118 <span class="comment">!  Initializations.</span>
    3183. 

  3183. <a name="l03119"></a>03119 <span class="comment">!</span>
    3184. 

  3184. <a name="l03120"></a>03120   l = int ( abs ( omega ) )
    3185. 

  3185. <a name="l03121"></a>03121   dl = 2 * l + 1
    3186. 

  3186. <a name="l03122"></a>03122   cycle = dl * pi / abs ( omega )
    3187. 

  3187. <a name="l03123"></a>03123   ier = 0
    3188. 

  3188. <a name="l03124"></a>03124   ktmin = 0
    3189. 

  3189. <a name="l03125"></a>03125   neval = 0
    3190. 

  3190. <a name="l03126"></a>03126   numrl2 = 0
    3191. 

  3191. <a name="l03127"></a>03127   nres = 0
    3192. 

  3192. <a name="l03128"></a>03128   c1 = a
    3193. 

  3193. <a name="l03129"></a>03129   c2 = cycle+a
    3194. 

  3194. <a name="l03130"></a>03130   p1 = 1.0e+00-p
    3195. 

  3195. <a name="l03131"></a>03131   eps = epsabs
    3196. 

  3196. <a name="l03132"></a>03132 
    3197. 

  3197. <a name="l03133"></a>03133   <span class="keyword">if</span> ( epsabs &gt; tiny ( epsabs ) / p1 ) <span class="keyword">then</span>
    3198. 

  3198. <a name="l03134"></a>03134     eps = epsabs * p1
    3199. 

  3199. <a name="l03135"></a>03135   <span class="keyword">end if</span>
    3200. 

  3200. <a name="l03136"></a>03136 
    3201. 

  3201. <a name="l03137"></a>03137   ep = eps
    3202. 

  3202. <a name="l03138"></a>03138   fact = 1.0e+00
    3203. 

  3203. <a name="l03139"></a>03139   correc = 0.0e+00
    3204. 

  3204. <a name="l03140"></a>03140   abserr = 0.0e+00
    3205. 

  3205. <a name="l03141"></a>03141   errsum = 0.0e+00
    3206. 

  3206. <a name="l03142"></a>03142 
    3207. 

  3207. <a name="l03143"></a>03143   <span class="keyword">do</span> lst = 1, limlst
    3208. 

  3208. <a name="l03144"></a>03144 <span class="comment">!</span>
    3209. 

  3209. <a name="l03145"></a>03145 <span class="comment">!  Integrate over current subinterval.</span>
    3210. 

  3210. <a name="l03146"></a>03146 <span class="comment">!</span>
    3211. 

  3211. <a name="l03147"></a>03147 <span class="comment">!   dla = lst</span>
    3212. 

  3212. <a name="l03148"></a>03148     epsa = eps * fact
    3213. 

  3213. <a name="l03149"></a>03149 
    3214. 

  3214. <a name="l03150"></a>03150     call <a class="code" href="quadpack_8f90.html#abbba06307e0e8c4daa2651945570ba1c">qfour </a>( f, c1, c2, omega, integr, epsa, 0.0e+00, limit, lst, maxp1, &amp;
    3215. 

  3215. <a name="l03151"></a>03151       rslst(lst), erlst(lst), nev, ierlst(lst), alist, blist, rlist, elist, &amp;
    3216. 

  3216. <a name="l03152"></a>03152       iord, nnlog, momcom, chebmo )
    3217. 

  3217. <a name="l03153"></a>03153 
    3218. 

  3218. <a name="l03154"></a>03154     neval = neval + nev
    3219. 

  3219. <a name="l03155"></a>03155     fact = fact * p
    3220. 

  3220. <a name="l03156"></a>03156     errsum = errsum + erlst(lst)
    3221. 

  3221. <a name="l03157"></a>03157     drl = 5.0e+01 * abs(rslst(lst))
    3222. 

  3222. <a name="l03158"></a>03158 <span class="comment">!</span>
    3223. 

  3223. <a name="l03159"></a>03159 <span class="comment">!  Test on accuracy with partial sum.</span>
    3224. 

  3224. <a name="l03160"></a>03160 <span class="comment">!</span>
    3225. 

  3225. <a name="l03161"></a>03161     <span class="keyword">if</span> ((errsum+drl) &lt;= epsabs.and.lst &gt;= 6) <span class="keyword">then</span>
    3226. 

  3226. <a name="l03162"></a>03162       go to 80
    3227. 

  3227. <a name="l03163"></a>03163     <span class="keyword">end if</span>
    3228. 

  3228. <a name="l03164"></a>03164 
    3229. 

  3229. <a name="l03165"></a>03165     correc = max ( correc,erlst(lst))
    3230. 

  3230. <a name="l03166"></a>03166 
    3231. 

  3231. <a name="l03167"></a>03167     <span class="keyword">if</span> ( ierlst(lst) /= 0 ) <span class="keyword">then</span>
    3232. 

  3232. <a name="l03168"></a>03168       eps = max ( ep,correc*p1)
    3233. 

  3233. <a name="l03169"></a>03169       ier = 7
    3234. 

  3234. <a name="l03170"></a>03170     <span class="keyword">end if</span>
    3235. 

  3235. <a name="l03171"></a>03171 
    3236. 

  3236. <a name="l03172"></a>03172     <span class="keyword">if</span> ( ier == 7 .and. (errsum+drl) &lt;= correc*1.0e+01.and. lst &gt; 5) go to 80
    3237. 

  3237. <a name="l03173"></a>03173 
    3238. 

  3238. <a name="l03174"></a>03174     numrl2 = numrl2+1
    3239. 

  3239. <a name="l03175"></a>03175 
    3240. 

  3240. <a name="l03176"></a>03176     <span class="keyword">if</span> ( lst &lt;= 1 ) <span class="keyword">then</span>
    3241. 

  3241. <a name="l03177"></a>03177       psum(1) = rslst(1)
    3242. 

  3242. <a name="l03178"></a>03178       go to 40
    3243. 

  3243. <a name="l03179"></a>03179     <span class="keyword">end if</span>
    3244. 

  3244. <a name="l03180"></a>03180 
    3245. 

  3245. <a name="l03181"></a>03181     psum(numrl2) = psum(ll) + rslst(lst)
    3246. 

  3246. <a name="l03182"></a>03182 
    3247. 

  3247. <a name="l03183"></a>03183     <span class="keyword">if</span> ( lst == 2 ) <span class="keyword">then</span>
    3248. 

  3248. <a name="l03184"></a>03184       go to 40
    3249. 

  3249. <a name="l03185"></a>03185     <span class="keyword">end if</span>
    3250. 

  3250. <a name="l03186"></a>03186 <span class="comment">!</span>
    3251. 

  3251. <a name="l03187"></a>03187 <span class="comment">!  Test on maximum number of subintervals</span>
    3252. 

  3252. <a name="l03188"></a>03188 <span class="comment">!</span>
    3253. 

  3253. <a name="l03189"></a>03189     <span class="keyword">if</span> ( lst == limlst ) <span class="keyword">then</span>
    3254. 

  3254. <a name="l03190"></a>03190       ier = 8
    3255. 

  3255. <a name="l03191"></a>03191     <span class="keyword">end if</span>
    3256. 

  3256. <a name="l03192"></a>03192 <span class="comment">!</span>
    3257. 

  3257. <a name="l03193"></a>03193 <span class="comment">!  Perform new extrapolation</span>
    3258. 

  3258. <a name="l03194"></a>03194 <span class="comment">!</span>
    3259. 

  3259. <a name="l03195"></a>03195     call <a class="code" href="quadpack_8f90.html#a5a75101d080f224c63adde98a0e64386">qextr </a>( numrl2, psum, reseps, abseps, res3la, nres )
    3260. 

  3260. <a name="l03196"></a>03196 <span class="comment">!</span>
    3261. 

  3261. <a name="l03197"></a>03197 <span class="comment">!  Test whether extrapolated result is influenced by roundoff</span>
    3262. 

  3262. <a name="l03198"></a>03198 <span class="comment">!</span>
    3263. 

  3263. <a name="l03199"></a>03199     ktmin = ktmin + 1
    3264. 

  3264. <a name="l03200"></a>03200 
    3265. 

  3265. <a name="l03201"></a>03201     <span class="keyword">if</span> ( ktmin &gt;= 15 .and. abserr &lt;= 1.0e-03 * (errsum+drl) ) <span class="keyword">then</span>
    3266. 

  3266. <a name="l03202"></a>03202       ier = 9
    3267. 

  3267. <a name="l03203"></a>03203     <span class="keyword">end  if</span>
    3268. 

  3268. <a name="l03204"></a>03204 
    3269. 

  3269. <a name="l03205"></a>03205     <span class="keyword">if</span> ( abseps &lt;= abserr .or. lst == 3 ) <span class="keyword">then</span>
    3270. 

  3270. <a name="l03206"></a>03206 
    3271. 

  3271. <a name="l03207"></a>03207       abserr = abseps
    3272. 

  3272. <a name="l03208"></a>03208       result = reseps
    3273. 

  3273. <a name="l03209"></a>03209       ktmin = 0
    3274. 

  3274. <a name="l03210"></a>03210 <span class="comment">!</span>
    3275. 

  3275. <a name="l03211"></a>03211 <span class="comment">!  If IER is not 0, check whether direct result (partial</span>
    3276. 

  3276. <a name="l03212"></a>03212 <span class="comment">!  sum) or extrapolated result yields the best integral</span>
    3277. 

  3277. <a name="l03213"></a>03213 <span class="comment">!  approximation</span>
    3278. 

  3278. <a name="l03214"></a>03214 <span class="comment">!</span>
    3279. 

  3279. <a name="l03215"></a>03215       <span class="keyword">if</span> ( ( abserr + 1.0e+01 * correc ) &lt;= epsabs ) <span class="keyword">then</span>
    3280. 

  3280. <a name="l03216"></a>03216         exit
    3281. 

  3281. <a name="l03217"></a>03217       <span class="keyword">end if</span>
    3282. 

  3282. <a name="l03218"></a>03218 
    3283. 

  3283. <a name="l03219"></a>03219       <span class="keyword">if</span> ( abserr &lt;= epsabs .and. 1.0e+01 * correc &gt;= epsabs ) <span class="keyword">then</span>
    3284. 

  3284. <a name="l03220"></a>03220         exit
    3285. 

  3285. <a name="l03221"></a>03221       <span class="keyword">end if</span>
    3286. 

  3286. <a name="l03222"></a>03222 
    3287. 

  3287. <a name="l03223"></a>03223     <span class="keyword">end if</span>
    3288. 

  3288. <a name="l03224"></a>03224 
    3289. 

  3289. <a name="l03225"></a>03225     <span class="keyword">if</span> ( ier /= 0 .and. ier /= 7 ) <span class="keyword">then</span>
    3290. 

  3290. <a name="l03226"></a>03226       exit
    3291. 

  3291. <a name="l03227"></a>03227     <span class="keyword">end if</span>
    3292. 

  3292. <a name="l03228"></a>03228 
    3293. 

  3293. <a name="l03229"></a>03229 40  continue
    3294. 

  3294. <a name="l03230"></a>03230 
    3295. 

  3295. <a name="l03231"></a>03231     ll = numrl2
    3296. 

  3296. <a name="l03232"></a>03232     c1 = c2
    3297. 

  3297. <a name="l03233"></a>03233     c2 = c2+cycle
    3298. 

  3298. <a name="l03234"></a>03234 
    3299. 

  3299. <a name="l03235"></a>03235   <span class="keyword">end do</span>
    3300. 

  3300. <a name="l03236"></a>03236 <span class="comment">!</span>
    3301. 

  3301. <a name="l03237"></a>03237 <span class="comment">!  Set final result and error estimate.</span>
    3302. 

  3302. <a name="l03238"></a>03238 <span class="comment">!</span>
    3303. 

  3303. <a name="l03239"></a>03239 <span class="comment">!60 continue</span>
    3304. 

  3304. <a name="l03240"></a>03240 
    3305. 

  3305. <a name="l03241"></a>03241   abserr = abserr + 1.0e+01 * correc
    3306. 

  3306. <a name="l03242"></a>03242 
    3307. 

  3307. <a name="l03243"></a>03243   <span class="keyword">if</span> ( ier == 0 ) <span class="keyword">then</span>
    3308. 

  3308. <a name="l03244"></a>03244     return
    3309. 

  3309. <a name="l03245"></a>03245   <span class="keyword">end if</span>
    3310. 

  3310. <a name="l03246"></a>03246 
    3311. 

  3311. <a name="l03247"></a>03247   <span class="keyword">if</span> ( result /= 0.0e+00 .and. psum(numrl2) /= 0.0e+00) go to 70
    3312. 

  3312. <a name="l03248"></a>03248 
    3313. 

  3313. <a name="l03249"></a>03249   <span class="keyword">if</span> ( abserr &gt; errsum ) <span class="keyword">then</span>
    3314. 

  3314. <a name="l03250"></a>03250     go to 80
    3315. 

  3315. <a name="l03251"></a>03251   <span class="keyword">end if</span>
    3316. 

  3316. <a name="l03252"></a>03252 
    3317. 

  3317. <a name="l03253"></a>03253   <span class="keyword">if</span> ( psum(numrl2) == 0.0e+00 ) <span class="keyword">then</span>
    3318. 

  3318. <a name="l03254"></a>03254     return
    3319. 

  3319. <a name="l03255"></a>03255   <span class="keyword">end if</span>
    3320. 

  3320. <a name="l03256"></a>03256 
    3321. 

  3321. <a name="l03257"></a>03257 70 continue
    3322. 

  3322. <a name="l03258"></a>03258 
    3323. 

  3323. <a name="l03259"></a>03259   <span class="keyword">if</span> ( abserr / abs(result) &lt;= (errsum+drl)/abs(psum(numrl2)) ) <span class="keyword">then</span>
    3324. 

  3324. <a name="l03260"></a>03260 
    3325. 

  3325. <a name="l03261"></a>03261     <span class="keyword">if</span> ( ier &gt;= 1 .and. ier /= 7 ) <span class="keyword">then</span>
    3326. 

  3326. <a name="l03262"></a>03262       abserr = abserr + drl
    3327. 

  3327. <a name="l03263"></a>03263     <span class="keyword">end if</span>
    3328. 

  3328. <a name="l03264"></a>03264 
    3329. 

  3329. <a name="l03265"></a>03265     return
    3330. 

  3330. <a name="l03266"></a>03266 
    3331. 

  3331. <a name="l03267"></a>03267   <span class="keyword">end if</span>
    3332. 

  3332. <a name="l03268"></a>03268 
    3333. 

  3333. <a name="l03269"></a>03269 80 continue
    3334. 

  3334. <a name="l03270"></a>03270 
    3335. 

  3335. <a name="l03271"></a>03271   result = psum(numrl2)
    3336. 

  3336. <a name="l03272"></a>03272   abserr = errsum + drl
    3337. 

  3337. <a name="l03273"></a>03273 
    3338. 

  3338. <a name="l03274"></a>03274   return
    3339. 

  3339. <a name="l03275"></a>03275 <span class="keyword">end</span>
    3340. 

  3340. <a name="l03276"></a><a class="code" href="quadpack_8f90.html#aaa4f15baf0dadd3383219f0d42a62752">03276</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#aaa4f15baf0dadd3383219f0d42a62752">qawo</a> ( f, a, b, omega, integr, epsabs, epsrel, result, abserr, &amp;
    3341. 

  3341. <a name="l03277"></a>03277   neval, ier )
    3342. 

  3342. <a name="l03278"></a>03278 
    3343. 

  3343. <a name="l03279"></a>03279 <span class="comment">!*****************************************************************************80</span>
    3344. 

  3344. <a name="l03280"></a>03280 <span class="comment">!</span>
    3345. 

  3345. <a name="l03281"></a>03281 <span class="comment">!! QAWO computes the integrals of oscillatory integrands.</span>
    3346. 

  3346. <a name="l03282"></a>03282 <span class="comment">!</span>
    3347. 

  3347. <a name="l03283"></a>03283 <span class="comment">!  Discussion:</span>
    3348. 

  3348. <a name="l03284"></a>03284 <span class="comment">!</span>
    3349. 

  3349. <a name="l03285"></a>03285 <span class="comment">!    The routine calculates an approximation RESULT to a given</span>
    3350. 

  3350. <a name="l03286"></a>03286 <span class="comment">!    definite integral</span>
    3351. 

  3351. <a name="l03287"></a>03287 <span class="comment">!      I = Integral ( A &lt;= X &lt;= B ) F(X) * cos ( OMEGA * X ) dx</span>
    3352. 

  3352. <a name="l03288"></a>03288 <span class="comment">!    or </span>
    3353. 

  3353. <a name="l03289"></a>03289 <span class="comment">!      I = Integral ( A &lt;= X &lt;= B ) F(X) * sin ( OMEGA * X ) dx</span>
    3354. 

  3354. <a name="l03290"></a>03290 <span class="comment">!    hopefully satisfying following claim for accuracy</span>
    3355. 

  3355. <a name="l03291"></a>03291 <span class="comment">!      | I - RESULT | &lt;= max ( epsabs, epsrel * |I| ).</span>
    3356. 

  3356. <a name="l03292"></a>03292 <span class="comment">!</span>
    3357. 

  3357. <a name="l03293"></a>03293 <span class="comment">!  Author:</span>
    3358. 

  3358. <a name="l03294"></a>03294 <span class="comment">!</span>
    3359. 

  3359. <a name="l03295"></a>03295 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    3360. 

  3360. <a name="l03296"></a>03296 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    3361. 

  3361. <a name="l03297"></a>03297 <span class="comment">!</span>
    3362. 

  3362. <a name="l03298"></a>03298 <span class="comment">!  Reference:</span>
    3363. 

  3363. <a name="l03299"></a>03299 <span class="comment">!</span>
    3364. 

  3364. <a name="l03300"></a>03300 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    3365. 

  3365. <a name="l03301"></a>03301 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    3366. 

  3366. <a name="l03302"></a>03302 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    3367. 

  3367. <a name="l03303"></a>03303 <span class="comment">!    Springer Verlag, 1983</span>
    3368. 

  3368. <a name="l03304"></a>03304 <span class="comment">!</span>
    3369. 

  3369. <a name="l03305"></a>03305 <span class="comment">!  Parameters:</span>
    3370. 

  3370. <a name="l03306"></a>03306 <span class="comment">!</span>
    3371. 

  3371. <a name="l03307"></a>03307 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    3372. 

  3372. <a name="l03308"></a>03308 <span class="comment">!      function f ( x )</span>
    3373. 

  3373. <a name="l03309"></a>03309 <span class="comment">!      real f</span>
    3374. 

  3374. <a name="l03310"></a>03310 <span class="comment">!      real x</span>
    3375. 

  3375. <a name="l03311"></a>03311 <span class="comment">!    which evaluates the integrand function.</span>
    3376. 

  3376. <a name="l03312"></a>03312 <span class="comment">!</span>
    3377. 

  3377. <a name="l03313"></a>03313 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    3378. 

  3378. <a name="l03314"></a>03314 <span class="comment">!</span>
    3379. 

  3379. <a name="l03315"></a>03315 <span class="comment">!    Input, real OMEGA, the parameter in the weight function.</span>
    3380. 

  3380. <a name="l03316"></a>03316 <span class="comment">!</span>
    3381. 

  3381. <a name="l03317"></a>03317 <span class="comment">!    Input, integer INTEGR, specifies the weight function:</span>
    3382. 

  3382. <a name="l03318"></a>03318 <span class="comment">!    1, W(X) = cos ( OMEGA * X )</span>
    3383. 

  3383. <a name="l03319"></a>03319 <span class="comment">!    2, W(X) = sin ( OMEGA * X )</span>
    3384. 

  3384. <a name="l03320"></a>03320 <span class="comment">!</span>
    3385. 

  3385. <a name="l03321"></a>03321 <span class="comment">!    Input, real EPSABS, EPSREL, the absolute and relative accuracy requested.</span>
    3386. 

  3386. <a name="l03322"></a>03322 <span class="comment">!</span>
    3387. 

  3387. <a name="l03323"></a>03323 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    3388. 

  3388. <a name="l03324"></a>03324 <span class="comment">!</span>
    3389. 

  3389. <a name="l03325"></a>03325 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    3390. 

  3390. <a name="l03326"></a>03326 <span class="comment">!</span>
    3391. 

  3391. <a name="l03327"></a>03327 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    3392. 

  3392. <a name="l03328"></a>03328 <span class="comment">!</span>
    3393. 

  3393. <a name="l03329"></a>03329 <span class="comment">!            ier    - integer</span>
    3394. 

  3394. <a name="l03330"></a>03330 <span class="comment">!                     ier = 0 normal and reliable termination of the</span>
    3395. 

  3395. <a name="l03331"></a>03331 <span class="comment">!                             routine. it is assumed that the</span>
    3396. 

  3396. <a name="l03332"></a>03332 <span class="comment">!                             requested accuracy has been achieved.</span>
    3397. 

  3397. <a name="l03333"></a>03333 <span class="comment">!                   - ier &gt; 0 abnormal termination of the routine.</span>
    3398. 

  3398. <a name="l03334"></a>03334 <span class="comment">!                             the estimates for integral and error are</span>
    3399. 

  3399. <a name="l03335"></a>03335 <span class="comment">!                             less reliable. it is assumed that the</span>
    3400. 

  3400. <a name="l03336"></a>03336 <span class="comment">!                             requested accuracy has not been achieved.</span>
    3401. 

  3401. <a name="l03337"></a>03337 <span class="comment">!                     ier = 1 maximum number of subdivisions allowed</span>
    3402. 

  3402. <a name="l03338"></a>03338 <span class="comment">!                             (= leniw/2) has been achieved. one can</span>
    3403. 

  3403. <a name="l03339"></a>03339 <span class="comment">!                             allow more subdivisions by increasing the</span>
    3404. 

  3404. <a name="l03340"></a>03340 <span class="comment">!                             value of leniw (and taking the according</span>
    3405. 

  3405. <a name="l03341"></a>03341 <span class="comment">!                             dimension adjustments into account).</span>
    3406. 

  3406. <a name="l03342"></a>03342 <span class="comment">!                             however, if this yields no improvement it</span>
    3407. 

  3407. <a name="l03343"></a>03343 <span class="comment">!                             is advised to analyze the integrand in</span>
    3408. 

  3408. <a name="l03344"></a>03344 <span class="comment">!                             order to determine the integration</span>
    3409. 

  3409. <a name="l03345"></a>03345 <span class="comment">!                             difficulties. if the position of a local</span>
    3410. 

  3410. <a name="l03346"></a>03346 <span class="comment">!                             difficulty can be determined (e.g.</span>
    3411. 

  3411. <a name="l03347"></a>03347 <span class="comment">!                             singularity, discontinuity within the</span>
    3412. 

  3412. <a name="l03348"></a>03348 <span class="comment">!                             interval) one will probably gain from</span>
    3413. 

  3413. <a name="l03349"></a>03349 <span class="comment">!                             splitting up the interval at this point</span>
    3414. 

  3414. <a name="l03350"></a>03350 <span class="comment">!                             and calling the integrator on the</span>
    3415. 

  3415. <a name="l03351"></a>03351 <span class="comment">!                             subranges. if possible, an appropriate</span>
    3416. 

  3416. <a name="l03352"></a>03352 <span class="comment">!                             special-purpose integrator should</span>
    3417. 

  3417. <a name="l03353"></a>03353 <span class="comment">!                             be used which is designed for handling</span>
    3418. 

  3418. <a name="l03354"></a>03354 <span class="comment">!                             the type of difficulty involved.</span>
    3419. 

  3419. <a name="l03355"></a>03355 <span class="comment">!                         = 2 the occurrence of roundoff error is</span>
    3420. 

  3420. <a name="l03356"></a>03356 <span class="comment">!                             detected, which prevents the requested</span>
    3421. 

  3421. <a name="l03357"></a>03357 <span class="comment">!                             tolerance from being achieved.</span>
    3422. 

  3422. <a name="l03358"></a>03358 <span class="comment">!                             the error may be under-estimated.</span>
    3423. 

  3423. <a name="l03359"></a>03359 <span class="comment">!                         = 3 extremely bad integrand behavior occurs</span>
    3424. 

  3424. <a name="l03360"></a>03360 <span class="comment">!                             at some interior points of the integration</span>
    3425. 

  3425. <a name="l03361"></a>03361 <span class="comment">!                             interval.</span>
    3426. 

  3426. <a name="l03362"></a>03362 <span class="comment">!                         = 4 the algorithm does not converge. roundoff</span>
    3427. 

  3427. <a name="l03363"></a>03363 <span class="comment">!                             error is detected in the extrapolation</span>
    3428. 

  3428. <a name="l03364"></a>03364 <span class="comment">!                             table. it is presumed that the requested</span>
    3429. 

  3429. <a name="l03365"></a>03365 <span class="comment">!                             tolerance cannot be achieved due to</span>
    3430. 

  3430. <a name="l03366"></a>03366 <span class="comment">!                             roundoff in the extrapolation table,</span>
    3431. 

  3431. <a name="l03367"></a>03367 <span class="comment">!                             and that the returned result is the best</span>
    3432. 

  3432. <a name="l03368"></a>03368 <span class="comment">!                             which can be obtained.</span>
    3433. 

  3433. <a name="l03369"></a>03369 <span class="comment">!                         = 5 the integral is probably divergent, or</span>
    3434. 

  3434. <a name="l03370"></a>03370 <span class="comment">!                             slowly convergent. it must be noted that</span>
    3435. 

  3435. <a name="l03371"></a>03371 <span class="comment">!                             divergence can occur with any other value</span>
    3436. 

  3436. <a name="l03372"></a>03372 <span class="comment">!                             of ier.</span>
    3437. 

  3437. <a name="l03373"></a>03373 <span class="comment">!                         = 6 the input is invalid, because</span>
    3438. 

  3438. <a name="l03374"></a>03374 <span class="comment">!                             epsabs &lt; 0 and epsrel &lt; 0,</span>
    3439. 

  3439. <a name="l03375"></a>03375 <span class="comment">!                             result, abserr, neval are set to zero.</span>
    3440. 

  3440. <a name="l03376"></a>03376 <span class="comment">!</span>
    3441. 

  3441. <a name="l03377"></a>03377 <span class="comment">!  Local parameters:</span>
    3442. 

  3442. <a name="l03378"></a>03378 <span class="comment">!</span>
    3443. 

  3443. <a name="l03379"></a>03379 <span class="comment">!    limit is the maximum number of subintervals allowed in the</span>
    3444. 

  3444. <a name="l03380"></a>03380 <span class="comment">!    subdivision process of QFOUR. take care that limit &gt;= 1.</span>
    3445. 

  3445. <a name="l03381"></a>03381 <span class="comment">!</span>
    3446. 

  3446. <a name="l03382"></a>03382 <span class="comment">!    maxp1 gives an upper bound on the number of Chebyshev moments</span>
    3447. 

  3447. <a name="l03383"></a>03383 <span class="comment">!    which can be stored, i.e. for the intervals of lengths</span>
    3448. 

  3448. <a name="l03384"></a>03384 <span class="comment">!    abs(b-a)*2**(-l), l = 0, 1, ... , maxp1-2. take care that</span>
    3449. 

  3449. <a name="l03385"></a>03385 <span class="comment">!    maxp1 &gt;= 1.</span>
    3450. 

  3450. <a name="l03386"></a>03386 
    3451. 

  3451. <a name="l03387"></a>03387   <span class="keyword">implicit none</span>
    3452. 

  3452. <a name="l03388"></a>03388 
    3453. 

  3453. <a name="l03389"></a>03389   <span class="keywordtype">integer</span>, <span class="keywordtype">parameter</span> :: limit = 500
    3454. 

  3454. <a name="l03390"></a>03390   <span class="keywordtype">integer</span>, <span class="keywordtype">parameter</span> :: maxp1 = 21
    3455. 

  3455. <a name="l03391"></a>03391 
    3456. 

  3456. <a name="l03392"></a>03392   <span class="keywordtype">real</span> a
    3457. 

  3457. <a name="l03393"></a>03393   <span class="keywordtype">real</span> abserr
    3458. 

  3458. <a name="l03394"></a>03394   <span class="keywordtype">real</span> alist(limit)
    3459. 

  3459. <a name="l03395"></a>03395   <span class="keywordtype">real</span> b
    3460. 

  3460. <a name="l03396"></a>03396   <span class="keywordtype">real</span> blist(limit)
    3461. 

  3461. <a name="l03397"></a>03397   <span class="keywordtype">real</span> chebmo(maxp1,25)
    3462. 

  3462. <a name="l03398"></a>03398   <span class="keywordtype">real</span> elist(limit)
    3463. 

  3463. <a name="l03399"></a>03399   <span class="keywordtype">real</span> epsabs
    3464. 

  3464. <a name="l03400"></a>03400   <span class="keywordtype">real</span> epsrel
    3465. 

  3465. <a name="l03401"></a>03401   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    3466. 

  3466. <a name="l03402"></a>03402   <span class="keywordtype">integer</span> ier
    3467. 

  3467. <a name="l03403"></a>03403   <span class="keywordtype">integer</span> integr
    3468. 

  3468. <a name="l03404"></a>03404   <span class="keywordtype">integer</span> iord(limit)
    3469. 

  3469. <a name="l03405"></a>03405   <span class="keywordtype">integer</span> momcom
    3470. 

  3470. <a name="l03406"></a>03406   <span class="keywordtype">integer</span> neval
    3471. 

  3471. <a name="l03407"></a>03407   <span class="keywordtype">integer</span> nnlog(limit)
    3472. 

  3472. <a name="l03408"></a>03408   <span class="keywordtype">real</span> omega
    3473. 

  3473. <a name="l03409"></a>03409   <span class="keywordtype">real</span> result
    3474. 

  3474. <a name="l03410"></a>03410   <span class="keywordtype">real</span> rlist(limit)
    3475. 

  3475. <a name="l03411"></a>03411 
    3476. 

  3476. <a name="l03412"></a>03412   call <a class="code" href="quadpack_8f90.html#abbba06307e0e8c4daa2651945570ba1c">qfour </a>( f, a, b, omega, integr, epsabs, epsrel, limit, 1, maxp1, &amp;
    3477. 

  3477. <a name="l03413"></a>03413     result, abserr, neval, ier, alist, blist, rlist, elist, iord, nnlog, &amp;
    3478. 

  3478. <a name="l03414"></a>03414     momcom, chebmo )
    3479. 

  3479. <a name="l03415"></a>03415 
    3480. 

  3480. <a name="l03416"></a>03416   return
    3481. 

  3481. <a name="l03417"></a>03417 <span class="keyword">end</span>
    3482. 

  3482. <a name="l03418"></a><a class="code" href="quadpack_8f90.html#a22846cb66e25f85c2eaa2b7fc1d6c81b">03418</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a22846cb66e25f85c2eaa2b7fc1d6c81b">qaws</a> ( f, a, b, alfa, beta, integr, epsabs, epsrel, result, &amp;
    3483. 

  3483. <a name="l03419"></a>03419   abserr, neval, ier )
    3484. 

  3484. <a name="l03420"></a>03420 
    3485. 

  3485. <a name="l03421"></a>03421 <span class="comment">!*****************************************************************************80</span>
    3486. 

  3486. <a name="l03422"></a>03422 <span class="comment">!</span>
    3487. 

  3487. <a name="l03423"></a>03423 <span class="comment">!! QAWS estimates integrals with algebraico-logarithmic endpoint singularities.</span>
    3488. 

  3488. <a name="l03424"></a>03424 <span class="comment">!</span>
    3489. 

  3489. <a name="l03425"></a>03425 <span class="comment">!  Discussion:</span>
    3490. 

  3490. <a name="l03426"></a>03426 <span class="comment">!</span>
    3491. 

  3491. <a name="l03427"></a>03427 <span class="comment">!    This routine calculates an approximation RESULT to a given</span>
    3492. 

  3492. <a name="l03428"></a>03428 <span class="comment">!    definite integral   </span>
    3493. 

  3493. <a name="l03429"></a>03429 <span class="comment">!      I = integral of f*w over (a,b) </span>
    3494. 

  3494. <a name="l03430"></a>03430 <span class="comment">!    where w shows a singular behavior at the end points, see parameter</span>
    3495. 

  3495. <a name="l03431"></a>03431 <span class="comment">!    integr, hopefully satisfying following claim for accuracy</span>
    3496. 

  3496. <a name="l03432"></a>03432 <span class="comment">!      abs(i-result) &lt;= max(epsabs,epsrel*abs(i)).</span>
    3497. 

  3497. <a name="l03433"></a>03433 <span class="comment">!</span>
    3498. 

  3498. <a name="l03434"></a>03434 <span class="comment">!  Author:</span>
    3499. 

  3499. <a name="l03435"></a>03435 <span class="comment">!</span>
    3500. 

  3500. <a name="l03436"></a>03436 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    3501. 

  3501. <a name="l03437"></a>03437 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    3502. 

  3502. <a name="l03438"></a>03438 <span class="comment">!</span>
    3503. 

  3503. <a name="l03439"></a>03439 <span class="comment">!  Reference:</span>
    3504. 

  3504. <a name="l03440"></a>03440 <span class="comment">!</span>
    3505. 

  3505. <a name="l03441"></a>03441 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    3506. 

  3506. <a name="l03442"></a>03442 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    3507. 

  3507. <a name="l03443"></a>03443 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    3508. 

  3508. <a name="l03444"></a>03444 <span class="comment">!    Springer Verlag, 1983</span>
    3509. 

  3509. <a name="l03445"></a>03445 <span class="comment">!</span>
    3510. 

  3510. <a name="l03446"></a>03446 <span class="comment">!  Parameters:</span>
    3511. 

  3511. <a name="l03447"></a>03447 <span class="comment">!</span>
    3512. 

  3512. <a name="l03448"></a>03448 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    3513. 

  3513. <a name="l03449"></a>03449 <span class="comment">!      function f ( x )</span>
    3514. 

  3514. <a name="l03450"></a>03450 <span class="comment">!      real f</span>
    3515. 

  3515. <a name="l03451"></a>03451 <span class="comment">!      real x</span>
    3516. 

  3516. <a name="l03452"></a>03452 <span class="comment">!    which evaluates the integrand function.</span>
    3517. 

  3517. <a name="l03453"></a>03453 <span class="comment">!</span>
    3518. 

  3518. <a name="l03454"></a>03454 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    3519. 

  3519. <a name="l03455"></a>03455 <span class="comment">!</span>
    3520. 

  3520. <a name="l03456"></a>03456 <span class="comment">!    Input, real ALFA, BETA, parameters used in the weight function.</span>
    3521. 

  3521. <a name="l03457"></a>03457 <span class="comment">!    ALFA and BETA should be greater than -1.</span>
    3522. 

  3522. <a name="l03458"></a>03458 <span class="comment">!</span>
    3523. 

  3523. <a name="l03459"></a>03459 <span class="comment">!    Input, integer INTEGR, indicates which weight function is to be used</span>
    3524. 

  3524. <a name="l03460"></a>03460 <span class="comment">!    = 1  (x-a)**alfa*(b-x)**beta</span>
    3525. 

  3525. <a name="l03461"></a>03461 <span class="comment">!    = 2  (x-a)**alfa*(b-x)**beta*log(x-a)</span>
    3526. 

  3526. <a name="l03462"></a>03462 <span class="comment">!    = 3  (x-a)**alfa*(b-x)**beta*log(b-x)</span>
    3527. 

  3527. <a name="l03463"></a>03463 <span class="comment">!    = 4  (x-a)**alfa*(b-x)**beta*log(x-a)*log(b-x)</span>
    3528. 

  3528. <a name="l03464"></a>03464 <span class="comment">!</span>
    3529. 

  3529. <a name="l03465"></a>03465 <span class="comment">!    Input, real EPSABS, EPSREL, the absolute and relative accuracy requested.</span>
    3530. 

  3530. <a name="l03466"></a>03466 <span class="comment">!</span>
    3531. 

  3531. <a name="l03467"></a>03467 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    3532. 

  3532. <a name="l03468"></a>03468 <span class="comment">!</span>
    3533. 

  3533. <a name="l03469"></a>03469 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    3534. 

  3534. <a name="l03470"></a>03470 <span class="comment">!</span>
    3535. 

  3535. <a name="l03471"></a>03471 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    3536. 

  3536. <a name="l03472"></a>03472 <span class="comment">!</span>
    3537. 

  3537. <a name="l03473"></a>03473 <span class="comment">!            ier    - integer</span>
    3538. 

  3538. <a name="l03474"></a>03474 <span class="comment">!                     ier = 0 normal and reliable termination of the</span>
    3539. 

  3539. <a name="l03475"></a>03475 <span class="comment">!                             routine. it is assumed that the requested</span>
    3540. 

  3540. <a name="l03476"></a>03476 <span class="comment">!                             accuracy has been achieved.</span>
    3541. 

  3541. <a name="l03477"></a>03477 <span class="comment">!                     ier &gt; 0 abnormal termination of the routine</span>
    3542. 

  3542. <a name="l03478"></a>03478 <span class="comment">!                             the estimates for the integral and error</span>
    3543. 

  3543. <a name="l03479"></a>03479 <span class="comment">!                             are less reliable. it is assumed that the</span>
    3544. 

  3544. <a name="l03480"></a>03480 <span class="comment">!                             requested accuracy has not been achieved.</span>
    3545. 

  3545. <a name="l03481"></a>03481 <span class="comment">!                     ier = 1 maximum number of subdivisions allowed</span>
    3546. 

  3546. <a name="l03482"></a>03482 <span class="comment">!                             has been achieved. one can allow more</span>
    3547. 

  3547. <a name="l03483"></a>03483 <span class="comment">!                             subdivisions by increasing the data value</span>
    3548. 

  3548. <a name="l03484"></a>03484 <span class="comment">!                             of limit in qaws (and taking the according</span>
    3549. 

  3549. <a name="l03485"></a>03485 <span class="comment">!                             dimension adjustments into account).</span>
    3550. 

  3550. <a name="l03486"></a>03486 <span class="comment">!                             however, if this yields no improvement it</span>
    3551. 

  3551. <a name="l03487"></a>03487 <span class="comment">!                             is advised to analyze the integrand, in</span>
    3552. 

  3552. <a name="l03488"></a>03488 <span class="comment">!                             order to determine the integration</span>
    3553. 

  3553. <a name="l03489"></a>03489 <span class="comment">!                             difficulties which prevent the requested</span>
    3554. 

  3554. <a name="l03490"></a>03490 <span class="comment">!                             tolerance from being achieved. in case of</span>
    3555. 

  3555. <a name="l03491"></a>03491 <span class="comment">!                             a jump discontinuity or a local</span>
    3556. 

  3556. <a name="l03492"></a>03492 <span class="comment">!                             singularity of algebraico-logarithmic type</span>
    3557. 

  3557. <a name="l03493"></a>03493 <span class="comment">!                             at one or more interior points of the</span>
    3558. 

  3558. <a name="l03494"></a>03494 <span class="comment">!                             integration range, one should proceed by</span>
    3559. 

  3559. <a name="l03495"></a>03495 <span class="comment">!                             splitting up the interval at these points</span>
    3560. 

  3560. <a name="l03496"></a>03496 <span class="comment">!                             and calling the integrator on the</span>
    3561. 

  3561. <a name="l03497"></a>03497 <span class="comment">!                             subranges.</span>
    3562. 

  3562. <a name="l03498"></a>03498 <span class="comment">!                         = 2 the occurrence of roundoff error is</span>
    3563. 

  3563. <a name="l03499"></a>03499 <span class="comment">!                             detected, which prevents the requested</span>
    3564. 

  3564. <a name="l03500"></a>03500 <span class="comment">!                             tolerance from being achieved.</span>
    3565. 

  3565. <a name="l03501"></a>03501 <span class="comment">!                         = 3 extremely bad integrand behavior occurs</span>
    3566. 

  3566. <a name="l03502"></a>03502 <span class="comment">!                             at some points of the integration</span>
    3567. 

  3567. <a name="l03503"></a>03503 <span class="comment">!                             interval.</span>
    3568. 

  3568. <a name="l03504"></a>03504 <span class="comment">!                         = 6 the input is invalid, because</span>
    3569. 

  3569. <a name="l03505"></a>03505 <span class="comment">!                             b &lt;= a or alfa &lt;= (-1) or beta &lt;= (-1) or</span>
    3570. 

  3570. <a name="l03506"></a>03506 <span class="comment">!                             integr &lt; 1 or integr &gt; 4 or</span>
    3571. 

  3571. <a name="l03507"></a>03507 <span class="comment">!                             epsabs &lt; 0 and epsrel &lt; 0,</span>
    3572. 

  3572. <a name="l03508"></a>03508 <span class="comment">!                             result, abserr, neval are set to zero.</span>
    3573. 

  3573. <a name="l03509"></a>03509 <span class="comment">!</span>
    3574. 

  3574. <a name="l03510"></a>03510 <span class="comment">!  Local parameters:</span>
    3575. 

  3575. <a name="l03511"></a>03511 <span class="comment">!</span>
    3576. 

  3576. <a name="l03512"></a>03512 <span class="comment">!    LIMIT is the maximum number of subintervals allowed in the</span>
    3577. 

  3577. <a name="l03513"></a>03513 <span class="comment">!    subdivision process of qawse. take care that limit &gt;= 2.</span>
    3578. 

  3578. <a name="l03514"></a>03514 <span class="comment">!</span>
    3579. 

  3579. <a name="l03515"></a>03515   <span class="keyword">implicit none</span>
    3580. 

  3580. <a name="l03516"></a>03516 
    3581. 

  3581. <a name="l03517"></a>03517   <span class="keywordtype">integer</span>, <span class="keywordtype">parameter</span> :: limit = 500
    3582. 

  3582. <a name="l03518"></a>03518 
    3583. 

  3583. <a name="l03519"></a>03519   <span class="keywordtype">real</span> a
    3584. 

  3584. <a name="l03520"></a>03520   <span class="keywordtype">real</span> abserr
    3585. 

  3585. <a name="l03521"></a>03521   <span class="keywordtype">real</span> alfa
    3586. 

  3586. <a name="l03522"></a>03522   <span class="keywordtype">real</span> alist(limit)
    3587. 

  3587. <a name="l03523"></a>03523   <span class="keywordtype">real</span> b
    3588. 

  3588. <a name="l03524"></a>03524   <span class="keywordtype">real</span> blist(limit)
    3589. 

  3589. <a name="l03525"></a>03525   <span class="keywordtype">real</span> beta
    3590. 

  3590. <a name="l03526"></a>03526   <span class="keywordtype">real</span> elist(limit)
    3591. 

  3591. <a name="l03527"></a>03527   <span class="keywordtype">real</span> epsabs
    3592. 

  3592. <a name="l03528"></a>03528   <span class="keywordtype">real</span> epsrel
    3593. 

  3593. <a name="l03529"></a>03529   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    3594. 

  3594. <a name="l03530"></a>03530   <span class="keywordtype">integer</span> ier
    3595. 

  3595. <a name="l03531"></a>03531   <span class="keywordtype">integer</span> integr
    3596. 

  3596. <a name="l03532"></a>03532   <span class="keywordtype">integer</span> iord(limit)
    3597. 

  3597. <a name="l03533"></a>03533   <span class="keywordtype">integer</span> last
    3598. 

  3598. <a name="l03534"></a>03534   <span class="keywordtype">integer</span> neval
    3599. 

  3599. <a name="l03535"></a>03535   <span class="keywordtype">real</span> result
    3600. 

  3600. <a name="l03536"></a>03536   <span class="keywordtype">real</span> rlist(limit)
    3601. 

  3601. <a name="l03537"></a>03537 
    3602. 

  3602. <a name="l03538"></a>03538   call <a class="code" href="quadpack_8f90.html#adf40beeb87b948ed57824e42c023f518">qawse </a>( f, a, b, alfa, beta, integr, epsabs, epsrel, limit, result, &amp;
    3603. 

  3603. <a name="l03539"></a>03539     abserr, neval, ier, alist, blist, rlist, elist, iord, last )
    3604. 

  3604. <a name="l03540"></a>03540 
    3605. 

  3605. <a name="l03541"></a>03541   return
    3606. 

  3606. <a name="l03542"></a>03542 <span class="keyword">end</span>
    3607. 

  3607. <a name="l03543"></a><a class="code" href="quadpack_8f90.html#adf40beeb87b948ed57824e42c023f518">03543</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#adf40beeb87b948ed57824e42c023f518">qawse</a> ( f, a, b, alfa, beta, integr, epsabs, epsrel, limit, &amp;
    3608. 

  3608. <a name="l03544"></a>03544   result, abserr, neval, ier, alist, blist, rlist, elist, iord, last )
    3609. 

  3609. <a name="l03545"></a>03545 
    3610. 

  3610. <a name="l03546"></a>03546 <span class="comment">!*****************************************************************************80</span>
    3611. 

  3611. <a name="l03547"></a>03547 <span class="comment">!</span>
    3612. 

  3612. <a name="l03548"></a>03548 <span class="comment">!! QAWSE estimates integrals with algebraico-logarithmic endpoint singularities.</span>
    3613. 

  3613. <a name="l03549"></a>03549 <span class="comment">!</span>
    3614. 

  3614. <a name="l03550"></a>03550 <span class="comment">!  Discussion:</span>
    3615. 

  3615. <a name="l03551"></a>03551 <span class="comment">!</span>
    3616. 

  3616. <a name="l03552"></a>03552 <span class="comment">!    This routine calculates an approximation RESULT to an integral</span>
    3617. 

  3617. <a name="l03553"></a>03553 <span class="comment">!      I = integral of F(X) * W(X) over (a,b), </span>
    3618. 

  3618. <a name="l03554"></a>03554 <span class="comment">!    where W(X) shows a singular behavior at the endpoints, hopefully </span>
    3619. 

  3619. <a name="l03555"></a>03555 <span class="comment">!    satisfying:</span>
    3620. 

  3620. <a name="l03556"></a>03556 <span class="comment">!      | I - RESULT | &lt;= max ( epsabs, epsrel * |I| ).</span>
    3621. 

  3621. <a name="l03557"></a>03557 <span class="comment">!</span>
    3622. 

  3622. <a name="l03558"></a>03558 <span class="comment">!  Author:</span>
    3623. 

  3623. <a name="l03559"></a>03559 <span class="comment">!</span>
    3624. 

  3624. <a name="l03560"></a>03560 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    3625. 

  3625. <a name="l03561"></a>03561 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    3626. 

  3626. <a name="l03562"></a>03562 <span class="comment">!</span>
    3627. 

  3627. <a name="l03563"></a>03563 <span class="comment">!  Reference:</span>
    3628. 

  3628. <a name="l03564"></a>03564 <span class="comment">!</span>
    3629. 

  3629. <a name="l03565"></a>03565 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    3630. 

  3630. <a name="l03566"></a>03566 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    3631. 

  3631. <a name="l03567"></a>03567 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    3632. 

  3632. <a name="l03568"></a>03568 <span class="comment">!    Springer Verlag, 1983</span>
    3633. 

  3633. <a name="l03569"></a>03569 <span class="comment">!</span>
    3634. 

  3634. <a name="l03570"></a>03570 <span class="comment">!  Parameters:</span>
    3635. 

  3635. <a name="l03571"></a>03571 <span class="comment">!</span>
    3636. 

  3636. <a name="l03572"></a>03572 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    3637. 

  3637. <a name="l03573"></a>03573 <span class="comment">!      function f ( x )</span>
    3638. 

  3638. <a name="l03574"></a>03574 <span class="comment">!      real f</span>
    3639. 

  3639. <a name="l03575"></a>03575 <span class="comment">!      real x</span>
    3640. 

  3640. <a name="l03576"></a>03576 <span class="comment">!    which evaluates the integrand function.</span>
    3641. 

  3641. <a name="l03577"></a>03577 <span class="comment">!</span>
    3642. 

  3642. <a name="l03578"></a>03578 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    3643. 

  3643. <a name="l03579"></a>03579 <span class="comment">!</span>
    3644. 

  3644. <a name="l03580"></a>03580 <span class="comment">!    Input, real ALFA, BETA, parameters used in the weight function.</span>
    3645. 

  3645. <a name="l03581"></a>03581 <span class="comment">!    ALFA and BETA should be greater than -1.</span>
    3646. 

  3646. <a name="l03582"></a>03582 <span class="comment">!</span>
    3647. 

  3647. <a name="l03583"></a>03583 <span class="comment">!    Input, integer INTEGR, indicates which weight function is used:</span>
    3648. 

  3648. <a name="l03584"></a>03584 <span class="comment">!    = 1  (x-a)**alfa*(b-x)**beta</span>
    3649. 

  3649. <a name="l03585"></a>03585 <span class="comment">!    = 2  (x-a)**alfa*(b-x)**beta*log(x-a)</span>
    3650. 

  3650. <a name="l03586"></a>03586 <span class="comment">!    = 3  (x-a)**alfa*(b-x)**beta*log(b-x)</span>
    3651. 

  3651. <a name="l03587"></a>03587 <span class="comment">!    = 4  (x-a)**alfa*(b-x)**beta*log(x-a)*log(b-x)</span>
    3652. 

  3652. <a name="l03588"></a>03588 <span class="comment">!</span>
    3653. 

  3653. <a name="l03589"></a>03589 <span class="comment">!    Input, real EPSABS, EPSREL, the absolute and relative accuracy requested.</span>
    3654. 

  3654. <a name="l03590"></a>03590 <span class="comment">!</span>
    3655. 

  3655. <a name="l03591"></a>03591 <span class="comment">!    Input, integer LIMIT, an upper bound on the number of subintervals</span>
    3656. 

  3656. <a name="l03592"></a>03592 <span class="comment">!    in the partition of (A,B), LIMIT &gt;= 2.  If LIMIT &lt; 2, the routine </span>
    3657. 

  3657. <a name="l03593"></a>03593 <span class="comment">!     will end with IER = 6.</span>
    3658. 

  3658. <a name="l03594"></a>03594 <span class="comment">!</span>
    3659. 

  3659. <a name="l03595"></a>03595 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    3660. 

  3660. <a name="l03596"></a>03596 <span class="comment">!</span>
    3661. 

  3661. <a name="l03597"></a>03597 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    3662. 

  3662. <a name="l03598"></a>03598 <span class="comment">!</span>
    3663. 

  3663. <a name="l03599"></a>03599 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    3664. 

  3664. <a name="l03600"></a>03600 <span class="comment">!</span>
    3665. 

  3665. <a name="l03601"></a>03601 <span class="comment">!            ier    - integer</span>
    3666. 

  3666. <a name="l03602"></a>03602 <span class="comment">!                     ier = 0 normal and reliable termination of the</span>
    3667. 

  3667. <a name="l03603"></a>03603 <span class="comment">!                             routine. it is assumed that the requested</span>
    3668. 

  3668. <a name="l03604"></a>03604 <span class="comment">!                             accuracy has been achieved.</span>
    3669. 

  3669. <a name="l03605"></a>03605 <span class="comment">!                     ier &gt; 0 abnormal termination of the routine</span>
    3670. 

  3670. <a name="l03606"></a>03606 <span class="comment">!                             the estimates for the integral and error</span>
    3671. 

  3671. <a name="l03607"></a>03607 <span class="comment">!                             are less reliable. it is assumed that the</span>
    3672. 

  3672. <a name="l03608"></a>03608 <span class="comment">!                             requested accuracy has not been achieved.</span>
    3673. 

  3673. <a name="l03609"></a>03609 <span class="comment">!                         = 1 maximum number of subdivisions allowed</span>
    3674. 

  3674. <a name="l03610"></a>03610 <span class="comment">!                             has been achieved. one can allow more</span>
    3675. 

  3675. <a name="l03611"></a>03611 <span class="comment">!                             subdivisions by increasing the value of</span>
    3676. 

  3676. <a name="l03612"></a>03612 <span class="comment">!                             limit. however, if this yields no</span>
    3677. 

  3677. <a name="l03613"></a>03613 <span class="comment">!                             improvement it is advised to analyze the</span>
    3678. 

  3678. <a name="l03614"></a>03614 <span class="comment">!                             integrand, in order to determine the</span>
    3679. 

  3679. <a name="l03615"></a>03615 <span class="comment">!                             integration difficulties which prevent</span>
    3680. 

  3680. <a name="l03616"></a>03616 <span class="comment">!                             the requested tolerance from being</span>
    3681. 

  3681. <a name="l03617"></a>03617 <span class="comment">!                             achieved. in case of a jump discontinuity</span>
    3682. 

  3682. <a name="l03618"></a>03618 <span class="comment">!                             or a local singularity of algebraico-</span>
    3683. 

  3683. <a name="l03619"></a>03619 <span class="comment">!                             logarithmic type at one or more interior</span>
    3684. 

  3684. <a name="l03620"></a>03620 <span class="comment">!                             points of the integration range, one</span>
    3685. 

  3685. <a name="l03621"></a>03621 <span class="comment">!                             should proceed by splitting up the</span>
    3686. 

  3686. <a name="l03622"></a>03622 <span class="comment">!                             interval at these points and calling the</span>
    3687. 

  3687. <a name="l03623"></a>03623 <span class="comment">!                             integrator on the subranges.</span>
    3688. 

  3688. <a name="l03624"></a>03624 <span class="comment">!                         = 2 the occurrence of roundoff error is</span>
    3689. 

  3689. <a name="l03625"></a>03625 <span class="comment">!                             detected, which prevents the requested</span>
    3690. 

  3690. <a name="l03626"></a>03626 <span class="comment">!                             tolerance from being achieved.</span>
    3691. 

  3691. <a name="l03627"></a>03627 <span class="comment">!                         = 3 extremely bad integrand behavior occurs</span>
    3692. 

  3692. <a name="l03628"></a>03628 <span class="comment">!                             at some points of the integration</span>
    3693. 

  3693. <a name="l03629"></a>03629 <span class="comment">!                             interval.</span>
    3694. 

  3694. <a name="l03630"></a>03630 <span class="comment">!                         = 6 the input is invalid, because</span>
    3695. 

  3695. <a name="l03631"></a>03631 <span class="comment">!                             b &lt;= a or alfa &lt;= (-1) or beta &lt;= (-1) or</span>
    3696. 

  3696. <a name="l03632"></a>03632 <span class="comment">!                             integr &lt; 1 or integr &gt; 4, or</span>
    3697. 

  3697. <a name="l03633"></a>03633 <span class="comment">!                             epsabs &lt; 0 and epsrel &lt; 0,</span>
    3698. 

  3698. <a name="l03634"></a>03634 <span class="comment">!                             or limit &lt; 2.</span>
    3699. 

  3699. <a name="l03635"></a>03635 <span class="comment">!                             result, abserr, neval, rlist(1), elist(1),</span>
    3700. 

  3700. <a name="l03636"></a>03636 <span class="comment">!                             iord(1) and last are set to zero.</span>
    3701. 

  3701. <a name="l03637"></a>03637 <span class="comment">!                             alist(1) and blist(1) are set to a and b</span>
    3702. 

  3702. <a name="l03638"></a>03638 <span class="comment">!                             respectively.</span>
    3703. 

  3703. <a name="l03639"></a>03639 <span class="comment">!</span>
    3704. 

  3704. <a name="l03640"></a>03640 <span class="comment">!    Workspace, real ALIST(LIMIT), BLIST(LIMIT), contains in entries 1 </span>
    3705. 

  3705. <a name="l03641"></a>03641 <span class="comment">!    through LAST the left and right ends of the partition subintervals.</span>
    3706. 

  3706. <a name="l03642"></a>03642 <span class="comment">!</span>
    3707. 

  3707. <a name="l03643"></a>03643 <span class="comment">!    Workspace, real RLIST(LIMIT), contains in entries 1 through LAST</span>
    3708. 

  3708. <a name="l03644"></a>03644 <span class="comment">!    the integral approximations on the subintervals.</span>
    3709. 

  3709. <a name="l03645"></a>03645 <span class="comment">!</span>
    3710. 

  3710. <a name="l03646"></a>03646 <span class="comment">!    Workspace, real ELIST(LIMIT), contains in entries 1 through LAST</span>
    3711. 

  3711. <a name="l03647"></a>03647 <span class="comment">!    the absolute error estimates on the subintervals.</span>
    3712. 

  3712. <a name="l03648"></a>03648 <span class="comment">!</span>
    3713. 

  3713. <a name="l03649"></a>03649 <span class="comment">!            iord   - integer</span>
    3714. 

  3714. <a name="l03650"></a>03650 <span class="comment">!                     vector of dimension at least limit, the first k</span>
    3715. 

  3715. <a name="l03651"></a>03651 <span class="comment">!                     elements of which are pointers to the error</span>
    3716. 

  3716. <a name="l03652"></a>03652 <span class="comment">!                     estimates over the subintervals, so that</span>
    3717. 

  3717. <a name="l03653"></a>03653 <span class="comment">!                     elist(iord(1)), ..., elist(iord(k)) with k = last</span>
    3718. 

  3718. <a name="l03654"></a>03654 <span class="comment">!                     if last &lt;= (limit/2+2), and k = limit+1-last</span>
    3719. 

  3719. <a name="l03655"></a>03655 <span class="comment">!                     otherwise, form a decreasing sequence.</span>
    3720. 

  3720. <a name="l03656"></a>03656 <span class="comment">!</span>
    3721. 

  3721. <a name="l03657"></a>03657 <span class="comment">!    Output, integer LAST, the number of subintervals actually produced in </span>
    3722. 

  3722. <a name="l03658"></a>03658 <span class="comment">!    the subdivision process.</span>
    3723. 

  3723. <a name="l03659"></a>03659 <span class="comment">!</span>
    3724. 

  3724. <a name="l03660"></a>03660 <span class="comment">!  Local parameters:</span>
    3725. 

  3725. <a name="l03661"></a>03661 <span class="comment">!</span>
    3726. 

  3726. <a name="l03662"></a>03662 <span class="comment">!           alist     - list of left end points of all subintervals</span>
    3727. 

  3727. <a name="l03663"></a>03663 <span class="comment">!                       considered up to now</span>
    3728. 

  3728. <a name="l03664"></a>03664 <span class="comment">!           blist     - list of right end points of all subintervals</span>
    3729. 

  3729. <a name="l03665"></a>03665 <span class="comment">!                       considered up to now</span>
    3730. 

  3730. <a name="l03666"></a>03666 <span class="comment">!           rlist(i)  - approximation to the integral over</span>
    3731. 

  3731. <a name="l03667"></a>03667 <span class="comment">!                       (alist(i),blist(i))</span>
    3732. 

  3732. <a name="l03668"></a>03668 <span class="comment">!           elist(i)  - error estimate applying to rlist(i)</span>
    3733. 

  3733. <a name="l03669"></a>03669 <span class="comment">!           maxerr    - pointer to the interval with largest error</span>
    3734. 

  3734. <a name="l03670"></a>03670 <span class="comment">!                       estimate</span>
    3735. 

  3735. <a name="l03671"></a>03671 <span class="comment">!           errmax    - elist(maxerr)</span>
    3736. 

  3736. <a name="l03672"></a>03672 <span class="comment">!           area      - sum of the integrals over the subintervals</span>
    3737. 

  3737. <a name="l03673"></a>03673 <span class="comment">!           errsum    - sum of the errors over the subintervals</span>
    3738. 

  3738. <a name="l03674"></a>03674 <span class="comment">!           errbnd    - requested accuracy max(epsabs,epsrel*</span>
    3739. 

  3739. <a name="l03675"></a>03675 <span class="comment">!                       abs(result))</span>
    3740. 

  3740. <a name="l03676"></a>03676 <span class="comment">!           *****1    - variable for the left subinterval</span>
    3741. 

  3741. <a name="l03677"></a>03677 <span class="comment">!           *****2    - variable for the right subinterval</span>
    3742. 

  3742. <a name="l03678"></a>03678 <span class="comment">!           last      - index for subdivision</span>
    3743. 

  3743. <a name="l03679"></a>03679 <span class="comment">!</span>
    3744. 

  3744. <a name="l03680"></a>03680   <span class="keyword">implicit none</span>
    3745. 

  3745. <a name="l03681"></a>03681 
    3746. 

  3746. <a name="l03682"></a>03682   <span class="keywordtype">integer</span> limit
    3747. 

  3747. <a name="l03683"></a>03683 
    3748. 

  3748. <a name="l03684"></a>03684   <span class="keywordtype">real</span> a
    3749. 

  3749. <a name="l03685"></a>03685   <span class="keywordtype">real</span> abserr
    3750. 

  3750. <a name="l03686"></a>03686   <span class="keywordtype">real</span> alfa
    3751. 

  3751. <a name="l03687"></a>03687   <span class="keywordtype">real</span> alist(limit)
    3752. 

  3752. <a name="l03688"></a>03688   <span class="keywordtype">real</span> area
    3753. 

  3753. <a name="l03689"></a>03689   <span class="keywordtype">real</span> area1
    3754. 

  3754. <a name="l03690"></a>03690   <span class="keywordtype">real</span> area12
    3755. 

  3755. <a name="l03691"></a>03691   <span class="keywordtype">real</span> area2
    3756. 

  3756. <a name="l03692"></a>03692   <span class="keywordtype">real</span> a1
    3757. 

  3757. <a name="l03693"></a>03693   <span class="keywordtype">real</span> a2
    3758. 

  3758. <a name="l03694"></a>03694   <span class="keywordtype">real</span> b
    3759. 

  3759. <a name="l03695"></a>03695   <span class="keywordtype">real</span> beta
    3760. 

  3760. <a name="l03696"></a>03696   <span class="keywordtype">real</span> blist(limit)
    3761. 

  3761. <a name="l03697"></a>03697   <span class="keywordtype">real</span> b1
    3762. 

  3762. <a name="l03698"></a>03698   <span class="keywordtype">real</span> b2
    3763. 

  3763. <a name="l03699"></a>03699   <span class="keywordtype">real</span> centre
    3764. 

  3764. <a name="l03700"></a>03700   <span class="keywordtype">real</span> elist(limit)
    3765. 

  3765. <a name="l03701"></a>03701   <span class="keywordtype">real</span> epsabs
    3766. 

  3766. <a name="l03702"></a>03702   <span class="keywordtype">real</span> epsrel
    3767. 

  3767. <a name="l03703"></a>03703   <span class="keywordtype">real</span> errbnd
    3768. 

  3768. <a name="l03704"></a>03704   <span class="keywordtype">real</span> errmax
    3769. 

  3769. <a name="l03705"></a>03705   <span class="keywordtype">real</span> error1
    3770. 

  3770. <a name="l03706"></a>03706   <span class="keywordtype">real</span> erro12
    3771. 

  3771. <a name="l03707"></a>03707   <span class="keywordtype">real</span> error2
    3772. 

  3772. <a name="l03708"></a>03708   <span class="keywordtype">real</span> errsum
    3773. 

  3773. <a name="l03709"></a>03709   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    3774. 

  3774. <a name="l03710"></a>03710   <span class="keywordtype">integer</span> ier
    3775. 

  3775. <a name="l03711"></a>03711   <span class="keywordtype">integer</span> integr
    3776. 

  3776. <a name="l03712"></a>03712   <span class="keywordtype">integer</span> iord(limit)
    3777. 

  3777. <a name="l03713"></a>03713   <span class="keywordtype">integer</span> iroff1
    3778. 

  3778. <a name="l03714"></a>03714   <span class="keywordtype">integer</span> iroff2
    3779. 

  3779. <a name="l03715"></a>03715   <span class="keywordtype">integer</span> last
    3780. 

  3780. <a name="l03716"></a>03716   <span class="keywordtype">integer</span> maxerr
    3781. 

  3781. <a name="l03717"></a>03717   <span class="keywordtype">integer</span> nev
    3782. 

  3782. <a name="l03718"></a>03718   <span class="keywordtype">integer</span> neval
    3783. 

  3783. <a name="l03719"></a>03719   <span class="keywordtype">integer</span> nrmax
    3784. 

  3784. <a name="l03720"></a>03720   <span class="keywordtype">real</span> resas1
    3785. 

  3785. <a name="l03721"></a>03721   <span class="keywordtype">real</span> resas2
    3786. 

  3786. <a name="l03722"></a>03722   <span class="keywordtype">real</span> result
    3787. 

  3787. <a name="l03723"></a>03723   <span class="keywordtype">real</span> rg(25)
    3788. 

  3788. <a name="l03724"></a>03724   <span class="keywordtype">real</span> rh(25)
    3789. 

  3789. <a name="l03725"></a>03725   <span class="keywordtype">real</span> ri(25)
    3790. 

  3790. <a name="l03726"></a>03726   <span class="keywordtype">real</span> rj(25)
    3791. 

  3791. <a name="l03727"></a>03727   <span class="keywordtype">real</span> rlist(limit)
    3792. 

  3792. <a name="l03728"></a>03728 <span class="comment">!</span>
    3793. 

  3793. <a name="l03729"></a>03729 <span class="comment">!  Test on validity of parameters.</span>
    3794. 

  3794. <a name="l03730"></a>03730 <span class="comment">!</span>
    3795. 

  3795. <a name="l03731"></a>03731   ier = 0
    3796. 

  3796. <a name="l03732"></a>03732   neval = 0
    3797. 

  3797. <a name="l03733"></a>03733   last = 0
    3798. 

  3798. <a name="l03734"></a>03734   rlist(1) = 0.0e+00
    3799. 

  3799. <a name="l03735"></a>03735   elist(1) = 0.0e+00
    3800. 

  3800. <a name="l03736"></a>03736   iord(1) = 0
    3801. 

  3801. <a name="l03737"></a>03737   result = 0.0e+00
    3802. 

  3802. <a name="l03738"></a>03738   abserr = 0.0e+00
    3803. 

  3803. <a name="l03739"></a>03739 
    3804. 

  3804. <a name="l03740"></a>03740   <span class="keyword">if</span> ( b &lt;= a .or. &amp;
    3805. 

  3805. <a name="l03741"></a>03741     (epsabs &lt; 0.0e+00 .and. epsrel &lt; 0.0e+00) .or. &amp;
    3806. 

  3806. <a name="l03742"></a>03742     alfa &lt;= (-1.0e+00) .or. &amp;
    3807. 

  3807. <a name="l03743"></a>03743     beta &lt;= (-1.0e+00) .or. &amp;
    3808. 

  3808. <a name="l03744"></a>03744     integr &lt; 1 .or. &amp;
    3809. 

  3809. <a name="l03745"></a>03745     integr &gt; 4 .or. &amp;
    3810. 

  3810. <a name="l03746"></a>03746     limit &lt; 2 ) <span class="keyword">then</span>
    3811. 

  3811. <a name="l03747"></a>03747     ier = 6
    3812. 

  3812. <a name="l03748"></a>03748     return
    3813. 

  3813. <a name="l03749"></a>03749   <span class="keyword">end if</span>
    3814. 

  3814. <a name="l03750"></a>03750 <span class="comment">!</span>
    3815. 

  3815. <a name="l03751"></a>03751 <span class="comment">!  Compute the modified Chebyshev moments.</span>
    3816. 

  3816. <a name="l03752"></a>03752 <span class="comment">!</span>
    3817. 

  3817. <a name="l03753"></a>03753   call <a class="code" href="quadpack_8f90.html#aa732651ae77f9486d6e3d17999c699ab">qmomo </a>( alfa, beta, ri, rj, rg, rh, integr )
    3818. 

  3818. <a name="l03754"></a>03754 <span class="comment">!</span>
    3819. 

  3819. <a name="l03755"></a>03755 <span class="comment">!  Integrate over the intervals (a,(a+b)/2) and ((a+b)/2,b).</span>
    3820. 

  3820. <a name="l03756"></a>03756 <span class="comment">!</span>
    3821. 

  3821. <a name="l03757"></a>03757   centre = 5.0e-01 * ( b + a )
    3822. 

  3822. <a name="l03758"></a>03758 
    3823. 

  3823. <a name="l03759"></a>03759   call <a class="code" href="quadpack_8f90.html#a034546450320f53f05096bc12af9b5bc">qc25s </a>( f, a, b, a, centre, alfa, beta, ri, rj, rg, rh, area1, &amp;
    3824. 

  3824. <a name="l03760"></a>03760     error1, resas1, integr, nev )
    3825. 

  3825. <a name="l03761"></a>03761 
    3826. 

  3826. <a name="l03762"></a>03762   neval = nev
    3827. 

  3827. <a name="l03763"></a>03763 
    3828. 

  3828. <a name="l03764"></a>03764   call <a class="code" href="quadpack_8f90.html#a034546450320f53f05096bc12af9b5bc">qc25s </a>( f, a, b, centre, b, alfa, beta, ri, rj, rg, rh, area2, &amp;
    3829. 

  3829. <a name="l03765"></a>03765     error2, resas2, integr, nev )
    3830. 

  3830. <a name="l03766"></a>03766 
    3831. 

  3831. <a name="l03767"></a>03767   last = 2
    3832. 

  3832. <a name="l03768"></a>03768   neval = neval+nev
    3833. 

  3833. <a name="l03769"></a>03769   result = area1+area2
    3834. 

  3834. <a name="l03770"></a>03770   abserr = error1+error2
    3835. 

  3835. <a name="l03771"></a>03771 <span class="comment">!</span>
    3836. 

  3836. <a name="l03772"></a>03772 <span class="comment">!  Test on accuracy.</span>
    3837. 

  3837. <a name="l03773"></a>03773 <span class="comment">!</span>
    3838. 

  3838. <a name="l03774"></a>03774   errbnd = max ( epsabs, epsrel * abs ( result ) )
    3839. 

  3839. <a name="l03775"></a>03775 <span class="comment">!</span>
    3840. 

  3840. <a name="l03776"></a>03776 <span class="comment">!  Initialization.</span>
    3841. 

  3841. <a name="l03777"></a>03777 <span class="comment">!</span>
    3842. 

  3842. <a name="l03778"></a>03778   <span class="keyword">if</span> ( error2 &lt;= error1 ) <span class="keyword">then</span>
    3843. 

  3843. <a name="l03779"></a>03779     alist(1) = a
    3844. 

  3844. <a name="l03780"></a>03780     alist(2) = centre
    3845. 

  3845. <a name="l03781"></a>03781     blist(1) = centre
    3846. 

  3846. <a name="l03782"></a>03782     blist(2) = b
    3847. 

  3847. <a name="l03783"></a>03783     rlist(1) = area1
    3848. 

  3848. <a name="l03784"></a>03784     rlist(2) = area2
    3849. 

  3849. <a name="l03785"></a>03785     elist(1) = error1
    3850. 

  3850. <a name="l03786"></a>03786     elist(2) = error2
    3851. 

  3851. <a name="l03787"></a>03787   <span class="keyword">else</span>
    3852. 

  3852. <a name="l03788"></a>03788     alist(1) = centre
    3853. 

  3853. <a name="l03789"></a>03789     alist(2) = a
    3854. 

  3854. <a name="l03790"></a>03790     blist(1) = b
    3855. 

  3855. <a name="l03791"></a>03791     blist(2) = centre
    3856. 

  3856. <a name="l03792"></a>03792     rlist(1) = area2
    3857. 

  3857. <a name="l03793"></a>03793     rlist(2) = area1
    3858. 

  3858. <a name="l03794"></a>03794     elist(1) = error2
    3859. 

  3859. <a name="l03795"></a>03795     elist(2) = error1
    3860. 

  3860. <a name="l03796"></a>03796   <span class="keyword">end if</span>
    3861. 

  3861. <a name="l03797"></a>03797 
    3862. 

  3862. <a name="l03798"></a>03798   iord(1) = 1
    3863. 

  3863. <a name="l03799"></a>03799   iord(2) = 2
    3864. 

  3864. <a name="l03800"></a>03800 
    3865. 

  3865. <a name="l03801"></a>03801   <span class="keyword">if</span> ( limit == 2 ) <span class="keyword">then</span>
    3866. 

  3866. <a name="l03802"></a>03802     ier = 1
    3867. 

  3867. <a name="l03803"></a>03803     return
    3868. 

  3868. <a name="l03804"></a>03804   <span class="keyword">end if</span>
    3869. 

  3869. <a name="l03805"></a>03805 
    3870. 

  3870. <a name="l03806"></a>03806   <span class="keyword">if</span> ( abserr &lt;= errbnd ) <span class="keyword">then</span>
    3871. 

  3871. <a name="l03807"></a>03807     return
    3872. 

  3872. <a name="l03808"></a>03808   <span class="keyword">end if</span>
    3873. 

  3873. <a name="l03809"></a>03809 
    3874. 

  3874. <a name="l03810"></a>03810   errmax = elist(1)
    3875. 

  3875. <a name="l03811"></a>03811   maxerr = 1
    3876. 

  3876. <a name="l03812"></a>03812   nrmax = 1
    3877. 

  3877. <a name="l03813"></a>03813   area = result
    3878. 

  3878. <a name="l03814"></a>03814   errsum = abserr
    3879. 

  3879. <a name="l03815"></a>03815   iroff1 = 0
    3880. 

  3880. <a name="l03816"></a>03816   iroff2 = 0
    3881. 

  3881. <a name="l03817"></a>03817 
    3882. 

  3882. <a name="l03818"></a>03818   <span class="keyword">do</span> last = 3, limit
    3883. 

  3883. <a name="l03819"></a>03819 <span class="comment">!</span>
    3884. 

  3884. <a name="l03820"></a>03820 <span class="comment">!  Bisect the subinterval with largest error estimate.</span>
    3885. 

  3885. <a name="l03821"></a>03821 <span class="comment">!</span>
    3886. 

  3886. <a name="l03822"></a>03822     a1 = alist(maxerr)
    3887. 

  3887. <a name="l03823"></a>03823     b1 = 5.0e-01 * ( alist(maxerr) + blist(maxerr) )
    3888. 

  3888. <a name="l03824"></a>03824     a2 = b1
    3889. 

  3889. <a name="l03825"></a>03825     b2 = blist(maxerr)
    3890. 

  3890. <a name="l03826"></a>03826 
    3891. 

  3891. <a name="l03827"></a>03827     call <a class="code" href="quadpack_8f90.html#a034546450320f53f05096bc12af9b5bc">qc25s </a>( f, a, b, a1, b1, alfa, beta, ri, rj, rg, rh, area1, &amp;
    3892. 

  3892. <a name="l03828"></a>03828       error1, resas1, integr, nev )
    3893. 

  3893. <a name="l03829"></a>03829 
    3894. 

  3894. <a name="l03830"></a>03830     neval = neval + nev
    3895. 

  3895. <a name="l03831"></a>03831 
    3896. 

  3896. <a name="l03832"></a>03832     call <a class="code" href="quadpack_8f90.html#a034546450320f53f05096bc12af9b5bc">qc25s </a>( f, a, b, a2, b2, alfa, beta, ri, rj, rg, rh, area2, &amp;
    3897. 

  3897. <a name="l03833"></a>03833       error2, resas2, integr, nev )
    3898. 

  3898. <a name="l03834"></a>03834 
    3899. 

  3899. <a name="l03835"></a>03835     neval = neval + nev
    3900. 

  3900. <a name="l03836"></a>03836 <span class="comment">!</span>
    3901. 

  3901. <a name="l03837"></a>03837 <span class="comment">!  Improve previous approximations integral and error and</span>
    3902. 

  3902. <a name="l03838"></a>03838 <span class="comment">!  test for accuracy.</span>
    3903. 

  3903. <a name="l03839"></a>03839 <span class="comment">!</span>
    3904. 

  3904. <a name="l03840"></a>03840     area12 = area1+area2
    3905. 

  3905. <a name="l03841"></a>03841     erro12 = error1+error2
    3906. 

  3906. <a name="l03842"></a>03842     errsum = errsum+erro12-errmax
    3907. 

  3907. <a name="l03843"></a>03843     area = area+area12-rlist(maxerr)
    3908. 

  3908. <a name="l03844"></a>03844 <span class="comment">!</span>
    3909. 

  3909. <a name="l03845"></a>03845 <span class="comment">!  Test for roundoff error.</span>
    3910. 

  3910. <a name="l03846"></a>03846 <span class="comment">!</span>
    3911. 

  3911. <a name="l03847"></a>03847     <span class="keyword">if</span> ( a /= a1 .and. b /= b2 ) <span class="keyword">then</span>
    3912. 

  3912. <a name="l03848"></a>03848 
    3913. 

  3913. <a name="l03849"></a>03849       <span class="keyword">if</span> ( resas1 /= error1 .and. resas2 /= error2 ) <span class="keyword">then</span>
    3914. 

  3914. <a name="l03850"></a>03850 
    3915. 

  3915. <a name="l03851"></a>03851         <span class="keyword">if</span> ( abs ( rlist(maxerr) - area12 ) &lt; 1.0e-05 * abs ( area12 ) &amp;
    3916. 

  3916. <a name="l03852"></a>03852           .and.erro12 &gt;= 9.9e-01*errmax ) <span class="keyword">then</span>
    3917. 

  3917. <a name="l03853"></a>03853           iroff1 = iroff1 + 1
    3918. 

  3918. <a name="l03854"></a>03854         <span class="keyword">end if</span>
    3919. 

  3919. <a name="l03855"></a>03855 
    3920. 

  3920. <a name="l03856"></a>03856         <span class="keyword">if</span> ( last &gt; 10 .and. erro12 &gt; errmax ) <span class="keyword">then</span>
    3921. 

  3921. <a name="l03857"></a>03857           iroff2 = iroff2 + 1
    3922. 

  3922. <a name="l03858"></a>03858         <span class="keyword">end if</span>
    3923. 

  3923. <a name="l03859"></a>03859 
    3924. 

  3924. <a name="l03860"></a>03860       <span class="keyword">end if</span>
    3925. 

  3925. <a name="l03861"></a>03861 
    3926. 

  3926. <a name="l03862"></a>03862     <span class="keyword">end if</span>
    3927. 

  3927. <a name="l03863"></a>03863 
    3928. 

  3928. <a name="l03864"></a>03864     rlist(maxerr) = area1
    3929. 

  3929. <a name="l03865"></a>03865     rlist(last) = area2
    3930. 

  3930. <a name="l03866"></a>03866 <span class="comment">!</span>
    3931. 

  3931. <a name="l03867"></a>03867 <span class="comment">!  Test on accuracy.</span>
    3932. 

  3932. <a name="l03868"></a>03868 <span class="comment">!</span>
    3933. 

  3933. <a name="l03869"></a>03869     errbnd = max ( epsabs, epsrel * abs ( area ) )
    3934. 

  3934. <a name="l03870"></a>03870 
    3935. 

  3935. <a name="l03871"></a>03871     <span class="keyword">if</span> ( errsum &gt; errbnd ) <span class="keyword">then</span>
    3936. 

  3936. <a name="l03872"></a>03872 <span class="comment">!</span>
    3937. 

  3937. <a name="l03873"></a>03873 <span class="comment">!  Set error flag in the case that the number of interval</span>
    3938. 

  3938. <a name="l03874"></a>03874 <span class="comment">!  bisections exceeds limit.</span>
    3939. 

  3939. <a name="l03875"></a>03875 <span class="comment">!</span>
    3940. 

  3940. <a name="l03876"></a>03876       <span class="keyword">if</span> ( last == limit ) <span class="keyword">then</span>
    3941. 

  3941. <a name="l03877"></a>03877         ier = 1
    3942. 

  3942. <a name="l03878"></a>03878       <span class="keyword">end if</span>
    3943. 

  3943. <a name="l03879"></a>03879 <span class="comment">!</span>
    3944. 

  3944. <a name="l03880"></a>03880 <span class="comment">!  Set error flag in the case of roundoff error.</span>
    3945. 

  3945. <a name="l03881"></a>03881 <span class="comment">!</span>
    3946. 

  3946. <a name="l03882"></a>03882       <span class="keyword">if</span> ( iroff1 &gt;= 6 .or. iroff2 &gt;= 20 ) <span class="keyword">then</span>
    3947. 

  3947. <a name="l03883"></a>03883         ier = 2
    3948. 

  3948. <a name="l03884"></a>03884      <span class="keyword">end if</span>
    3949. 

  3949. <a name="l03885"></a>03885 <span class="comment">!</span>
    3950. 

  3950. <a name="l03886"></a>03886 <span class="comment">!  Set error flag in the case of bad integrand behavior</span>
    3951. 

  3951. <a name="l03887"></a>03887 <span class="comment">!  at interior points of integration range.</span>
    3952. 

  3952. <a name="l03888"></a>03888 <span class="comment">!</span>
    3953. 

  3953. <a name="l03889"></a>03889       <span class="keyword">if</span> ( max ( abs(a1),abs(b2)) &lt;= (1.0e+00+1.0e+03* epsilon ( a1 ) )* &amp;
    3954. 

  3954. <a name="l03890"></a>03890         ( abs(a2) + 1.0e+03* tiny ( a2) )) <span class="keyword">then</span>
    3955. 

  3955. <a name="l03891"></a>03891         ier = 3
    3956. 

  3956. <a name="l03892"></a>03892       <span class="keyword">end if</span>
    3957. 

  3957. <a name="l03893"></a>03893 
    3958. 

  3958. <a name="l03894"></a>03894     <span class="keyword">end if</span>
    3959. 

  3959. <a name="l03895"></a>03895 <span class="comment">!</span>
    3960. 

  3960. <a name="l03896"></a>03896 <span class="comment">!  Append the newly-created intervals to the list.</span>
    3961. 

  3961. <a name="l03897"></a>03897 <span class="comment">!</span>
    3962. 

  3962. <a name="l03898"></a>03898     <span class="keyword">if</span> ( error2 &lt;= error1 ) <span class="keyword">then</span>
    3963. 

  3963. <a name="l03899"></a>03899       alist(last) = a2
    3964. 

  3964. <a name="l03900"></a>03900       blist(maxerr) = b1
    3965. 

  3965. <a name="l03901"></a>03901       blist(last) = b2
    3966. 

  3966. <a name="l03902"></a>03902       elist(maxerr) = error1
    3967. 

  3967. <a name="l03903"></a>03903       elist(last) = error2
    3968. 

  3968. <a name="l03904"></a>03904     <span class="keyword">else</span>
    3969. 

  3969. <a name="l03905"></a>03905       alist(maxerr) = a2
    3970. 

  3970. <a name="l03906"></a>03906       alist(last) = a1
    3971. 

  3971. <a name="l03907"></a>03907       blist(last) = b1
    3972. 

  3972. <a name="l03908"></a>03908       rlist(maxerr) = area2
    3973. 

  3973. <a name="l03909"></a>03909       rlist(last) = area1
    3974. 

  3974. <a name="l03910"></a>03910       elist(maxerr) = error2
    3975. 

  3975. <a name="l03911"></a>03911       elist(last) = error1
    3976. 

  3976. <a name="l03912"></a>03912     <span class="keyword">end if</span>
    3977. 

  3977. <a name="l03913"></a>03913 <span class="comment">!</span>
    3978. 

  3978. <a name="l03914"></a>03914 <span class="comment">!  Call QSORT to maintain the descending ordering</span>
    3979. 

  3979. <a name="l03915"></a>03915 <span class="comment">!  in the list of error estimates and select the subinterval</span>
    3980. 

  3980. <a name="l03916"></a>03916 <span class="comment">!  with largest error estimate (to be bisected next).</span>
    3981. 

  3981. <a name="l03917"></a>03917 <span class="comment">!</span>
    3982. 

  3982. <a name="l03918"></a>03918     call <a class="code" href="quadpack_8f90.html#a55e08a684c5a6315fb37dd0fdc66d8e6">qsort </a>( limit, last, maxerr, errmax, elist, iord, nrmax )
    3983. 

  3983. <a name="l03919"></a>03919 
    3984. 

  3984. <a name="l03920"></a>03920     <span class="keyword">if</span> ( ier /= 0 .or. errsum &lt;= errbnd ) <span class="keyword">then</span>
    3985. 

  3985. <a name="l03921"></a>03921       exit
    3986. 

  3986. <a name="l03922"></a>03922     <span class="keyword">end if</span>
    3987. 

  3987. <a name="l03923"></a>03923 
    3988. 

  3988. <a name="l03924"></a>03924   <span class="keyword">end do</span>
    3989. 

  3989. <a name="l03925"></a>03925 <span class="comment">!</span>
    3990. 

  3990. <a name="l03926"></a>03926 <span class="comment">!  Compute final result.</span>
    3991. 

  3991. <a name="l03927"></a>03927 <span class="comment">!</span>
    3992. 

  3992. <a name="l03928"></a>03928   result = sum ( rlist(1:last) )
    3993. 

  3993. <a name="l03929"></a>03929 
    3994. 

  3994. <a name="l03930"></a>03930   abserr = errsum
    3995. 

  3995. <a name="l03931"></a>03931 
    3996. 

  3996. <a name="l03932"></a>03932   return
    3997. 

  3997. <a name="l03933"></a>03933 <span class="keyword">end</span>
    3998. 

  3998. <a name="l03934"></a><a class="code" href="quadpack_8f90.html#af8148c1623b7cf59159c491cfb1856f4">03934</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#af8148c1623b7cf59159c491cfb1856f4">qc25c</a> ( f, a, b, c, result, abserr, krul, neval )
    3999. 

  3999. <a name="l03935"></a>03935 
    4000. 

  4000. <a name="l03936"></a>03936 <span class="comment">!*****************************************************************************80</span>
    4001. 

  4001. <a name="l03937"></a>03937 <span class="comment">!</span>
    4002. 

  4002. <a name="l03938"></a>03938 <span class="comment">!! QC25C returns integration rules for Cauchy Principal Value integrals.</span>
    4003. 

  4003. <a name="l03939"></a>03939 <span class="comment">!</span>
    4004. 

  4004. <a name="l03940"></a>03940 <span class="comment">!  Discussion:</span>
    4005. 

  4005. <a name="l03941"></a>03941 <span class="comment">!</span>
    4006. 

  4006. <a name="l03942"></a>03942 <span class="comment">!    This routine estimates </span>
    4007. 

  4007. <a name="l03943"></a>03943 <span class="comment">!      I = integral of F(X) * W(X) over (a,b) </span>
    4008. 

  4008. <a name="l03944"></a>03944 <span class="comment">!    with error estimate, where </span>
    4009. 

  4009. <a name="l03945"></a>03945 <span class="comment">!      w(x) = 1/(x-c)</span>
    4010. 

  4010. <a name="l03946"></a>03946 <span class="comment">!</span>
    4011. 

  4011. <a name="l03947"></a>03947 <span class="comment">!  Author:</span>
    4012. 

  4012. <a name="l03948"></a>03948 <span class="comment">!</span>
    4013. 

  4013. <a name="l03949"></a>03949 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    4014. 

  4014. <a name="l03950"></a>03950 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    4015. 

  4015. <a name="l03951"></a>03951 <span class="comment">!</span>
    4016. 

  4016. <a name="l03952"></a>03952 <span class="comment">!  Reference:</span>
    4017. 

  4017. <a name="l03953"></a>03953 <span class="comment">!</span>
    4018. 

  4018. <a name="l03954"></a>03954 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    4019. 

  4019. <a name="l03955"></a>03955 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    4020. 

  4020. <a name="l03956"></a>03956 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    4021. 

  4021. <a name="l03957"></a>03957 <span class="comment">!    Springer Verlag, 1983</span>
    4022. 

  4022. <a name="l03958"></a>03958 <span class="comment">!</span>
    4023. 

  4023. <a name="l03959"></a>03959 <span class="comment">!  Parameters:</span>
    4024. 

  4024. <a name="l03960"></a>03960 <span class="comment">!</span>
    4025. 

  4025. <a name="l03961"></a>03961 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    4026. 

  4026. <a name="l03962"></a>03962 <span class="comment">!      function f ( x )</span>
    4027. 

  4027. <a name="l03963"></a>03963 <span class="comment">!      real f</span>
    4028. 

  4028. <a name="l03964"></a>03964 <span class="comment">!      real x</span>
    4029. 

  4029. <a name="l03965"></a>03965 <span class="comment">!    which evaluates the integrand function.</span>
    4030. 

  4030. <a name="l03966"></a>03966 <span class="comment">!</span>
    4031. 

  4031. <a name="l03967"></a>03967 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    4032. 

  4032. <a name="l03968"></a>03968 <span class="comment">!</span>
    4033. 

  4033. <a name="l03969"></a>03969 <span class="comment">!    Input, real C, the parameter in the weight function.</span>
    4034. 

  4034. <a name="l03970"></a>03970 <span class="comment">!</span>
    4035. 

  4035. <a name="l03971"></a>03971 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    4036. 

  4036. <a name="l03972"></a>03972 <span class="comment">!    RESULT is computed by using a generalized Clenshaw-Curtis method if</span>
    4037. 

  4037. <a name="l03973"></a>03973 <span class="comment">!    C lies within ten percent of the integration interval.  In the </span>
    4038. 

  4038. <a name="l03974"></a>03974 <span class="comment">!    other case the 15-point Kronrod rule obtained by optimal addition</span>
    4039. 

  4039. <a name="l03975"></a>03975 <span class="comment">!    of abscissae to the 7-point Gauss rule, is applied.</span>
    4040. 

  4040. <a name="l03976"></a>03976 <span class="comment">!</span>
    4041. 

  4041. <a name="l03977"></a>03977 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    4042. 

  4042. <a name="l03978"></a>03978 <span class="comment">!</span>
    4043. 

  4043. <a name="l03979"></a>03979 <span class="comment">!           krul   - integer</span>
    4044. 

  4044. <a name="l03980"></a>03980 <span class="comment">!                    key which is decreased by 1 if the 15-point</span>
    4045. 

  4045. <a name="l03981"></a>03981 <span class="comment">!                    Gauss-Kronrod scheme has been used</span>
    4046. 

  4046. <a name="l03982"></a>03982 <span class="comment">!</span>
    4047. 

  4047. <a name="l03983"></a>03983 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    4048. 

  4048. <a name="l03984"></a>03984 <span class="comment">!</span>
    4049. 

  4049. <a name="l03985"></a>03985 <span class="comment">!  Local parameters:</span>
    4050. 

  4050. <a name="l03986"></a>03986 <span class="comment">!</span>
    4051. 

  4051. <a name="l03987"></a>03987 <span class="comment">!           fval   - value of the function f at the points</span>
    4052. 

  4052. <a name="l03988"></a>03988 <span class="comment">!                    cos(k*pi/24),  k = 0, ..., 24</span>
    4053. 

  4053. <a name="l03989"></a>03989 <span class="comment">!           cheb12 - Chebyshev series expansion coefficients, for the</span>
    4054. 

  4054. <a name="l03990"></a>03990 <span class="comment">!                    function f, of degree 12</span>
    4055. 

  4055. <a name="l03991"></a>03991 <span class="comment">!           cheb24 - Chebyshev series expansion coefficients, for the</span>
    4056. 

  4056. <a name="l03992"></a>03992 <span class="comment">!                    function f, of degree 24</span>
    4057. 

  4057. <a name="l03993"></a>03993 <span class="comment">!           res12  - approximation to the integral corresponding to the</span>
    4058. 

  4058. <a name="l03994"></a>03994 <span class="comment">!                    use of cheb12</span>
    4059. 

  4059. <a name="l03995"></a>03995 <span class="comment">!           res24  - approximation to the integral corresponding to the</span>
    4060. 

  4060. <a name="l03996"></a>03996 <span class="comment">!                    use of cheb24</span>
    4061. 

  4061. <a name="l03997"></a>03997 <span class="comment">!           qwgtc  - external function subprogram defining the weight</span>
    4062. 

  4062. <a name="l03998"></a>03998 <span class="comment">!                    function</span>
    4063. 

  4063. <a name="l03999"></a>03999 <span class="comment">!           hlgth  - half-length of the interval</span>
    4064. 

  4064. <a name="l04000"></a>04000 <span class="comment">!           centr  - mid point of the interval</span>
    4065. 

  4065. <a name="l04001"></a>04001 <span class="comment">!</span>
    4066. 

  4066. <a name="l04002"></a>04002   <span class="keyword">implicit none</span>
    4067. 

  4067. <a name="l04003"></a>04003 
    4068. 

  4068. <a name="l04004"></a>04004   <span class="keywordtype">real</span> a
    4069. 

  4069. <a name="l04005"></a>04005   <span class="keywordtype">real</span> abserr
    4070. 

  4070. <a name="l04006"></a>04006   <span class="keywordtype">real</span> ak22
    4071. 

  4071. <a name="l04007"></a>04007   <span class="keywordtype">real</span> amom0
    4072. 

  4072. <a name="l04008"></a>04008   <span class="keywordtype">real</span> amom1
    4073. 

  4073. <a name="l04009"></a>04009   <span class="keywordtype">real</span> amom2
    4074. 

  4074. <a name="l04010"></a>04010   <span class="keywordtype">real</span> b
    4075. 

  4075. <a name="l04011"></a>04011   <span class="keywordtype">real</span> c
    4076. 

  4076. <a name="l04012"></a>04012   <span class="keywordtype">real</span> cc
    4077. 

  4077. <a name="l04013"></a>04013   <span class="keywordtype">real</span> centr
    4078. 

  4078. <a name="l04014"></a>04014   <span class="keywordtype">real</span> cheb12(13)
    4079. 

  4079. <a name="l04015"></a>04015   <span class="keywordtype">real</span> cheb24(25)
    4080. 

  4080. <a name="l04016"></a>04016   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    4081. 

  4081. <a name="l04017"></a>04017   <span class="keywordtype">real</span> fval(25)
    4082. 

  4082. <a name="l04018"></a>04018   <span class="keywordtype">real</span> hlgth
    4083. 

  4083. <a name="l04019"></a>04019   <span class="keywordtype">integer</span> i
    4084. 

  4084. <a name="l04020"></a>04020   <span class="keywordtype">integer</span> isym
    4085. 

  4085. <a name="l04021"></a>04021   <span class="keywordtype">integer</span> k 
    4086. 

  4086. <a name="l04022"></a>04022   <span class="keywordtype">integer</span> kp
    4087. 

  4087. <a name="l04023"></a>04023   <span class="keywordtype">integer</span> krul
    4088. 

  4088. <a name="l04024"></a>04024   <span class="keywordtype">integer</span> neval
    4089. 

  4089. <a name="l04025"></a>04025   <span class="keywordtype">real</span> p2
    4090. 

  4090. <a name="l04026"></a>04026   <span class="keywordtype">real</span> p3
    4091. 

  4091. <a name="l04027"></a>04027   <span class="keywordtype">real</span> p4
    4092. 

  4092. <a name="l04028"></a>04028   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: qwgtc
    4093. 

  4093. <a name="l04029"></a>04029   <span class="keywordtype">real</span> resabs
    4094. 

  4094. <a name="l04030"></a>04030   <span class="keywordtype">real</span> resasc
    4095. 

  4095. <a name="l04031"></a>04031   <span class="keywordtype">real</span> result
    4096. 

  4096. <a name="l04032"></a>04032   <span class="keywordtype">real</span> res12
    4097. 

  4097. <a name="l04033"></a>04033   <span class="keywordtype">real</span> res24
    4098. 

  4098. <a name="l04034"></a>04034   <span class="keywordtype">real</span> u
    4099. 

  4099. <a name="l04035"></a>04035   <span class="keywordtype">real</span>, <span class="keywordtype">parameter</span>, <span class="keywordtype">dimension ( 11 )</span> :: x = (/ 
    4100. 

  4100. <a name="l04036"></a>04036     9.914448613738104e-01, 9.659258262890683e-01, 
    4101. 

  4101. <a name="l04037"></a>04037     9.238795325112868e-01, 8.660254037844386e-01, 
    4102. 

  4102. <a name="l04038"></a>04038     7.933533402912352e-01, 7.071067811865475e-01, 
    4103. 

  4103. <a name="l04039"></a>04039     6.087614290087206e-01, 5.000000000000000e-01, 
    4104. 

  4104. <a name="l04040"></a>04040     3.826834323650898e-01, 2.588190451025208e-01, 
    4105. 

  4105. <a name="l04041"></a>04041     1.305261922200516e-01 /)
    4106. 

  4106. <a name="l04042"></a>04042 <span class="comment">!</span>
    4107. 

  4107. <a name="l04043"></a>04043 <span class="comment">!  Check the position of C.</span>
    4108. 

  4108. <a name="l04044"></a>04044 <span class="comment">!</span>
    4109. 

  4109. <a name="l04045"></a>04045   cc = ( 2.0e+00 * c - b - a ) / ( b - a )
    4110. 

  4110. <a name="l04046"></a>04046 <span class="comment">!</span>
    4111. 

  4111. <a name="l04047"></a>04047 <span class="comment">!  Apply the 15-point Gauss-Kronrod scheme.</span>
    4112. 

  4112. <a name="l04048"></a>04048 <span class="comment">!</span>
    4113. 

  4113. <a name="l04049"></a>04049   <span class="keyword">if</span> ( abs ( cc ) &gt;= 1.1e+00 ) <span class="keyword">then</span>
    4114. 

  4114. <a name="l04050"></a>04050     krul = krul - 1
    4115. 

  4115. <a name="l04051"></a>04051     call <a class="code" href="quadpack_8f90.html#a0c083838940925726abd5bc85fa29587">qk15w </a>( f, qwgtc, c, p2, p3, p4, kp, a, b, result, abserr, &amp;
    4116. 

  4116. <a name="l04052"></a>04052       resabs, resasc )
    4117. 

  4117. <a name="l04053"></a>04053     neval = 15
    4118. 

  4118. <a name="l04054"></a>04054     <span class="keyword">if</span> ( resasc == abserr ) <span class="keyword">then</span>
    4119. 

  4119. <a name="l04055"></a>04055       krul = krul+1
    4120. 

  4120. <a name="l04056"></a>04056     <span class="keyword">end if</span>
    4121. 

  4121. <a name="l04057"></a>04057     return
    4122. 

  4122. <a name="l04058"></a>04058   <span class="keyword">end if</span>
    4123. 

  4123. <a name="l04059"></a>04059 <span class="comment">!</span>
    4124. 

  4124. <a name="l04060"></a>04060 <span class="comment">!  Use the generalized Clenshaw-Curtis method.</span>
    4125. 

  4125. <a name="l04061"></a>04061 <span class="comment">!</span>
    4126. 

  4126. <a name="l04062"></a>04062   hlgth = 5.0e-01 * ( b - a )
    4127. 

  4127. <a name="l04063"></a>04063   centr = 5.0e-01 * ( b + a )
    4128. 

  4128. <a name="l04064"></a>04064   neval = 25
    4129. 

  4129. <a name="l04065"></a>04065   fval(1) = 5.0e-01 * f(hlgth+centr)
    4130. 

  4130. <a name="l04066"></a>04066   fval(13) = f(centr)
    4131. 

  4131. <a name="l04067"></a>04067   fval(25) = 5.0e-01 * f(centr-hlgth)
    4132. 

  4132. <a name="l04068"></a>04068 
    4133. 

  4133. <a name="l04069"></a>04069   <span class="keyword">do</span> i = 2, 12
    4134. 

  4134. <a name="l04070"></a>04070     u = hlgth * x(i-1)
    4135. 

  4135. <a name="l04071"></a>04071     isym = 26 - i
    4136. 

  4136. <a name="l04072"></a>04072     fval(i) = f(u+centr)
    4137. 

  4137. <a name="l04073"></a>04073     fval(isym) = f(centr-u)
    4138. 

  4138. <a name="l04074"></a>04074   <span class="keyword">end do</span>
    4139. 

  4139. <a name="l04075"></a>04075 <span class="comment">!</span>
    4140. 

  4140. <a name="l04076"></a>04076 <span class="comment">!  Compute the Chebyshev series expansion.</span>
    4141. 

  4141. <a name="l04077"></a>04077 <span class="comment">!</span>
    4142. 

  4142. <a name="l04078"></a>04078   call <a class="code" href="quadpack_8f90.html#ad5beefcfdb335ea68ccf8397536c8c36">qcheb </a>( x, fval, cheb12, cheb24 )
    4143. 

  4143. <a name="l04079"></a>04079 <span class="comment">!</span>
    4144. 

  4144. <a name="l04080"></a>04080 <span class="comment">!  The modified Chebyshev moments are computed by forward</span>
    4145. 

  4145. <a name="l04081"></a>04081 <span class="comment">!  recursion, using AMOM0 and AMOM1 as starting values.</span>
    4146. 

  4146. <a name="l04082"></a>04082 <span class="comment">!</span>
    4147. 

  4147. <a name="l04083"></a>04083   amom0 = log ( abs ( ( 1.0e+00 - cc ) / ( 1.0e+00 + cc ) ) )
    4148. 

  4148. <a name="l04084"></a>04084   amom1 = 2.0e+00 + cc * amom0
    4149. 

  4149. <a name="l04085"></a>04085   res12 = cheb12(1) * amom0 + cheb12(2) * amom1
    4150. 

  4150. <a name="l04086"></a>04086   res24 = cheb24(1) * amom0 + cheb24(2) * amom1
    4151. 

  4151. <a name="l04087"></a>04087 
    4152. 

  4152. <a name="l04088"></a>04088   <span class="keyword">do</span> k = 3, 13
    4153. 

  4153. <a name="l04089"></a>04089     amom2 = 2.0e+00 * cc * amom1 - amom0
    4154. 

  4154. <a name="l04090"></a>04090     ak22 = ( k - 2 ) * ( k - 2 )
    4155. 

  4155. <a name="l04091"></a>04091     <span class="keyword">if</span> ( ( k / 2 ) * 2 == k ) <span class="keyword">then</span>
    4156. 

  4156. <a name="l04092"></a>04092       amom2 = amom2 - 4.0e+00 / ( ak22 - 1.0e+00 )
    4157. 

  4157. <a name="l04093"></a>04093     <span class="keyword">end if</span>
    4158. 

  4158. <a name="l04094"></a>04094     res12 = res12 + cheb12(k) * amom2
    4159. 

  4159. <a name="l04095"></a>04095     res24 = res24 + cheb24(k) * amom2
    4160. 

  4160. <a name="l04096"></a>04096     amom0 = amom1
    4161. 

  4161. <a name="l04097"></a>04097     amom1 = amom2
    4162. 

  4162. <a name="l04098"></a>04098   <span class="keyword">end do</span>
    4163. 

  4163. <a name="l04099"></a>04099 
    4164. 

  4164. <a name="l04100"></a>04100   <span class="keyword">do</span> k = 14, 25
    4165. 

  4165. <a name="l04101"></a>04101     amom2 = 2.0e+00 * cc * amom1 - amom0
    4166. 

  4166. <a name="l04102"></a>04102     ak22 = ( k - 2 ) * ( k - 2 )
    4167. 

  4167. <a name="l04103"></a>04103     <span class="keyword">if</span> ( ( k / 2 ) * 2 == k ) <span class="keyword">then</span>
    4168. 

  4168. <a name="l04104"></a>04104       amom2 = amom2 - 4.0e+00 / ( ak22 - 1.0e+00 )
    4169. 

  4169. <a name="l04105"></a>04105     <span class="keyword">end if</span>
    4170. 

  4170. <a name="l04106"></a>04106     res24 = res24 + cheb24(k) * amom2
    4171. 

  4171. <a name="l04107"></a>04107     amom0 = amom1
    4172. 

  4172. <a name="l04108"></a>04108     amom1 = amom2
    4173. 

  4173. <a name="l04109"></a>04109   <span class="keyword">end do</span>
    4174. 

  4174. <a name="l04110"></a>04110 
    4175. 

  4175. <a name="l04111"></a>04111   result = res24
    4176. 

  4176. <a name="l04112"></a>04112   abserr = abs ( res24 - res12 )
    4177. 

  4177. <a name="l04113"></a>04113 
    4178. 

  4178. <a name="l04114"></a>04114   return
    4179. 

  4179. <a name="l04115"></a>04115 <span class="keyword">end</span>
    4180. 

  4180. <a name="l04116"></a><a class="code" href="quadpack_8f90.html#ab0843f4831942d2c9bf3430cb71aca06">04116</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#ab0843f4831942d2c9bf3430cb71aca06">qc25o</a> ( f, a, b, omega, integr, nrmom, maxp1, ksave, result, &amp;
    4181. 

  4181. <a name="l04117"></a>04117   abserr, neval, resabs, resasc, momcom, chebmo )
    4182. 

  4182. <a name="l04118"></a>04118 
    4183. 

  4183. <a name="l04119"></a>04119 <span class="comment">!*****************************************************************************80</span>
    4184. 

  4184. <a name="l04120"></a>04120 <span class="comment">!</span>
    4185. 

  4185. <a name="l04121"></a>04121 <span class="comment">!! QC25O returns integration rules for integrands with a COS or SIN factor.</span>
    4186. 

  4186. <a name="l04122"></a>04122 <span class="comment">!</span>
    4187. 

  4187. <a name="l04123"></a>04123 <span class="comment">!  Discussion:</span>
    4188. 

  4188. <a name="l04124"></a>04124 <span class="comment">!</span>
    4189. 

  4189. <a name="l04125"></a>04125 <span class="comment">!    This routine estimates the integral</span>
    4190. 

  4190. <a name="l04126"></a>04126 <span class="comment">!      I = integral of f(x) * w(x) over (a,b)</span>
    4191. 

  4191. <a name="l04127"></a>04127 <span class="comment">!    where</span>
    4192. 

  4192. <a name="l04128"></a>04128 <span class="comment">!      w(x) = cos(omega*x)</span>
    4193. 

  4193. <a name="l04129"></a>04129 <span class="comment">!    or </span>
    4194. 

  4194. <a name="l04130"></a>04130 <span class="comment">!      w(x) = sin(omega*x),</span>
    4195. 

  4195. <a name="l04131"></a>04131 <span class="comment">!    and estimates</span>
    4196. 

  4196. <a name="l04132"></a>04132 <span class="comment">!      J = integral ( A &lt;= X &lt;= B ) |F(X)| dx.</span>
    4197. 

  4197. <a name="l04133"></a>04133 <span class="comment">!</span>
    4198. 

  4198. <a name="l04134"></a>04134 <span class="comment">!    For small values of OMEGA or small intervals (a,b) the 15-point</span>
    4199. 

  4199. <a name="l04135"></a>04135 <span class="comment">!    Gauss-Kronrod rule is used.  In all other cases a generalized</span>
    4200. 

  4200. <a name="l04136"></a>04136 <span class="comment">!    Clenshaw-Curtis method is used, that is, a truncated Chebyshev </span>
    4201. 

  4201. <a name="l04137"></a>04137 <span class="comment">!    expansion of the function F is computed on (a,b), so that the </span>
    4202. 

  4202. <a name="l04138"></a>04138 <span class="comment">!    integrand can be written as a sum of terms of the form W(X)*T(K,X), </span>
    4203. 

  4203. <a name="l04139"></a>04139 <span class="comment">!    where T(K,X) is the Chebyshev polynomial of degree K.  The Chebyshev</span>
    4204. 

  4204. <a name="l04140"></a>04140 <span class="comment">!    moments are computed with use of a linear recurrence relation.</span>
    4205. 

  4205. <a name="l04141"></a>04141 <span class="comment">!</span>
    4206. 

  4206. <a name="l04142"></a>04142 <span class="comment">!  Author:</span>
    4207. 

  4207. <a name="l04143"></a>04143 <span class="comment">!</span>
    4208. 

  4208. <a name="l04144"></a>04144 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    4209. 

  4209. <a name="l04145"></a>04145 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    4210. 

  4210. <a name="l04146"></a>04146 <span class="comment">!</span>
    4211. 

  4211. <a name="l04147"></a>04147 <span class="comment">!  Reference:</span>
    4212. 

  4212. <a name="l04148"></a>04148 <span class="comment">!</span>
    4213. 

  4213. <a name="l04149"></a>04149 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    4214. 

  4214. <a name="l04150"></a>04150 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    4215. 

  4215. <a name="l04151"></a>04151 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    4216. 

  4216. <a name="l04152"></a>04152 <span class="comment">!    Springer Verlag, 1983</span>
    4217. 

  4217. <a name="l04153"></a>04153 <span class="comment">!</span>
    4218. 

  4218. <a name="l04154"></a>04154 <span class="comment">!  Parameters:</span>
    4219. 

  4219. <a name="l04155"></a>04155 <span class="comment">!</span>
    4220. 

  4220. <a name="l04156"></a>04156 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    4221. 

  4221. <a name="l04157"></a>04157 <span class="comment">!      function f ( x )</span>
    4222. 

  4222. <a name="l04158"></a>04158 <span class="comment">!      real f</span>
    4223. 

  4223. <a name="l04159"></a>04159 <span class="comment">!      real x</span>
    4224. 

  4224. <a name="l04160"></a>04160 <span class="comment">!    which evaluates the integrand function.</span>
    4225. 

  4225. <a name="l04161"></a>04161 <span class="comment">!</span>
    4226. 

  4226. <a name="l04162"></a>04162 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    4227. 

  4227. <a name="l04163"></a>04163 <span class="comment">!</span>
    4228. 

  4228. <a name="l04164"></a>04164 <span class="comment">!    Input, real OMEGA, the parameter in the weight function.</span>
    4229. 

  4229. <a name="l04165"></a>04165 <span class="comment">!</span>
    4230. 

  4230. <a name="l04166"></a>04166 <span class="comment">!    Input, integer INTEGR, indicates which weight function is to be used</span>
    4231. 

  4231. <a name="l04167"></a>04167 <span class="comment">!    = 1, w(x) = cos(omega*x)</span>
    4232. 

  4232. <a name="l04168"></a>04168 <span class="comment">!    = 2, w(x) = sin(omega*x)</span>
    4233. 

  4233. <a name="l04169"></a>04169 <span class="comment">!</span>
    4234. 

  4234. <a name="l04170"></a>04170 <span class="comment">!    ?, integer NRMOM, the length of interval (a,b) is equal to the length</span>
    4235. 

  4235. <a name="l04171"></a>04171 <span class="comment">!    of the original integration interval divided by</span>
    4236. 

  4236. <a name="l04172"></a>04172 <span class="comment">!    2**nrmom (we suppose that the routine is used in an</span>
    4237. 

  4237. <a name="l04173"></a>04173 <span class="comment">!    adaptive integration process, otherwise set</span>
    4238. 

  4238. <a name="l04174"></a>04174 <span class="comment">!    nrmom = 0).  nrmom must be zero at the first call.</span>
    4239. 

  4239. <a name="l04175"></a>04175 <span class="comment">!</span>
    4240. 

  4240. <a name="l04176"></a>04176 <span class="comment">!           maxp1  - integer</span>
    4241. 

  4241. <a name="l04177"></a>04177 <span class="comment">!                    gives an upper bound on the number of Chebyshev</span>
    4242. 

  4242. <a name="l04178"></a>04178 <span class="comment">!                    moments which can be stored, i.e. for the intervals</span>
    4243. 

  4243. <a name="l04179"></a>04179 <span class="comment">!                    of lengths abs(bb-aa)*2**(-l), l = 0,1,2, ...,</span>
    4244. 

  4244. <a name="l04180"></a>04180 <span class="comment">!                    maxp1-2.</span>
    4245. 

  4245. <a name="l04181"></a>04181 <span class="comment">!</span>
    4246. 

  4246. <a name="l04182"></a>04182 <span class="comment">!           ksave  - integer</span>
    4247. 

  4247. <a name="l04183"></a>04183 <span class="comment">!                    key which is one when the moments for the</span>
    4248. 

  4248. <a name="l04184"></a>04184 <span class="comment">!                    current interval have been computed</span>
    4249. 

  4249. <a name="l04185"></a>04185 <span class="comment">!</span>
    4250. 

  4250. <a name="l04186"></a>04186 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    4251. 

  4251. <a name="l04187"></a>04187 <span class="comment">!</span>
    4252. 

  4252. <a name="l04188"></a>04188 <span class="comment">!           abserr - real</span>
    4253. 

  4253. <a name="l04189"></a>04189 <span class="comment">!                    estimate of the modulus of the absolute</span>
    4254. 

  4254. <a name="l04190"></a>04190 <span class="comment">!                    error, which should equal or exceed abs(i-result)</span>
    4255. 

  4255. <a name="l04191"></a>04191 <span class="comment">!</span>
    4256. 

  4256. <a name="l04192"></a>04192 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    4257. 

  4257. <a name="l04193"></a>04193 <span class="comment">!</span>
    4258. 

  4258. <a name="l04194"></a>04194 <span class="comment">!    Output, real RESABS, approximation to the integral J.</span>
    4259. 

  4259. <a name="l04195"></a>04195 <span class="comment">!</span>
    4260. 

  4260. <a name="l04196"></a>04196 <span class="comment">!    Output, real RESASC, approximation to the integral of abs(F-I/(B-A)).</span>
    4261. 

  4261. <a name="l04197"></a>04197 <span class="comment">!</span>
    4262. 

  4262. <a name="l04198"></a>04198 <span class="comment">!         on entry and return</span>
    4263. 

  4263. <a name="l04199"></a>04199 <span class="comment">!           momcom - integer</span>
    4264. 

  4264. <a name="l04200"></a>04200 <span class="comment">!                    for each interval length we need to compute</span>
    4265. 

  4265. <a name="l04201"></a>04201 <span class="comment">!                    the Chebyshev moments. momcom counts the number</span>
    4266. 

  4266. <a name="l04202"></a>04202 <span class="comment">!                    of intervals for which these moments have already</span>
    4267. 

  4267. <a name="l04203"></a>04203 <span class="comment">!                    been computed. if nrmom &lt; momcom or ksave = 1,</span>
    4268. 

  4268. <a name="l04204"></a>04204 <span class="comment">!                    the Chebyshev moments for the interval (a,b)</span>
    4269. 

  4269. <a name="l04205"></a>04205 <span class="comment">!                    have already been computed and stored, otherwise</span>
    4270. 

  4270. <a name="l04206"></a>04206 <span class="comment">!                    we compute them and we increase momcom.</span>
    4271. 

  4271. <a name="l04207"></a>04207 <span class="comment">!</span>
    4272. 

  4272. <a name="l04208"></a>04208 <span class="comment">!           chebmo - real</span>
    4273. 

  4273. <a name="l04209"></a>04209 <span class="comment">!                    array of dimension at least (maxp1,25) containing</span>
    4274. 

  4274. <a name="l04210"></a>04210 <span class="comment">!                    the modified Chebyshev moments for the first momcom</span>
    4275. 

  4275. <a name="l04211"></a>04211 <span class="comment">!                    interval lengths</span>
    4276. 

  4276. <a name="l04212"></a>04212 <span class="comment">!</span>
    4277. 

  4277. <a name="l04213"></a>04213 <span class="comment">!  Local parameters:</span>
    4278. 

  4278. <a name="l04214"></a>04214 <span class="comment">!</span>
    4279. 

  4279. <a name="l04215"></a>04215 <span class="comment">!    maxp1 gives an upper bound</span>
    4280. 

  4280. <a name="l04216"></a>04216 <span class="comment">!           on the number of Chebyshev moments which can be</span>
    4281. 

  4281. <a name="l04217"></a>04217 <span class="comment">!           computed, i.e. for the interval (bb-aa), ...,</span>
    4282. 

  4282. <a name="l04218"></a>04218 <span class="comment">!           (bb-aa)/2**(maxp1-2).</span>
    4283. 

  4283. <a name="l04219"></a>04219 <span class="comment">!           should this number be altered, the first dimension of</span>
    4284. 

  4284. <a name="l04220"></a>04220 <span class="comment">!           chebmo needs to be adapted.</span>
    4285. 

  4285. <a name="l04221"></a>04221 <span class="comment">!</span>
    4286. 

  4286. <a name="l04222"></a>04222 <span class="comment">!    x contains the values cos(k*pi/24)</span>
    4287. 

  4287. <a name="l04223"></a>04223 <span class="comment">!           k = 1, ...,11, to be used for the Chebyshev expansion of f</span>
    4288. 

  4288. <a name="l04224"></a>04224 <span class="comment">!</span>
    4289. 

  4289. <a name="l04225"></a>04225 <span class="comment">!           centr  - mid point of the integration interval</span>
    4290. 

  4290. <a name="l04226"></a>04226 <span class="comment">!           hlgth  - half length of the integration interval</span>
    4291. 

  4291. <a name="l04227"></a>04227 <span class="comment">!           fval   - value of the function f at the points</span>
    4292. 

  4292. <a name="l04228"></a>04228 <span class="comment">!                    (b-a)*0.5*cos(k*pi/12) + (b+a)*0.5</span>
    4293. 

  4293. <a name="l04229"></a>04229 <span class="comment">!                    k = 0, ...,24</span>
    4294. 

  4294. <a name="l04230"></a>04230 <span class="comment">!           cheb12 - coefficients of the Chebyshev series expansion</span>
    4295. 

  4295. <a name="l04231"></a>04231 <span class="comment">!                    of degree 12, for the function f, in the</span>
    4296. 

  4296. <a name="l04232"></a>04232 <span class="comment">!                    interval (a,b)</span>
    4297. 

  4297. <a name="l04233"></a>04233 <span class="comment">!           cheb24 - coefficients of the Chebyshev series expansion</span>
    4298. 

  4298. <a name="l04234"></a>04234 <span class="comment">!                    of degree 24, for the function f, in the</span>
    4299. 

  4299. <a name="l04235"></a>04235 <span class="comment">!                    interval (a,b)</span>
    4300. 

  4300. <a name="l04236"></a>04236 <span class="comment">!           resc12 - approximation to the integral of</span>
    4301. 

  4301. <a name="l04237"></a>04237 <span class="comment">!                    cos(0.5*(b-a)*omega*x)*f(0.5*(b-a)*x+0.5*(b+a))</span>
    4302. 

  4302. <a name="l04238"></a>04238 <span class="comment">!                    over (-1,+1), using the Chebyshev series</span>
    4303. 

  4303. <a name="l04239"></a>04239 <span class="comment">!                    expansion of degree 12</span>
    4304. 

  4304. <a name="l04240"></a>04240 <span class="comment">!           resc24 - approximation to the same integral, using the</span>
    4305. 

  4305. <a name="l04241"></a>04241 <span class="comment">!                    Chebyshev series expansion of degree 24</span>
    4306. 

  4306. <a name="l04242"></a>04242 <span class="comment">!           ress12 - the analogue of resc12 for the sine</span>
    4307. 

  4307. <a name="l04243"></a>04243 <span class="comment">!           ress24 - the analogue of resc24 for the sine</span>
    4308. 

  4308. <a name="l04244"></a>04244 <span class="comment">!</span>
    4309. 

  4309. <a name="l04245"></a>04245   <span class="keyword">implicit none</span>
    4310. 

  4310. <a name="l04246"></a>04246 
    4311. 

  4311. <a name="l04247"></a>04247   <span class="keywordtype">integer</span> maxp1
    4312. 

  4312. <a name="l04248"></a>04248 
    4313. 

  4313. <a name="l04249"></a>04249   <span class="keywordtype">real</span> a
    4314. 

  4314. <a name="l04250"></a>04250   <span class="keywordtype">real</span> abserr
    4315. 

  4315. <a name="l04251"></a>04251   <span class="keywordtype">real</span> ac
    4316. 

  4316. <a name="l04252"></a>04252   <span class="keywordtype">real</span> an
    4317. 

  4317. <a name="l04253"></a>04253   <span class="keywordtype">real</span> an2
    4318. 

  4318. <a name="l04254"></a>04254   <span class="keywordtype">real</span> as
    4319. 

  4319. <a name="l04255"></a>04255   <span class="keywordtype">real</span> asap
    4320. 

  4320. <a name="l04256"></a>04256   <span class="keywordtype">real</span> ass
    4321. 

  4321. <a name="l04257"></a>04257   <span class="keywordtype">real</span> b
    4322. 

  4322. <a name="l04258"></a>04258   <span class="keywordtype">real</span> centr
    4323. 

  4323. <a name="l04259"></a>04259   <span class="keywordtype">real</span> chebmo(maxp1,25)
    4324. 

  4324. <a name="l04260"></a>04260   <span class="keywordtype">real</span> cheb12(13)
    4325. 

  4325. <a name="l04261"></a>04261   <span class="keywordtype">real</span> cheb24(25)
    4326. 

  4326. <a name="l04262"></a>04262   <span class="keywordtype">real</span> conc
    4327. 

  4327. <a name="l04263"></a>04263   <span class="keywordtype">real</span> cons
    4328. 

  4328. <a name="l04264"></a>04264   <span class="keywordtype">real</span> cospar
    4329. 

  4329. <a name="l04265"></a>04265   <span class="keywordtype">real</span> d(28)
    4330. 

  4330. <a name="l04266"></a>04266   <span class="keywordtype">real</span> d1(28)
    4331. 

  4331. <a name="l04267"></a>04267   <span class="keywordtype">real</span> d2(28)
    4332. 

  4332. <a name="l04268"></a>04268   <span class="keywordtype">real</span> d3(28)
    4333. 

  4333. <a name="l04269"></a>04269   <span class="keywordtype">real</span> estc
    4334. 

  4334. <a name="l04270"></a>04270   <span class="keywordtype">real</span> ests
    4335. 

  4335. <a name="l04271"></a>04271   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    4336. 

  4336. <a name="l04272"></a>04272   <span class="keywordtype">real</span> fval(25)
    4337. 

  4337. <a name="l04273"></a>04273   <span class="keywordtype">real</span> hlgth
    4338. 

  4338. <a name="l04274"></a>04274   <span class="keywordtype">integer</span> i
    4339. 

  4339. <a name="l04275"></a>04275   <span class="keywordtype">integer</span> integr
    4340. 

  4340. <a name="l04276"></a>04276   <span class="keywordtype">integer</span> isym
    4341. 

  4341. <a name="l04277"></a>04277   <span class="keywordtype">integer</span> j
    4342. 

  4342. <a name="l04278"></a>04278   <span class="keywordtype">integer</span> k
    4343. 

  4343. <a name="l04279"></a>04279   <span class="keywordtype">integer</span> ksave
    4344. 

  4344. <a name="l04280"></a>04280   <span class="keywordtype">integer</span> m
    4345. 

  4345. <a name="l04281"></a>04281   <span class="keywordtype">integer</span> momcom
    4346. 

  4346. <a name="l04282"></a>04282   <span class="keywordtype">integer</span> neval
    4347. 

  4347. <a name="l04283"></a>04283   <span class="keywordtype">integer</span>, <span class="keywordtype">parameter</span> :: nmac = 28
    4348. 

  4348. <a name="l04284"></a>04284   <span class="keywordtype">integer</span> noeq1
    4349. 

  4349. <a name="l04285"></a>04285   <span class="keywordtype">integer</span> noequ
    4350. 

  4350. <a name="l04286"></a>04286   <span class="keywordtype">integer</span> nrmom
    4351. 

  4351. <a name="l04287"></a>04287   <span class="keywordtype">real</span> omega
    4352. 

  4352. <a name="l04288"></a>04288   <span class="keywordtype">real</span> parint
    4353. 

  4353. <a name="l04289"></a>04289   <span class="keywordtype">real</span> par2
    4354. 

  4354. <a name="l04290"></a>04290   <span class="keywordtype">real</span> par22
    4355. 

  4355. <a name="l04291"></a>04291   <span class="keywordtype">real</span> p2
    4356. 

  4356. <a name="l04292"></a>04292   <span class="keywordtype">real</span> p3
    4357. 

  4357. <a name="l04293"></a>04293   <span class="keywordtype">real</span> p4
    4358. 

  4358. <a name="l04294"></a>04294   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: qwgto
    4359. 

  4359. <a name="l04295"></a>04295   <span class="keywordtype">real</span> resabs
    4360. 

  4360. <a name="l04296"></a>04296   <span class="keywordtype">real</span> resasc
    4361. 

  4361. <a name="l04297"></a>04297   <span class="keywordtype">real</span> resc12
    4362. 

  4362. <a name="l04298"></a>04298   <span class="keywordtype">real</span> resc24
    4363. 

  4363. <a name="l04299"></a>04299   <span class="keywordtype">real</span> ress12
    4364. 

  4364. <a name="l04300"></a>04300   <span class="keywordtype">real</span> ress24
    4365. 

  4365. <a name="l04301"></a>04301   <span class="keywordtype">real</span> result
    4366. 

  4366. <a name="l04302"></a>04302   <span class="keywordtype">real</span> sinpar
    4367. 

  4367. <a name="l04303"></a>04303   <span class="keywordtype">real</span> v(28)
    4368. 

  4368. <a name="l04304"></a>04304   <span class="keywordtype">real</span>, <span class="keywordtype">dimension ( 11 )</span> :: x = (/ 
    4369. 

  4369. <a name="l04305"></a>04305     9.914448613738104e-01,     9.659258262890683e-01, 
    4370. 

  4370. <a name="l04306"></a>04306     9.238795325112868e-01,     8.660254037844386e-01, 
    4371. 

  4371. <a name="l04307"></a>04307     7.933533402912352e-01,     7.071067811865475e-01, 
    4372. 

  4372. <a name="l04308"></a>04308     6.087614290087206e-01,     5.000000000000000e-01, 
    4373. 

  4373. <a name="l04309"></a>04309     3.826834323650898e-01,     2.588190451025208e-01, 
    4374. 

  4374. <a name="l04310"></a>04310     1.305261922200516e-01 /)
    4375. 

  4375. <a name="l04311"></a>04311 
    4376. 

  4376. <a name="l04312"></a>04312   centr = 5.0e-01 * ( b + a )
    4377. 

  4377. <a name="l04313"></a>04313   hlgth = 5.0e-01 * ( b - a )
    4378. 

  4378. <a name="l04314"></a>04314   parint = omega * hlgth
    4379. 

  4379. <a name="l04315"></a>04315 <span class="comment">!</span>
    4380. 

  4380. <a name="l04316"></a>04316 <span class="comment">!  Compute the integral using the 15-point Gauss-Kronrod</span>
    4381. 

  4381. <a name="l04317"></a>04317 <span class="comment">!  formula if the value of the parameter in the integrand</span>
    4382. 

  4382. <a name="l04318"></a>04318 <span class="comment">!  is small or if the length of the integration interval</span>
    4383. 

  4383. <a name="l04319"></a>04319 <span class="comment">!  is less than (bb-aa)/2**(maxp1-2), where (aa,bb) is the</span>
    4384. 

  4384. <a name="l04320"></a>04320 <span class="comment">!  original integration interval.</span>
    4385. 

  4385. <a name="l04321"></a>04321 <span class="comment">!</span>
    4386. 

  4386. <a name="l04322"></a>04322   <span class="keyword">if</span> ( abs ( parint ) &lt;= 2.0e+00 ) <span class="keyword">then</span>
    4387. 

  4387. <a name="l04323"></a>04323 
    4388. 

  4388. <a name="l04324"></a>04324     call <a class="code" href="quadpack_8f90.html#a0c083838940925726abd5bc85fa29587">qk15w </a>( f, qwgto, omega, p2, p3, p4, integr, a, b, result, &amp;
    4389. 

  4389. <a name="l04325"></a>04325       abserr, resabs, resasc )
    4390. 

  4390. <a name="l04326"></a>04326 
    4391. 

  4391. <a name="l04327"></a>04327     neval = 15
    4392. 

  4392. <a name="l04328"></a>04328     return
    4393. 

  4393. <a name="l04329"></a>04329 
    4394. 

  4394. <a name="l04330"></a>04330   <span class="keyword">end if</span>
    4395. 

  4395. <a name="l04331"></a>04331 <span class="comment">!</span>
    4396. 

  4396. <a name="l04332"></a>04332 <span class="comment">!  Compute the integral using the generalized clenshaw-curtis method.</span>
    4397. 

  4397. <a name="l04333"></a>04333 <span class="comment">!</span>
    4398. 

  4398. <a name="l04334"></a>04334   conc = hlgth * cos(centr*omega)
    4399. 

  4399. <a name="l04335"></a>04335   cons = hlgth * sin(centr*omega)
    4400. 

  4400. <a name="l04336"></a>04336   resasc = huge ( resasc )
    4401. 

  4401. <a name="l04337"></a>04337   neval = 25
    4402. 

  4402. <a name="l04338"></a>04338 <span class="comment">!</span>
    4403. 

  4403. <a name="l04339"></a>04339 <span class="comment">!  Check whether the Chebyshev moments for this interval</span>
    4404. 

  4404. <a name="l04340"></a>04340 <span class="comment">!  have already been computed.</span>
    4405. 

  4405. <a name="l04341"></a>04341 <span class="comment">!</span>
    4406. 

  4406. <a name="l04342"></a>04342   <span class="keyword">if</span> ( nrmom &lt; momcom .or. ksave == 1 ) <span class="keyword">then</span>
    4407. 

  4407. <a name="l04343"></a>04343     go to 140
    4408. 

  4408. <a name="l04344"></a>04344   <span class="keyword">end if</span>
    4409. 

  4409. <a name="l04345"></a>04345 <span class="comment">!</span>
    4410. 

  4410. <a name="l04346"></a>04346 <span class="comment">!  Compute a new set of Chebyshev moments.</span>
    4411. 

  4411. <a name="l04347"></a>04347 <span class="comment">!</span>
    4412. 

  4412. <a name="l04348"></a>04348   m = momcom + 1
    4413. 

  4413. <a name="l04349"></a>04349   par2 = parint * parint
    4414. 

  4414. <a name="l04350"></a>04350   par22 = par2 + 2.0e+00
    4415. 

  4415. <a name="l04351"></a>04351   sinpar = sin(parint)
    4416. 

  4416. <a name="l04352"></a>04352   cospar = cos(parint)
    4417. 

  4417. <a name="l04353"></a>04353 <span class="comment">!</span>
    4418. 

  4418. <a name="l04354"></a>04354 <span class="comment">!  Compute the Chebyshev moments with respect to cosine.</span>
    4419. 

  4419. <a name="l04355"></a>04355 <span class="comment">!</span>
    4420. 

  4420. <a name="l04356"></a>04356   v(1) = 2.0e+00 * sinpar / parint
    4421. 

  4421. <a name="l04357"></a>04357   v(2) = (8.0e+00*cospar+(par2+par2-8.0e+00)*sinpar/ parint)/par2
    4422. 

  4422. <a name="l04358"></a>04358   v(3) = (3.2e+01*(par2-1.2e+01)*cospar+(2.0e+00* &amp;
    4423. 

  4423. <a name="l04359"></a>04359      ((par2-8.0e+01)*par2+1.92e+02)*sinpar)/ &amp;
    4424. 

  4424. <a name="l04360"></a>04360      parint)/(par2*par2)
    4425. 

  4425. <a name="l04361"></a>04361   ac = 8.0e+00*cospar
    4426. 

  4426. <a name="l04362"></a>04362   as = 2.4e+01*parint*sinpar
    4427. 

  4427. <a name="l04363"></a>04363 
    4428. 

  4428. <a name="l04364"></a>04364   <span class="keyword">if</span> ( abs ( parint ) &gt; 2.4e+01 ) <span class="keyword">then</span>
    4429. 

  4429. <a name="l04365"></a>04365     go to 70
    4430. 

  4430. <a name="l04366"></a>04366   <span class="keyword">end if</span>
    4431. 

  4431. <a name="l04367"></a>04367 <span class="comment">!</span>
    4432. 

  4432. <a name="l04368"></a>04368 <span class="comment">!  Compute the Chebyshev moments as the solutions of a boundary value </span>
    4433. 

  4433. <a name="l04369"></a>04369 <span class="comment">!  problem with one initial value (v(3)) and one end value computed</span>
    4434. 

  4434. <a name="l04370"></a>04370 <span class="comment">!  using an asymptotic formula.</span>
    4435. 

  4435. <a name="l04371"></a>04371 <span class="comment">!</span>
    4436. 

  4436. <a name="l04372"></a>04372   noequ = nmac-3
    4437. 

  4437. <a name="l04373"></a>04373   noeq1 = noequ-1
    4438. 

  4438. <a name="l04374"></a>04374   an = 6.0e+00
    4439. 

  4439. <a name="l04375"></a>04375 
    4440. 

  4440. <a name="l04376"></a>04376   <span class="keyword">do</span> k = 1, noeq1
    4441. 

  4441. <a name="l04377"></a>04377     an2 = an*an
    4442. 

  4442. <a name="l04378"></a>04378     d(k) = -2.0e+00*(an2-4.0e+00) * (par22-an2-an2)
    4443. 

  4443. <a name="l04379"></a>04379     d2(k) = (an-1.0e+00)*(an-2.0e+00) * par2
    4444. 

  4444. <a name="l04380"></a>04380     d1(k) = (an+3.0e+00)*(an+4.0e+00) * par2
    4445. 

  4445. <a name="l04381"></a>04381     v(k+3) = as-(an2-4.0e+00) * ac
    4446. 

  4446. <a name="l04382"></a>04382     an = an+2.0e+00
    4447. 

  4447. <a name="l04383"></a>04383   <span class="keyword">end do</span>
    4448. 

  4448. <a name="l04384"></a>04384 
    4449. 

  4449. <a name="l04385"></a>04385   an2 = an*an
    4450. 

  4450. <a name="l04386"></a>04386   d(noequ) = -2.0e+00*(an2-4.0e+00) * (par22-an2-an2)
    4451. 

  4451. <a name="l04387"></a>04387   v(noequ+3) = as - ( an2 - 4.0e+00 ) * ac
    4452. 

  4452. <a name="l04388"></a>04388   v(4) = v(4) - 5.6e+01 * par2 * v(3)
    4453. 

  4453. <a name="l04389"></a>04389   ass = parint * sinpar
    4454. 

  4454. <a name="l04390"></a>04390   asap = (((((2.10e+02*par2-1.0e+00)*cospar-(1.05e+02*par2 &amp;
    4455. 

  4455. <a name="l04391"></a>04391      -6.3e+01)*ass)/an2-(1.0e+00-1.5e+01*par2)*cospar &amp;
    4456. 

  4456. <a name="l04392"></a>04392      +1.5e+01*ass)/an2-cospar+3.0e+00*ass)/an2-cospar)/an2
    4457. 

  4457. <a name="l04393"></a>04393   v(noequ+3) = v(noequ+3)-2.0e+00*asap*par2*(an-1.0e+00)* &amp;
    4458. 

  4458. <a name="l04394"></a>04394       (an-2.0e+00)
    4459. 

  4459. <a name="l04395"></a>04395 <span class="comment">!</span>
    4460. 

  4460. <a name="l04396"></a>04396 <span class="comment">!  Solve the tridiagonal system by means of Gaussian</span>
    4461. 

  4461. <a name="l04397"></a>04397 <span class="comment">!  elimination with partial pivoting.</span>
    4462. 

  4462. <a name="l04398"></a>04398 <span class="comment">!</span>
    4463. 

  4463. <a name="l04399"></a>04399   d3(1:noequ) = 0.0e+00
    4464. 

  4464. <a name="l04400"></a>04400 
    4465. 

  4465. <a name="l04401"></a>04401   d2(noequ) = 0.0e+00
    4466. 

  4466. <a name="l04402"></a>04402 
    4467. 

  4467. <a name="l04403"></a>04403   <span class="keyword">do</span> i = 1, noeq1
    4468. 

  4468. <a name="l04404"></a>04404 
    4469. 

  4469. <a name="l04405"></a>04405     <span class="keyword">if</span> ( abs(d1(i)) &gt; abs(d(i)) ) <span class="keyword">then</span>
    4470. 

  4470. <a name="l04406"></a>04406       an = d1(i)
    4471. 

  4471. <a name="l04407"></a>04407       d1(i) = d(i)
    4472. 

  4472. <a name="l04408"></a>04408       d(i) = an
    4473. 

  4473. <a name="l04409"></a>04409       an = d2(i)
    4474. 

  4474. <a name="l04410"></a>04410       d2(i) = d(i+1)
    4475. 

  4475. <a name="l04411"></a>04411       d(i+1) = an
    4476. 

  4476. <a name="l04412"></a>04412       d3(i) = d2(i+1)
    4477. 

  4477. <a name="l04413"></a>04413       d2(i+1) = 0.0e+00
    4478. 

  4478. <a name="l04414"></a>04414       an = v(i+4)
    4479. 

  4479. <a name="l04415"></a>04415       v(i+4) = v(i+3)
    4480. 

  4480. <a name="l04416"></a>04416       v(i+3) = an
    4481. 

  4481. <a name="l04417"></a>04417     <span class="keyword">end if</span>
    4482. 

  4482. <a name="l04418"></a>04418 
    4483. 

  4483. <a name="l04419"></a>04419     d(i+1) = d(i+1)-d2(i)*d1(i)/d(i)
    4484. 

  4484. <a name="l04420"></a>04420     d2(i+1) = d2(i+1)-d3(i)*d1(i)/d(i)
    4485. 

  4485. <a name="l04421"></a>04421     v(i+4) = v(i+4)-v(i+3)*d1(i)/d(i)
    4486. 

  4486. <a name="l04422"></a>04422 
    4487. 

  4487. <a name="l04423"></a>04423   <span class="keyword">end do</span>
    4488. 

  4488. <a name="l04424"></a>04424 
    4489. 

  4489. <a name="l04425"></a>04425   v(noequ+3) = v(noequ+3) / d(noequ)
    4490. 

  4490. <a name="l04426"></a>04426   v(noequ+2) = (v(noequ+2)-d2(noeq1)*v(noequ+3))/d(noeq1)
    4491. 

  4491. <a name="l04427"></a>04427 
    4492. 

  4492. <a name="l04428"></a>04428   <span class="keyword">do</span> i = 2, noeq1
    4493. 

  4493. <a name="l04429"></a>04429     k = noequ-i
    4494. 

  4494. <a name="l04430"></a>04430     v(k+3) = (v(k+3)-d3(k)*v(k+5)-d2(k)*v(k+4))/d(k)
    4495. 

  4495. <a name="l04431"></a>04431   <span class="keyword">end do</span>
    4496. 

  4496. <a name="l04432"></a>04432 
    4497. 

  4497. <a name="l04433"></a>04433   go to 90
    4498. 

  4498. <a name="l04434"></a>04434 <span class="comment">!</span>
    4499. 

  4499. <a name="l04435"></a>04435 <span class="comment">!  Compute the Chebyshev moments by means of forward recursion</span>
    4500. 

  4500. <a name="l04436"></a>04436 <span class="comment">!</span>
    4501. 

  4501. <a name="l04437"></a>04437 70 continue
    4502. 

  4502. <a name="l04438"></a>04438 
    4503. 

  4503. <a name="l04439"></a>04439   an = 4.0e+00
    4504. 

  4504. <a name="l04440"></a>04440 
    4505. 

  4505. <a name="l04441"></a>04441   <span class="keyword">do</span> i = 4, 13
    4506. 

  4506. <a name="l04442"></a>04442     an2 = an*an
    4507. 

  4507. <a name="l04443"></a>04443     v(i) = ((an2-4.0e+00)*(2.0e+00*(par22-an2-an2)*v(i-1)-ac) &amp;
    4508. 

  4508. <a name="l04444"></a>04444      +as-par2*(an+1.0e+00)*(an+2.0e+00)*v(i-2))/ &amp;
    4509. 

  4509. <a name="l04445"></a>04445      (par2*(an-1.0e+00)*(an-2.0e+00))
    4510. 

  4510. <a name="l04446"></a>04446     an = an+2.0e+00
    4511. 

  4511. <a name="l04447"></a>04447   <span class="keyword">end do</span>
    4512. 

  4512. <a name="l04448"></a>04448 
    4513. 

  4513. <a name="l04449"></a>04449 90 continue
    4514. 

  4514. <a name="l04450"></a>04450 
    4515. 

  4515. <a name="l04451"></a>04451   <span class="keyword">do</span> j = 1, 13
    4516. 

  4516. <a name="l04452"></a>04452     chebmo(m,2*j-1) = v(j)
    4517. 

  4517. <a name="l04453"></a>04453   <span class="keyword">end do</span>
    4518. 

  4518. <a name="l04454"></a>04454 <span class="comment">!</span>
    4519. 

  4519. <a name="l04455"></a>04455 <span class="comment">!  Compute the Chebyshev moments with respect to sine.</span>
    4520. 

  4520. <a name="l04456"></a>04456 <span class="comment">!</span>
    4521. 

  4521. <a name="l04457"></a>04457   v(1) = 2.0e+00*(sinpar-parint*cospar)/par2
    4522. 

  4522. <a name="l04458"></a>04458   v(2) = (1.8e+01-4.8e+01/par2)*sinpar/par2 &amp;
    4523. 

  4523. <a name="l04459"></a>04459      +(-2.0e+00+4.8e+01/par2)*cospar/parint
    4524. 

  4524. <a name="l04460"></a>04460   ac = -2.4e+01*parint*cospar
    4525. 

  4525. <a name="l04461"></a>04461   as = -8.0e+00*sinpar
    4526. 

  4526. <a name="l04462"></a>04462   chebmo(m,2) = v(1)
    4527. 

  4527. <a name="l04463"></a>04463   chebmo(m,4) = v(2)
    4528. 

  4528. <a name="l04464"></a>04464 
    4529. 

  4529. <a name="l04465"></a>04465   <span class="keyword">if</span> ( abs(parint) &lt;= 2.4e+01 ) <span class="keyword">then</span>
    4530. 

  4530. <a name="l04466"></a>04466 
    4531. 

  4531. <a name="l04467"></a>04467     <span class="keyword">do</span> k = 3, 12
    4532. 

  4532. <a name="l04468"></a>04468       an = k
    4533. 

  4533. <a name="l04469"></a>04469       chebmo(m,2*k) = -sinpar/(an*(2.0e+00*an-2.0e+00)) &amp;
    4534. 

  4534. <a name="l04470"></a>04470                  -2.5e-01*parint*(v(k+1)/an-v(k)/(an-1.0e+00))
    4535. 

  4535. <a name="l04471"></a>04471     <span class="keyword">end do</span>
    4536. 

  4536. <a name="l04472"></a>04472 <span class="comment">!</span>
    4537. 

  4537. <a name="l04473"></a>04473 <span class="comment">!  Compute the Chebyshev moments by means of forward recursion.</span>
    4538. 

  4538. <a name="l04474"></a>04474 <span class="comment">!</span>
    4539. 

  4539. <a name="l04475"></a>04475   <span class="keyword">else</span>
    4540. 

  4540. <a name="l04476"></a>04476 
    4541. 

  4541. <a name="l04477"></a>04477     an = 3.0e+00
    4542. 

  4542. <a name="l04478"></a>04478 
    4543. 

  4543. <a name="l04479"></a>04479     <span class="keyword">do</span> i = 3, 12
    4544. 

  4544. <a name="l04480"></a>04480       an2 = an*an
    4545. 

  4545. <a name="l04481"></a>04481       v(i) = ((an2-4.0e+00)*(2.0e+00*(par22-an2-an2)*v(i-1)+as) &amp;
    4546. 

  4546. <a name="l04482"></a>04482        +ac-par2*(an+1.0e+00)*(an+2.0e+00)*v(i-2)) &amp;
    4547. 

  4547. <a name="l04483"></a>04483        /(par2*(an-1.0e+00)*(an-2.0e+00))
    4548. 

  4548. <a name="l04484"></a>04484       an = an+2.0e+00
    4549. 

  4549. <a name="l04485"></a>04485       chebmo(m,2*i) = v(i)
    4550. 

  4550. <a name="l04486"></a>04486     <span class="keyword">end do</span>
    4551. 

  4551. <a name="l04487"></a>04487 
    4552. 

  4552. <a name="l04488"></a>04488   <span class="keyword">end if</span>
    4553. 

  4553. <a name="l04489"></a>04489 
    4554. 

  4554. <a name="l04490"></a>04490 140 continue
    4555. 

  4555. <a name="l04491"></a>04491 
    4556. 

  4556. <a name="l04492"></a>04492   <span class="keyword">if</span> ( nrmom &lt; momcom ) <span class="keyword">then</span>
    4557. 

  4557. <a name="l04493"></a>04493     m = nrmom + 1
    4558. 

  4558. <a name="l04494"></a>04494   <span class="keyword">end if</span>
    4559. 

  4559. <a name="l04495"></a>04495 
    4560. 

  4560. <a name="l04496"></a>04496   <span class="keyword">if</span> ( momcom &lt; maxp1 - 1 .and. nrmom &gt;= momcom ) <span class="keyword">then</span>
    4561. 

  4561. <a name="l04497"></a>04497     momcom = momcom + 1
    4562. 

  4562. <a name="l04498"></a>04498   <span class="keyword">end if</span>
    4563. 

  4563. <a name="l04499"></a>04499 <span class="comment">!</span>
    4564. 

  4564. <a name="l04500"></a>04500 <span class="comment">!  Compute the coefficients of the Chebyshev expansions</span>
    4565. 

  4565. <a name="l04501"></a>04501 <span class="comment">!  of degrees 12 and 24 of the function F.</span>
    4566. 

  4566. <a name="l04502"></a>04502 <span class="comment">!</span>
    4567. 

  4567. <a name="l04503"></a>04503   fval(1) = 5.0e-01 * f(centr+hlgth)
    4568. 

  4568. <a name="l04504"></a>04504   fval(13) = f(centr)
    4569. 

  4569. <a name="l04505"></a>04505   fval(25) = 5.0e-01 * f(centr-hlgth)
    4570. 

  4570. <a name="l04506"></a>04506 
    4571. 

  4571. <a name="l04507"></a>04507   <span class="keyword">do</span> i = 2, 12
    4572. 

  4572. <a name="l04508"></a>04508     isym = 26-i
    4573. 

  4573. <a name="l04509"></a>04509     fval(i) = f(hlgth*x(i-1)+centr)
    4574. 

  4574. <a name="l04510"></a>04510     fval(isym) = f(centr-hlgth*x(i-1))
    4575. 

  4575. <a name="l04511"></a>04511   <span class="keyword">end do</span>
    4576. 

  4576. <a name="l04512"></a>04512 
    4577. 

  4577. <a name="l04513"></a>04513   call <a class="code" href="quadpack_8f90.html#ad5beefcfdb335ea68ccf8397536c8c36">qcheb </a>( x, fval, cheb12, cheb24 )
    4578. 

  4578. <a name="l04514"></a>04514 <span class="comment">!</span>
    4579. 

  4579. <a name="l04515"></a>04515 <span class="comment">!  Compute the integral and error estimates.</span>
    4580. 

  4580. <a name="l04516"></a>04516 <span class="comment">!</span>
    4581. 

  4581. <a name="l04517"></a>04517   resc12 = cheb12(13) * chebmo(m,13)
    4582. 

  4582. <a name="l04518"></a>04518   ress12 = 0.0e+00
    4583. 

  4583. <a name="l04519"></a>04519   estc = abs ( cheb24(25)*chebmo(m,25))+abs((cheb12(13)- &amp;
    4584. 

  4584. <a name="l04520"></a>04520     cheb24(13))*chebmo(m,13) )
    4585. 

  4585. <a name="l04521"></a>04521   ests = 0.0e+00
    4586. 

  4586. <a name="l04522"></a>04522   k = 11
    4587. 

  4587. <a name="l04523"></a>04523 
    4588. 

  4588. <a name="l04524"></a>04524   <span class="keyword">do</span> j = 1, 6
    4589. 

  4589. <a name="l04525"></a>04525     resc12 = resc12+cheb12(k)*chebmo(m,k)
    4590. 

  4590. <a name="l04526"></a>04526     ress12 = ress12+cheb12(k+1)*chebmo(m,k+1)
    4591. 

  4591. <a name="l04527"></a>04527     estc = estc+abs((cheb12(k)-cheb24(k))*chebmo(m,k))
    4592. 

  4592. <a name="l04528"></a>04528     ests = ests+abs((cheb12(k+1)-cheb24(k+1))*chebmo(m,k+1))
    4593. 

  4593. <a name="l04529"></a>04529     k = k-2
    4594. 

  4594. <a name="l04530"></a>04530   <span class="keyword">end do</span>
    4595. 

  4595. <a name="l04531"></a>04531 
    4596. 

  4596. <a name="l04532"></a>04532   resc24 = cheb24(25)*chebmo(m,25)
    4597. 

  4597. <a name="l04533"></a>04533   ress24 = 0.0e+00
    4598. 

  4598. <a name="l04534"></a>04534   resabs = abs(cheb24(25))
    4599. 

  4599. <a name="l04535"></a>04535   k = 23
    4600. 

  4600. <a name="l04536"></a>04536 
    4601. 

  4601. <a name="l04537"></a>04537   <span class="keyword">do</span> j = 1, 12
    4602. 

  4602. <a name="l04538"></a>04538 
    4603. 

  4603. <a name="l04539"></a>04539     resc24 = resc24+cheb24(k)*chebmo(m,k)
    4604. 

  4604. <a name="l04540"></a>04540     ress24 = ress24+cheb24(k+1)*chebmo(m,k+1)
    4605. 

  4605. <a name="l04541"></a>04541     resabs = resabs+abs(cheb24(k))+abs(cheb24(k+1))
    4606. 

  4606. <a name="l04542"></a>04542 
    4607. 

  4607. <a name="l04543"></a>04543     <span class="keyword">if</span> ( j &lt;= 5 ) <span class="keyword">then</span>
    4608. 

  4608. <a name="l04544"></a>04544       estc = estc+abs(cheb24(k)*chebmo(m,k))
    4609. 

  4609. <a name="l04545"></a>04545       ests = ests+abs(cheb24(k+1)*chebmo(m,k+1))
    4610. 

  4610. <a name="l04546"></a>04546     <span class="keyword">end if</span>
    4611. 

  4611. <a name="l04547"></a>04547 
    4612. 

  4612. <a name="l04548"></a>04548     k = k-2
    4613. 

  4613. <a name="l04549"></a>04549 
    4614. 

  4614. <a name="l04550"></a>04550   <span class="keyword">end do</span>
    4615. 

  4615. <a name="l04551"></a>04551 
    4616. 

  4616. <a name="l04552"></a>04552   resabs = resabs * abs ( hlgth )
    4617. 

  4617. <a name="l04553"></a>04553 
    4618. 

  4618. <a name="l04554"></a>04554   <span class="keyword">if</span> ( integr == 1 ) <span class="keyword">then</span>
    4619. 

  4619. <a name="l04555"></a>04555     result = conc * resc24-cons*ress24
    4620. 

  4620. <a name="l04556"></a>04556     abserr = abs ( conc * estc ) + abs ( cons * ests )
    4621. 

  4621. <a name="l04557"></a>04557   <span class="keyword">else</span>
    4622. 

  4622. <a name="l04558"></a>04558     result = conc*ress24+cons*resc24
    4623. 

  4623. <a name="l04559"></a>04559     abserr = abs(conc*ests)+abs(cons*estc)
    4624. 

  4624. <a name="l04560"></a>04560   <span class="keyword">end if</span>
    4625. 

  4625. <a name="l04561"></a>04561 
    4626. 

  4626. <a name="l04562"></a>04562   return
    4627. 

  4627. <a name="l04563"></a>04563 <span class="keyword">end</span>
    4628. 

  4628. <a name="l04564"></a><a class="code" href="quadpack_8f90.html#a034546450320f53f05096bc12af9b5bc">04564</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a034546450320f53f05096bc12af9b5bc">qc25s</a> ( f, a, b, bl, br, alfa, beta, ri, rj, rg, rh, result, &amp;
    4629. 

  4629. <a name="l04565"></a>04565   abserr, resasc, integr, neval )
    4630. 

  4630. <a name="l04566"></a>04566 
    4631. 

  4631. <a name="l04567"></a>04567 <span class="comment">!*****************************************************************************80</span>
    4632. 

  4632. <a name="l04568"></a>04568 <span class="comment">!</span>
    4633. 

  4633. <a name="l04569"></a>04569 <span class="comment">!! QC25S returns rules for algebraico-logarithmic end point singularities.</span>
    4634. 

  4634. <a name="l04570"></a>04570 <span class="comment">!</span>
    4635. 

  4635. <a name="l04571"></a>04571 <span class="comment">!  Discussion:</span>
    4636. 

  4636. <a name="l04572"></a>04572 <span class="comment">!</span>
    4637. 

  4637. <a name="l04573"></a>04573 <span class="comment">!    This routine computes </span>
    4638. 

  4638. <a name="l04574"></a>04574 <span class="comment">!      i = integral of F(X) * W(X) over (bl,br), </span>
    4639. 

  4639. <a name="l04575"></a>04575 <span class="comment">!    with error estimate, where the weight function W(X) has a singular</span>
    4640. 

  4640. <a name="l04576"></a>04576 <span class="comment">!    behavior of algebraico-logarithmic type at the points</span>
    4641. 

  4641. <a name="l04577"></a>04577 <span class="comment">!    a and/or b. </span>
    4642. 

  4642. <a name="l04578"></a>04578 <span class="comment">!</span>
    4643. 

  4643. <a name="l04579"></a>04579 <span class="comment">!    The interval (bl,br) is a subinterval of (a,b).</span>
    4644. 

  4644. <a name="l04580"></a>04580 <span class="comment">!</span>
    4645. 

  4645. <a name="l04581"></a>04581 <span class="comment">!  Author:</span>
    4646. 

  4646. <a name="l04582"></a>04582 <span class="comment">!</span>
    4647. 

  4647. <a name="l04583"></a>04583 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    4648. 

  4648. <a name="l04584"></a>04584 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    4649. 

  4649. <a name="l04585"></a>04585 <span class="comment">!</span>
    4650. 

  4650. <a name="l04586"></a>04586 <span class="comment">!  Reference:</span>
    4651. 

  4651. <a name="l04587"></a>04587 <span class="comment">!</span>
    4652. 

  4652. <a name="l04588"></a>04588 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    4653. 

  4653. <a name="l04589"></a>04589 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    4654. 

  4654. <a name="l04590"></a>04590 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    4655. 

  4655. <a name="l04591"></a>04591 <span class="comment">!    Springer Verlag, 1983</span>
    4656. 

  4656. <a name="l04592"></a>04592 <span class="comment">!</span>
    4657. 

  4657. <a name="l04593"></a>04593 <span class="comment">!  Parameters:</span>
    4658. 

  4658. <a name="l04594"></a>04594 <span class="comment">!</span>
    4659. 

  4659. <a name="l04595"></a>04595 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    4660. 

  4660. <a name="l04596"></a>04596 <span class="comment">!      function f ( x )</span>
    4661. 

  4661. <a name="l04597"></a>04597 <span class="comment">!      real f</span>
    4662. 

  4662. <a name="l04598"></a>04598 <span class="comment">!      real x</span>
    4663. 

  4663. <a name="l04599"></a>04599 <span class="comment">!    which evaluates the integrand function.</span>
    4664. 

  4664. <a name="l04600"></a>04600 <span class="comment">!</span>
    4665. 

  4665. <a name="l04601"></a>04601 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    4666. 

  4666. <a name="l04602"></a>04602 <span class="comment">!</span>
    4667. 

  4667. <a name="l04603"></a>04603 <span class="comment">!    Input, real BL, BR, the lower and upper limits of integration.</span>
    4668. 

  4668. <a name="l04604"></a>04604 <span class="comment">!    A &lt;= BL &lt; BR &lt;= B.</span>
    4669. 

  4669. <a name="l04605"></a>04605 <span class="comment">!</span>
    4670. 

  4670. <a name="l04606"></a>04606 <span class="comment">!    Input, real ALFA, BETA, parameters in the weight function.</span>
    4671. 

  4671. <a name="l04607"></a>04607 <span class="comment">!</span>
    4672. 

  4672. <a name="l04608"></a>04608 <span class="comment">!    Input, real RI(25), RJ(25), RG(25), RH(25), modified Chebyshev moments </span>
    4673. 

  4673. <a name="l04609"></a>04609 <span class="comment">!    for the application of the generalized Clenshaw-Curtis method,</span>
    4674. 

  4674. <a name="l04610"></a>04610 <span class="comment">!    computed in QMOMO.</span>
    4675. 

  4675. <a name="l04611"></a>04611 <span class="comment">!</span>
    4676. 

  4676. <a name="l04612"></a>04612 <span class="comment">!    Output, real RESULT, the estimated value of the integral, computed by </span>
    4677. 

  4677. <a name="l04613"></a>04613 <span class="comment">!    using a generalized clenshaw-curtis method if b1 = a or br = b.</span>
    4678. 

  4678. <a name="l04614"></a>04614 <span class="comment">!    In all other cases the 15-point Kronrod rule is applied, obtained by</span>
    4679. 

  4679. <a name="l04615"></a>04615 <span class="comment">!    optimal addition of abscissae to the 7-point Gauss rule.</span>
    4680. 

  4680. <a name="l04616"></a>04616 <span class="comment">!</span>
    4681. 

  4681. <a name="l04617"></a>04617 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    4682. 

  4682. <a name="l04618"></a>04618 <span class="comment">!</span>
    4683. 

  4683. <a name="l04619"></a>04619 <span class="comment">!    Output, real RESASC, approximation to the integral of abs(F*W-I/(B-A)).</span>
    4684. 

  4684. <a name="l04620"></a>04620 <span class="comment">!</span>
    4685. 

  4685. <a name="l04621"></a>04621 <span class="comment">!    Input, integer INTEGR,  determines the weight function</span>
    4686. 

  4686. <a name="l04622"></a>04622 <span class="comment">!    1, w(x) = (x-a)**alfa*(b-x)**beta</span>
    4687. 

  4687. <a name="l04623"></a>04623 <span class="comment">!    2, w(x) = (x-a)**alfa*(b-x)**beta*log(x-a)</span>
    4688. 

  4688. <a name="l04624"></a>04624 <span class="comment">!    3, w(x) = (x-a)**alfa*(b-x)**beta*log(b-x)</span>
    4689. 

  4689. <a name="l04625"></a>04625 <span class="comment">!    4, w(x) = (x-a)**alfa*(b-x)**beta*log(x-a)*log(b-x)</span>
    4690. 

  4690. <a name="l04626"></a>04626 <span class="comment">!</span>
    4691. 

  4691. <a name="l04627"></a>04627 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    4692. 

  4692. <a name="l04628"></a>04628 <span class="comment">!</span>
    4693. 

  4693. <a name="l04629"></a>04629 <span class="comment">!  Local Parameters:</span>
    4694. 

  4694. <a name="l04630"></a>04630 <span class="comment">!</span>
    4695. 

  4695. <a name="l04631"></a>04631 <span class="comment">!           fval   - value of the function f at the points</span>
    4696. 

  4696. <a name="l04632"></a>04632 <span class="comment">!                    (br-bl)*0.5*cos(k*pi/24)+(br+bl)*0.5</span>
    4697. 

  4697. <a name="l04633"></a>04633 <span class="comment">!                    k = 0, ..., 24</span>
    4698. 

  4698. <a name="l04634"></a>04634 <span class="comment">!           cheb12 - coefficients of the Chebyshev series expansion</span>
    4699. 

  4699. <a name="l04635"></a>04635 <span class="comment">!                    of degree 12, for the function f, in the interval</span>
    4700. 

  4700. <a name="l04636"></a>04636 <span class="comment">!                    (bl,br)</span>
    4701. 

  4701. <a name="l04637"></a>04637 <span class="comment">!           cheb24 - coefficients of the Chebyshev series expansion</span>
    4702. 

  4702. <a name="l04638"></a>04638 <span class="comment">!                    of degree 24, for the function f, in the interval</span>
    4703. 

  4703. <a name="l04639"></a>04639 <span class="comment">!                    (bl,br)</span>
    4704. 

  4704. <a name="l04640"></a>04640 <span class="comment">!           res12  - approximation to the integral obtained from cheb12</span>
    4705. 

  4705. <a name="l04641"></a>04641 <span class="comment">!           res24  - approximation to the integral obtained from cheb24</span>
    4706. 

  4706. <a name="l04642"></a>04642 <span class="comment">!           qwgts  - external function subprogram defining the four</span>
    4707. 

  4707. <a name="l04643"></a>04643 <span class="comment">!                    possible weight functions</span>
    4708. 

  4708. <a name="l04644"></a>04644 <span class="comment">!           hlgth  - half-length of the interval (bl,br)</span>
    4709. 

  4709. <a name="l04645"></a>04645 <span class="comment">!           centr  - mid point of the interval (bl,br)</span>
    4710. 

  4710. <a name="l04646"></a>04646 <span class="comment">!</span>
    4711. 

  4711. <a name="l04647"></a>04647 <span class="comment">!           the vector x contains the values cos(k*pi/24)</span>
    4712. 

  4712. <a name="l04648"></a>04648 <span class="comment">!           k = 1, ..., 11, to be used for the computation of the</span>
    4713. 

  4713. <a name="l04649"></a>04649 <span class="comment">!           Chebyshev series expansion of f.</span>
    4714. 

  4714. <a name="l04650"></a>04650 <span class="comment">!</span>
    4715. 

  4715. <a name="l04651"></a>04651   <span class="keyword">implicit none</span>
    4716. 

  4716. <a name="l04652"></a>04652 
    4717. 

  4717. <a name="l04653"></a>04653   <span class="keywordtype">real</span> a
    4718. 

  4718. <a name="l04654"></a>04654   <span class="keywordtype">real</span> abserr
    4719. 

  4719. <a name="l04655"></a>04655   <span class="keywordtype">real</span> alfa
    4720. 

  4720. <a name="l04656"></a>04656   <span class="keywordtype">real</span> b
    4721. 

  4721. <a name="l04657"></a>04657   <span class="keywordtype">real</span> beta
    4722. 

  4722. <a name="l04658"></a>04658   <span class="keywordtype">real</span> bl
    4723. 

  4723. <a name="l04659"></a>04659   <span class="keywordtype">real</span> br
    4724. 

  4724. <a name="l04660"></a>04660   <span class="keywordtype">real</span> centr
    4725. 

  4725. <a name="l04661"></a>04661   <span class="keywordtype">real</span> cheb12(13)
    4726. 

  4726. <a name="l04662"></a>04662   <span class="keywordtype">real</span> cheb24(25)
    4727. 

  4727. <a name="l04663"></a>04663   <span class="keywordtype">real</span> dc
    4728. 

  4728. <a name="l04664"></a>04664   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    4729. 

  4729. <a name="l04665"></a>04665   <span class="keywordtype">real</span> factor
    4730. 

  4730. <a name="l04666"></a>04666   <span class="keywordtype">real</span> fix
    4731. 

  4731. <a name="l04667"></a>04667   <span class="keywordtype">real</span> fval(25)
    4732. 

  4732. <a name="l04668"></a>04668   <span class="keywordtype">real</span> hlgth
    4733. 

  4733. <a name="l04669"></a>04669   <span class="keywordtype">integer</span> i
    4734. 

  4734. <a name="l04670"></a>04670   <span class="keywordtype">integer</span> integr
    4735. 

  4735. <a name="l04671"></a>04671   <span class="keywordtype">integer</span> isym
    4736. 

  4736. <a name="l04672"></a>04672   <span class="keywordtype">integer</span> neval
    4737. 

  4737. <a name="l04673"></a>04673   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: qwgts
    4738. 

  4738. <a name="l04674"></a>04674   <span class="keywordtype">real</span> resabs
    4739. 

  4739. <a name="l04675"></a>04675   <span class="keywordtype">real</span> resasc
    4740. 

  4740. <a name="l04676"></a>04676   <span class="keywordtype">real</span> result
    4741. 

  4741. <a name="l04677"></a>04677   <span class="keywordtype">real</span> res12
    4742. 

  4742. <a name="l04678"></a>04678   <span class="keywordtype">real</span> res24
    4743. 

  4743. <a name="l04679"></a>04679   <span class="keywordtype">real</span> rg(25)
    4744. 

  4744. <a name="l04680"></a>04680   <span class="keywordtype">real</span> rh(25)
    4745. 

  4745. <a name="l04681"></a>04681   <span class="keywordtype">real</span> ri(25)
    4746. 

  4746. <a name="l04682"></a>04682   <span class="keywordtype">real</span> rj(25)
    4747. 

  4747. <a name="l04683"></a>04683   <span class="keywordtype">real</span> u
    4748. 

  4748. <a name="l04684"></a>04684   <span class="keywordtype">real</span>, <span class="keywordtype">dimension ( 11 )</span> :: x = (/ 
    4749. 

  4749. <a name="l04685"></a>04685     9.914448613738104e-01,     9.659258262890683e-01, 
    4750. 

  4750. <a name="l04686"></a>04686     9.238795325112868e-01,     8.660254037844386e-01, 
    4751. 

  4751. <a name="l04687"></a>04687     7.933533402912352e-01,     7.071067811865475e-01, 
    4752. 

  4752. <a name="l04688"></a>04688     6.087614290087206e-01,     5.000000000000000e-01, 
    4753. 

  4753. <a name="l04689"></a>04689     3.826834323650898e-01,     2.588190451025208e-01, 
    4754. 

  4754. <a name="l04690"></a>04690     1.305261922200516e-01 /)
    4755. 

  4755. <a name="l04691"></a>04691 
    4756. 

  4756. <a name="l04692"></a>04692   neval = 25
    4757. 

  4757. <a name="l04693"></a>04693 
    4758. 

  4758. <a name="l04694"></a>04694   <span class="keyword">if</span> ( bl == a .and. (alfa /= 0.0e+00 .or. integr == 2 .or. integr == 4)) <span class="keyword">then</span>
    4759. 

  4759. <a name="l04695"></a>04695     go to 10
    4760. 

  4760. <a name="l04696"></a>04696   <span class="keyword">end if</span>
    4761. 

  4761. <a name="l04697"></a>04697 
    4762. 

  4762. <a name="l04698"></a>04698   <span class="keyword">if</span> ( br == b .and. (beta /= 0.0e+00 .or. integr == 3 .or. integr == 4)) &amp;
    4763. 

  4763. <a name="l04699"></a>04699     go to 140
    4764. 

  4764. <a name="l04700"></a>04700 <span class="comment">!</span>
    4765. 

  4765. <a name="l04701"></a>04701 <span class="comment">!  If a &gt; bl and b &lt; br, apply the 15-point Gauss-Kronrod scheme.</span>
    4766. 

  4766. <a name="l04702"></a>04702 <span class="comment">!</span>
    4767. 

  4767. <a name="l04703"></a>04703   call <a class="code" href="quadpack_8f90.html#a0c083838940925726abd5bc85fa29587">qk15w </a>( f, qwgts, a, b, alfa, beta, integr, bl, br, result, abserr, &amp;
    4768. 

  4768. <a name="l04704"></a>04704     resabs, resasc )
    4769. 

  4769. <a name="l04705"></a>04705 
    4770. 

  4770. <a name="l04706"></a>04706   neval = 15
    4771. 

  4771. <a name="l04707"></a>04707   return
    4772. 

  4772. <a name="l04708"></a>04708 <span class="comment">!</span>
    4773. 

  4773. <a name="l04709"></a>04709 <span class="comment">!  This part of the program is executed only if a = bl.</span>
    4774. 

  4774. <a name="l04710"></a>04710 <span class="comment">!</span>
    4775. 

  4775. <a name="l04711"></a>04711 <span class="comment">!  Compute the Chebyshev series expansion of the function</span>
    4776. 

  4776. <a name="l04712"></a>04712 <span class="comment">!  f1 = (0.5*(b+b-br-a)-0.5*(br-a)*x)**beta*f(0.5*(br-a)*x+0.5*(br+a))</span>
    4777. 

  4777. <a name="l04713"></a>04713 <span class="comment">!</span>
    4778. 

  4778. <a name="l04714"></a>04714 10 continue
    4779. 

  4779. <a name="l04715"></a>04715 
    4780. 

  4780. <a name="l04716"></a>04716   hlgth = 5.0e-01*(br-bl)
    4781. 

  4781. <a name="l04717"></a>04717   centr = 5.0e-01*(br+bl)
    4782. 

  4782. <a name="l04718"></a>04718   fix = b-centr
    4783. 

  4783. <a name="l04719"></a>04719   fval(1) = 5.0e-01*f(hlgth+centr)*(fix-hlgth)**beta
    4784. 

  4784. <a name="l04720"></a>04720   fval(13) = f(centr)*(fix**beta)
    4785. 

  4785. <a name="l04721"></a>04721   fval(25) = 5.0e-01*f(centr-hlgth)*(fix+hlgth)**beta
    4786. 

  4786. <a name="l04722"></a>04722 
    4787. 

  4787. <a name="l04723"></a>04723   <span class="keyword">do</span> i = 2, 12
    4788. 

  4788. <a name="l04724"></a>04724     u = hlgth*x(i-1)
    4789. 

  4789. <a name="l04725"></a>04725     isym = 26-i
    4790. 

  4790. <a name="l04726"></a>04726     fval(i) = f(u+centr)*(fix-u)**beta
    4791. 

  4791. <a name="l04727"></a>04727     fval(isym) = f(centr-u)*(fix+u)**beta
    4792. 

  4792. <a name="l04728"></a>04728   <span class="keyword">end do</span>
    4793. 

  4793. <a name="l04729"></a>04729 
    4794. 

  4794. <a name="l04730"></a>04730   factor = hlgth**(alfa+1.0e+00)
    4795. 

  4795. <a name="l04731"></a>04731   result = 0.0e+00
    4796. 

  4796. <a name="l04732"></a>04732   abserr = 0.0e+00
    4797. 

  4797. <a name="l04733"></a>04733   res12 = 0.0e+00
    4798. 

  4798. <a name="l04734"></a>04734   res24 = 0.0e+00
    4799. 

  4799. <a name="l04735"></a>04735 
    4800. 

  4800. <a name="l04736"></a>04736   <span class="keyword">if</span> ( integr &gt; 2 ) go to 70
    4801. 

  4801. <a name="l04737"></a>04737 
    4802. 

  4802. <a name="l04738"></a>04738   call <a class="code" href="quadpack_8f90.html#ad5beefcfdb335ea68ccf8397536c8c36">qcheb </a>( x, fval, cheb12, cheb24 )
    4803. 

  4803. <a name="l04739"></a>04739 <span class="comment">!</span>
    4804. 

  4804. <a name="l04740"></a>04740 <span class="comment">!  integr = 1  (or 2)</span>
    4805. 

  4805. <a name="l04741"></a>04741 <span class="comment">!</span>
    4806. 

  4806. <a name="l04742"></a>04742   <span class="keyword">do</span> i = 1, 13
    4807. 

  4807. <a name="l04743"></a>04743     res12 = res12+cheb12(i)*ri(i)
    4808. 

  4808. <a name="l04744"></a>04744     res24 = res24+cheb24(i)*ri(i)
    4809. 

  4809. <a name="l04745"></a>04745   <span class="keyword">end do</span>
    4810. 

  4810. <a name="l04746"></a>04746 
    4811. 

  4811. <a name="l04747"></a>04747   <span class="keyword">do</span> i = 14, 25
    4812. 

  4812. <a name="l04748"></a>04748     res24 = res24 + cheb24(i) * ri(i)
    4813. 

  4813. <a name="l04749"></a>04749   <span class="keyword">end do</span>
    4814. 

  4814. <a name="l04750"></a>04750 
    4815. 

  4815. <a name="l04751"></a>04751   <span class="keyword">if</span> ( integr == 1 ) go to 130
    4816. 

  4816. <a name="l04752"></a>04752 <span class="comment">!</span>
    4817. 

  4817. <a name="l04753"></a>04753 <span class="comment">!  integr = 2</span>
    4818. 

  4818. <a name="l04754"></a>04754 <span class="comment">!</span>
    4819. 

  4819. <a name="l04755"></a>04755   dc = log ( br - bl )
    4820. 

  4820. <a name="l04756"></a>04756   result = res24 * dc
    4821. 

  4821. <a name="l04757"></a>04757   abserr = abs((res24-res12)*dc)
    4822. 

  4822. <a name="l04758"></a>04758   res12 = 0.0e+00
    4823. 

  4823. <a name="l04759"></a>04759   res24 = 0.0e+00
    4824. 

  4824. <a name="l04760"></a>04760 
    4825. 

  4825. <a name="l04761"></a>04761   <span class="keyword">do</span> i = 1, 13
    4826. 

  4826. <a name="l04762"></a>04762     res12 = res12+cheb12(i)*rg(i)
    4827. 

  4827. <a name="l04763"></a>04763     res24 = res24+cheb24(i)*rg(i)
    4828. 

  4828. <a name="l04764"></a>04764   <span class="keyword">end do</span>
    4829. 

  4829. <a name="l04765"></a>04765 
    4830. 

  4830. <a name="l04766"></a>04766   <span class="keyword">do</span> i = 14, 25
    4831. 

  4831. <a name="l04767"></a>04767     res24 = res24+cheb24(i)*rg(i)
    4832. 

  4832. <a name="l04768"></a>04768   <span class="keyword">end do</span>
    4833. 

  4833. <a name="l04769"></a>04769 
    4834. 

  4834. <a name="l04770"></a>04770   go to 130
    4835. 

  4835. <a name="l04771"></a>04771 <span class="comment">!</span>
    4836. 

  4836. <a name="l04772"></a>04772 <span class="comment">!  Compute the Chebyshev series expansion of the function</span>
    4837. 

  4837. <a name="l04773"></a>04773 <span class="comment">!  F4 = f1*log(0.5*(b+b-br-a)-0.5*(br-a)*x)</span>
    4838. 

  4838. <a name="l04774"></a>04774 <span class="comment">!</span>
    4839. 

  4839. <a name="l04775"></a>04775 70 continue
    4840. 

  4840. <a name="l04776"></a>04776 
    4841. 

  4841. <a name="l04777"></a>04777   fval(1) = fval(1) * log ( fix - hlgth )
    4842. 

  4842. <a name="l04778"></a>04778   fval(13) = fval(13) * log ( fix )
    4843. 

  4843. <a name="l04779"></a>04779   fval(25) = fval(25) * log ( fix + hlgth )
    4844. 

  4844. <a name="l04780"></a>04780 
    4845. 

  4845. <a name="l04781"></a>04781   <span class="keyword">do</span> i = 2, 12
    4846. 

  4846. <a name="l04782"></a>04782     u = hlgth*x(i-1)
    4847. 

  4847. <a name="l04783"></a>04783     isym = 26-i
    4848. 

  4848. <a name="l04784"></a>04784     fval(i) = fval(i) * log ( fix - u )
    4849. 

  4849. <a name="l04785"></a>04785     fval(isym) = fval(isym) * log ( fix + u )
    4850. 

  4850. <a name="l04786"></a>04786   <span class="keyword">end do</span>
    4851. 

  4851. <a name="l04787"></a>04787 
    4852. 

  4852. <a name="l04788"></a>04788   call <a class="code" href="quadpack_8f90.html#ad5beefcfdb335ea68ccf8397536c8c36">qcheb </a>( x, fval, cheb12, cheb24 )
    4853. 

  4853. <a name="l04789"></a>04789 <span class="comment">!</span>
    4854. 

  4854. <a name="l04790"></a>04790 <span class="comment">!  integr = 3  (or 4)</span>
    4855. 

  4855. <a name="l04791"></a>04791 <span class="comment">!</span>
    4856. 

  4856. <a name="l04792"></a>04792   <span class="keyword">do</span> i = 1, 13
    4857. 

  4857. <a name="l04793"></a>04793     res12 = res12+cheb12(i)*ri(i)
    4858. 

  4858. <a name="l04794"></a>04794     res24 = res24+cheb24(i)*ri(i)
    4859. 

  4859. <a name="l04795"></a>04795   <span class="keyword">end do</span>
    4860. 

  4860. <a name="l04796"></a>04796 
    4861. 

  4861. <a name="l04797"></a>04797   <span class="keyword">do</span> i = 14, 25
    4862. 

  4862. <a name="l04798"></a>04798     res24 = res24+cheb24(i)*ri(i)
    4863. 

  4863. <a name="l04799"></a>04799   <span class="keyword">end do</span>
    4864. 

  4864. <a name="l04800"></a>04800 
    4865. 

  4865. <a name="l04801"></a>04801   <span class="keyword">if</span> ( integr == 3 ) <span class="keyword">then</span>
    4866. 

  4866. <a name="l04802"></a>04802     go to 130
    4867. 

  4867. <a name="l04803"></a>04803   <span class="keyword">end if</span>
    4868. 

  4868. <a name="l04804"></a>04804 <span class="comment">!</span>
    4869. 

  4869. <a name="l04805"></a>04805 <span class="comment">!  integr = 4</span>
    4870. 

  4870. <a name="l04806"></a>04806 <span class="comment">!</span>
    4871. 

  4871. <a name="l04807"></a>04807   dc = log ( br - bl )
    4872. 

  4872. <a name="l04808"></a>04808   result = res24*dc
    4873. 

  4873. <a name="l04809"></a>04809   abserr = abs((res24-res12)*dc)
    4874. 

  4874. <a name="l04810"></a>04810   res12 = 0.0e+00
    4875. 

  4875. <a name="l04811"></a>04811   res24 = 0.0e+00
    4876. 

  4876. <a name="l04812"></a>04812 
    4877. 

  4877. <a name="l04813"></a>04813   <span class="keyword">do</span> i = 1, 13
    4878. 

  4878. <a name="l04814"></a>04814     res12 = res12+cheb12(i)*rg(i)
    4879. 

  4879. <a name="l04815"></a>04815     res24 = res24+cheb24(i)*rg(i)
    4880. 

  4880. <a name="l04816"></a>04816   <span class="keyword">end do</span>
    4881. 

  4881. <a name="l04817"></a>04817 
    4882. 

  4882. <a name="l04818"></a>04818   <span class="keyword">do</span> i = 14, 25
    4883. 

  4883. <a name="l04819"></a>04819     res24 = res24+cheb24(i)*rg(i)
    4884. 

  4884. <a name="l04820"></a>04820   <span class="keyword">end do</span>
    4885. 

  4885. <a name="l04821"></a>04821 
    4886. 

  4886. <a name="l04822"></a>04822 130 continue
    4887. 

  4887. <a name="l04823"></a>04823 
    4888. 

  4888. <a name="l04824"></a>04824   result = (result+res24)*factor
    4889. 

  4889. <a name="l04825"></a>04825   abserr = (abserr+abs(res24-res12))*factor
    4890. 

  4890. <a name="l04826"></a>04826   go to 270
    4891. 

  4891. <a name="l04827"></a>04827 <span class="comment">!</span>
    4892. 

  4892. <a name="l04828"></a>04828 <span class="comment">!  This part of the program is executed only if b = br.</span>
    4893. 

  4893. <a name="l04829"></a>04829 <span class="comment">!</span>
    4894. 

  4894. <a name="l04830"></a>04830 <span class="comment">!  Compute the Chebyshev series expansion of the function</span>
    4895. 

  4895. <a name="l04831"></a>04831 <span class="comment">!  f2 = (0.5*(b+bl-a-a)+0.5*(b-bl)*x)**alfa*f(0.5*(b-bl)*x+0.5*(b+bl))</span>
    4896. 

  4896. <a name="l04832"></a>04832 <span class="comment">!</span>
    4897. 

  4897. <a name="l04833"></a>04833 140 continue
    4898. 

  4898. <a name="l04834"></a>04834 
    4899. 

  4899. <a name="l04835"></a>04835   hlgth = 5.0e-01*(br-bl)
    4900. 

  4900. <a name="l04836"></a>04836   centr = 5.0e-01*(br+bl)
    4901. 

  4901. <a name="l04837"></a>04837   fix = centr-a
    4902. 

  4902. <a name="l04838"></a>04838   fval(1) = 5.0e-01*f(hlgth+centr)*(fix+hlgth)**alfa
    4903. 

  4903. <a name="l04839"></a>04839   fval(13) = f(centr)*(fix**alfa)
    4904. 

  4904. <a name="l04840"></a>04840   fval(25) = 5.0e-01*f(centr-hlgth)*(fix-hlgth)**alfa
    4905. 

  4905. <a name="l04841"></a>04841 
    4906. 

  4906. <a name="l04842"></a>04842   <span class="keyword">do</span> i = 2, 12
    4907. 

  4907. <a name="l04843"></a>04843     u = hlgth*x(i-1)
    4908. 

  4908. <a name="l04844"></a>04844     isym = 26-i
    4909. 

  4909. <a name="l04845"></a>04845     fval(i) = f(u+centr)*(fix+u)**alfa
    4910. 

  4910. <a name="l04846"></a>04846     fval(isym) = f(centr-u)*(fix-u)**alfa
    4911. 

  4911. <a name="l04847"></a>04847   <span class="keyword">end do</span>
    4912. 

  4912. <a name="l04848"></a>04848 
    4913. 

  4913. <a name="l04849"></a>04849   factor = hlgth**(beta+1.0e+00)
    4914. 

  4914. <a name="l04850"></a>04850   result = 0.0e+00
    4915. 

  4915. <a name="l04851"></a>04851   abserr = 0.0e+00
    4916. 

  4916. <a name="l04852"></a>04852   res12 = 0.0e+00
    4917. 

  4917. <a name="l04853"></a>04853   res24 = 0.0e+00
    4918. 

  4918. <a name="l04854"></a>04854 
    4919. 

  4919. <a name="l04855"></a>04855   <span class="keyword">if</span> ( integr == 2 .or. integr == 4 ) <span class="keyword">then</span>
    4920. 

  4920. <a name="l04856"></a>04856     go to 200
    4921. 

  4921. <a name="l04857"></a>04857   <span class="keyword">end if</span>
    4922. 

  4922. <a name="l04858"></a>04858 <span class="comment">!</span>
    4923. 

  4923. <a name="l04859"></a>04859 <span class="comment">!  integr = 1  (or 3)</span>
    4924. 

  4924. <a name="l04860"></a>04860 <span class="comment">!</span>
    4925. 

  4925. <a name="l04861"></a>04861   call <a class="code" href="quadpack_8f90.html#ad5beefcfdb335ea68ccf8397536c8c36">qcheb </a>( x, fval, cheb12, cheb24 )
    4926. 

  4926. <a name="l04862"></a>04862 
    4927. 

  4927. <a name="l04863"></a>04863   <span class="keyword">do</span> i = 1, 13
    4928. 

  4928. <a name="l04864"></a>04864     res12 = res12+cheb12(i)*rj(i)
    4929. 

  4929. <a name="l04865"></a>04865     res24 = res24+cheb24(i)*rj(i)
    4930. 

  4930. <a name="l04866"></a>04866   <span class="keyword">end do</span>
    4931. 

  4931. <a name="l04867"></a>04867 
    4932. 

  4932. <a name="l04868"></a>04868   <span class="keyword">do</span> i = 14, 25
    4933. 

  4933. <a name="l04869"></a>04869     res24 = res24+cheb24(i)*rj(i)
    4934. 

  4934. <a name="l04870"></a>04870   <span class="keyword">end do</span>
    4935. 

  4935. <a name="l04871"></a>04871 
    4936. 

  4936. <a name="l04872"></a>04872   <span class="keyword">if</span> ( integr == 1 ) go to 260
    4937. 

  4937. <a name="l04873"></a>04873 <span class="comment">!</span>
    4938. 

  4938. <a name="l04874"></a>04874 <span class="comment">!  integr = 3</span>
    4939. 

  4939. <a name="l04875"></a>04875 <span class="comment">!</span>
    4940. 

  4940. <a name="l04876"></a>04876   dc = log ( br - bl )
    4941. 

  4941. <a name="l04877"></a>04877   result = res24*dc
    4942. 

  4942. <a name="l04878"></a>04878   abserr = abs((res24-res12)*dc)
    4943. 

  4943. <a name="l04879"></a>04879   res12 = 0.0e+00
    4944. 

  4944. <a name="l04880"></a>04880   res24 = 0.0e+00
    4945. 

  4945. <a name="l04881"></a>04881 
    4946. 

  4946. <a name="l04882"></a>04882   <span class="keyword">do</span> i = 1, 13
    4947. 

  4947. <a name="l04883"></a>04883     res12 = res12+cheb12(i)*rh(i)
    4948. 

  4948. <a name="l04884"></a>04884     res24 = res24+cheb24(i)*rh(i)
    4949. 

  4949. <a name="l04885"></a>04885   <span class="keyword">end do</span>
    4950. 

  4950. <a name="l04886"></a>04886 
    4951. 

  4951. <a name="l04887"></a>04887   <span class="keyword">do</span> i = 14, 25
    4952. 

  4952. <a name="l04888"></a>04888     res24 = res24+cheb24(i)*rh(i)
    4953. 

  4953. <a name="l04889"></a>04889   <span class="keyword">end do</span>
    4954. 

  4954. <a name="l04890"></a>04890 
    4955. 

  4955. <a name="l04891"></a>04891   go to 260
    4956. 

  4956. <a name="l04892"></a>04892 <span class="comment">!</span>
    4957. 

  4957. <a name="l04893"></a>04893 <span class="comment">!  Compute the Chebyshev series expansion of the function</span>
    4958. 

  4958. <a name="l04894"></a>04894 <span class="comment">!  f3 = f2*log(0.5*(b-bl)*x+0.5*(b+bl-a-a))</span>
    4959. 

  4959. <a name="l04895"></a>04895 <span class="comment">!</span>
    4960. 

  4960. <a name="l04896"></a>04896 200 continue
    4961. 

  4961. <a name="l04897"></a>04897 
    4962. 

  4962. <a name="l04898"></a>04898   fval(1) = fval(1) * log ( hlgth + fix )
    4963. 

  4963. <a name="l04899"></a>04899   fval(13) = fval(13) * log ( fix )
    4964. 

  4964. <a name="l04900"></a>04900   fval(25) = fval(25) * log ( fix - hlgth )
    4965. 

  4965. <a name="l04901"></a>04901 
    4966. 

  4966. <a name="l04902"></a>04902   <span class="keyword">do</span> i = 2, 12
    4967. 

  4967. <a name="l04903"></a>04903     u = hlgth*x(i-1)
    4968. 

  4968. <a name="l04904"></a>04904     isym = 26-i
    4969. 

  4969. <a name="l04905"></a>04905     fval(i) = fval(i) * log(u+fix)
    4970. 

  4970. <a name="l04906"></a>04906     fval(isym) = fval(isym) * log(fix-u)
    4971. 

  4971. <a name="l04907"></a>04907   <span class="keyword">end do</span>
    4972. 

  4972. <a name="l04908"></a>04908 
    4973. 

  4973. <a name="l04909"></a>04909   call <a class="code" href="quadpack_8f90.html#ad5beefcfdb335ea68ccf8397536c8c36">qcheb </a>( x, fval, cheb12, cheb24 )
    4974. 

  4974. <a name="l04910"></a>04910 <span class="comment">!</span>
    4975. 

  4975. <a name="l04911"></a>04911 <span class="comment">!  integr = 2  (or 4)</span>
    4976. 

  4976. <a name="l04912"></a>04912 <span class="comment">!</span>
    4977. 

  4977. <a name="l04913"></a>04913   <span class="keyword">do</span> i = 1, 13
    4978. 

  4978. <a name="l04914"></a>04914     res12 = res12+cheb12(i)*rj(i)
    4979. 

  4979. <a name="l04915"></a>04915     res24 = res24+cheb24(i)*rj(i)
    4980. 

  4980. <a name="l04916"></a>04916   <span class="keyword">end do</span>
    4981. 

  4981. <a name="l04917"></a>04917 
    4982. 

  4982. <a name="l04918"></a>04918   <span class="keyword">do</span> i = 14, 25
    4983. 

  4983. <a name="l04919"></a>04919     res24 = res24+cheb24(i)*rj(i)
    4984. 

  4984. <a name="l04920"></a>04920   <span class="keyword">end do</span>
    4985. 

  4985. <a name="l04921"></a>04921 
    4986. 

  4986. <a name="l04922"></a>04922   <span class="keyword">if</span> ( integr == 2 ) go to 260
    4987. 

  4987. <a name="l04923"></a>04923 
    4988. 

  4988. <a name="l04924"></a>04924   dc = log(br-bl)
    4989. 

  4989. <a name="l04925"></a>04925   result = res24*dc
    4990. 

  4990. <a name="l04926"></a>04926   abserr = abs((res24-res12)*dc)
    4991. 

  4991. <a name="l04927"></a>04927   res12 = 0.0e+00
    4992. 

  4992. <a name="l04928"></a>04928   res24 = 0.0e+00
    4993. 

  4993. <a name="l04929"></a>04929 <span class="comment">!</span>
    4994. 

  4994. <a name="l04930"></a>04930 <span class="comment">!  integr = 4</span>
    4995. 

  4995. <a name="l04931"></a>04931 <span class="comment">!</span>
    4996. 

  4996. <a name="l04932"></a>04932   <span class="keyword">do</span> i = 1, 13
    4997. 

  4997. <a name="l04933"></a>04933     res12 = res12+cheb12(i)*rh(i)
    4998. 

  4998. <a name="l04934"></a>04934     res24 = res24+cheb24(i)*rh(i)
    4999. 

  4999. <a name="l04935"></a>04935   <span class="keyword">end do</span>
    5000. 

  5000. <a name="l04936"></a>04936 
    5001. 

  5001. <a name="l04937"></a>04937   <span class="keyword">do</span> i = 14, 25
    5002. 

  5002. <a name="l04938"></a>04938     res24 = res24+cheb24(i)*rh(i)
    5003. 

  5003. <a name="l04939"></a>04939   <span class="keyword">end do</span>
    5004. 

  5004. <a name="l04940"></a>04940 
    5005. 

  5005. <a name="l04941"></a>04941 260 continue
    5006. 

  5006. <a name="l04942"></a>04942 
    5007. 

  5007. <a name="l04943"></a>04943   result = (result+res24)*factor
    5008. 

  5008. <a name="l04944"></a>04944   abserr = (abserr+abs(res24-res12))*factor
    5009. 

  5009. <a name="l04945"></a>04945 
    5010. 

  5010. <a name="l04946"></a>04946 270 continue
    5011. 

  5011. <a name="l04947"></a>04947 
    5012. 

  5012. <a name="l04948"></a>04948   return
    5013. 

  5013. <a name="l04949"></a>04949 <span class="keyword">end</span>
    5014. 

  5014. <a name="l04950"></a><a class="code" href="quadpack_8f90.html#ad5beefcfdb335ea68ccf8397536c8c36">04950</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#ad5beefcfdb335ea68ccf8397536c8c36">qcheb</a> ( x, fval, cheb12, cheb24 )
    5015. 

  5015. <a name="l04951"></a>04951 
    5016. 

  5016. <a name="l04952"></a>04952 <span class="comment">!*****************************************************************************80</span>
    5017. 

  5017. <a name="l04953"></a>04953 <span class="comment">!</span>
    5018. 

  5018. <a name="l04954"></a>04954 <span class="comment">!! QCHEB computes the Chebyshev series expansion.</span>
    5019. 

  5019. <a name="l04955"></a>04955 <span class="comment">!</span>
    5020. 

  5020. <a name="l04956"></a>04956 <span class="comment">!  Discussion:</span>
    5021. 

  5021. <a name="l04957"></a>04957 <span class="comment">!</span>
    5022. 

  5022. <a name="l04958"></a>04958 <span class="comment">!    This routine computes the Chebyshev series expansion</span>
    5023. 

  5023. <a name="l04959"></a>04959 <span class="comment">!    of degrees 12 and 24 of a function using a fast Fourier transform method</span>
    5024. 

  5024. <a name="l04960"></a>04960 <span class="comment">!</span>
    5025. 

  5025. <a name="l04961"></a>04961 <span class="comment">!      f(x) = sum(k=1, ...,13) (cheb12(k)*t(k-1,x)),</span>
    5026. 

  5026. <a name="l04962"></a>04962 <span class="comment">!      f(x) = sum(k=1, ...,25) (cheb24(k)*t(k-1,x)),</span>
    5027. 

  5027. <a name="l04963"></a>04963 <span class="comment">!</span>
    5028. 

  5028. <a name="l04964"></a>04964 <span class="comment">!    where T(K,X) is the Chebyshev polynomial of degree K.</span>
    5029. 

  5029. <a name="l04965"></a>04965 <span class="comment">!</span>
    5030. 

  5030. <a name="l04966"></a>04966 <span class="comment">!  Author:</span>
    5031. 

  5031. <a name="l04967"></a>04967 <span class="comment">!</span>
    5032. 

  5032. <a name="l04968"></a>04968 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    5033. 

  5033. <a name="l04969"></a>04969 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    5034. 

  5034. <a name="l04970"></a>04970 <span class="comment">!</span>
    5035. 

  5035. <a name="l04971"></a>04971 <span class="comment">!  Reference:</span>
    5036. 

  5036. <a name="l04972"></a>04972 <span class="comment">!</span>
    5037. 

  5037. <a name="l04973"></a>04973 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    5038. 

  5038. <a name="l04974"></a>04974 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    5039. 

  5039. <a name="l04975"></a>04975 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    5040. 

  5040. <a name="l04976"></a>04976 <span class="comment">!    Springer Verlag, 1983</span>
    5041. 

  5041. <a name="l04977"></a>04977 <span class="comment">!</span>
    5042. 

  5042. <a name="l04978"></a>04978 <span class="comment">!  Parameters:</span>
    5043. 

  5043. <a name="l04979"></a>04979 <span class="comment">!</span>
    5044. 

  5044. <a name="l04980"></a>04980 <span class="comment">!    Input, real X(11), contains the values of COS(K*PI/24), for K = 1 to 11.</span>
    5045. 

  5045. <a name="l04981"></a>04981 <span class="comment">!</span>
    5046. 

  5046. <a name="l04982"></a>04982 <span class="comment">!    Input/output, real FVAL(25), the function values at the points</span>
    5047. 

  5047. <a name="l04983"></a>04983 <span class="comment">!    (b+a+(b-a)*cos(k*pi/24))/2, k = 0, ...,24, where (a,b) is the </span>
    5048. 

  5048. <a name="l04984"></a>04984 <span class="comment">!    approximation interval.  FVAL(1) and FVAL(25) are divided by two</span>
    5049. 

  5049. <a name="l04985"></a>04985 <span class="comment">!    These values are destroyed at output.</span>
    5050. 

  5050. <a name="l04986"></a>04986 <span class="comment">!</span>
    5051. 

  5051. <a name="l04987"></a>04987 <span class="comment">!    Output, real CHEB12(13), the Chebyshev coefficients for degree 12.</span>
    5052. 

  5052. <a name="l04988"></a>04988 <span class="comment">!</span>
    5053. 

  5053. <a name="l04989"></a>04989 <span class="comment">!    Output, real CHEB24(25), the Chebyshev coefficients for degree 24.</span>
    5054. 

  5054. <a name="l04990"></a>04990 <span class="comment">!</span>
    5055. 

  5055. <a name="l04991"></a>04991   <span class="keyword">implicit none</span>
    5056. 

  5056. <a name="l04992"></a>04992 
    5057. 

  5057. <a name="l04993"></a>04993   <span class="keywordtype">real</span> alam
    5058. 

  5058. <a name="l04994"></a>04994   <span class="keywordtype">real</span> alam1
    5059. 

  5059. <a name="l04995"></a>04995   <span class="keywordtype">real</span> alam2
    5060. 

  5060. <a name="l04996"></a>04996   <span class="keywordtype">real</span> cheb12(13)
    5061. 

  5061. <a name="l04997"></a>04997   <span class="keywordtype">real</span> cheb24(25)
    5062. 

  5062. <a name="l04998"></a>04998   <span class="keywordtype">real</span> fval(25)
    5063. 

  5063. <a name="l04999"></a>04999   <span class="keywordtype">integer</span> i
    5064. 

  5064. <a name="l05000"></a>05000   <span class="keywordtype">integer</span> j
    5065. 

  5065. <a name="l05001"></a>05001   <span class="keywordtype">real</span> part1
    5066. 

  5066. <a name="l05002"></a>05002   <span class="keywordtype">real</span> part2
    5067. 

  5067. <a name="l05003"></a>05003   <span class="keywordtype">real</span> part3
    5068. 

  5068. <a name="l05004"></a>05004   <span class="keywordtype">real</span> v(12)
    5069. 

  5069. <a name="l05005"></a>05005   <span class="keywordtype">real</span> x(11)
    5070. 

  5070. <a name="l05006"></a>05006 
    5071. 

  5071. <a name="l05007"></a>05007   <span class="keyword">do</span> i = 1, 12
    5072. 

  5072. <a name="l05008"></a>05008     j = 26-i
    5073. 

  5073. <a name="l05009"></a>05009     v(i) = fval(i)-fval(j)
    5074. 

  5074. <a name="l05010"></a>05010     fval(i) = fval(i)+fval(j)
    5075. 

  5075. <a name="l05011"></a>05011   <span class="keyword">end do</span>
    5076. 

  5076. <a name="l05012"></a>05012 
    5077. 

  5077. <a name="l05013"></a>05013   alam1 = v(1)-v(9)
    5078. 

  5078. <a name="l05014"></a>05014   alam2 = x(6)*(v(3)-v(7)-v(11))
    5079. 

  5079. <a name="l05015"></a>05015   cheb12(4) = alam1+alam2
    5080. 

  5080. <a name="l05016"></a>05016   cheb12(10) = alam1-alam2
    5081. 

  5081. <a name="l05017"></a>05017   alam1 = v(2)-v(8)-v(10)
    5082. 

  5082. <a name="l05018"></a>05018   alam2 = v(4)-v(6)-v(12)
    5083. 

  5083. <a name="l05019"></a>05019   alam = x(3)*alam1+x(9)*alam2
    5084. 

  5084. <a name="l05020"></a>05020   cheb24(4) = cheb12(4)+alam
    5085. 

  5085. <a name="l05021"></a>05021   cheb24(22) = cheb12(4)-alam
    5086. 

  5086. <a name="l05022"></a>05022   alam = x(9)*alam1-x(3)*alam2
    5087. 

  5087. <a name="l05023"></a>05023   cheb24(10) = cheb12(10)+alam
    5088. 

  5088. <a name="l05024"></a>05024   cheb24(16) = cheb12(10)-alam
    5089. 

  5089. <a name="l05025"></a>05025   part1 = x(4)*v(5)
    5090. 

  5090. <a name="l05026"></a>05026   part2 = x(8)*v(9)
    5091. 

  5091. <a name="l05027"></a>05027   part3 = x(6)*v(7)
    5092. 

  5092. <a name="l05028"></a>05028   alam1 = v(1)+part1+part2
    5093. 

  5093. <a name="l05029"></a>05029   alam2 = x(2)*v(3)+part3+x(10)*v(11)
    5094. 

  5094. <a name="l05030"></a>05030   cheb12(2) = alam1+alam2
    5095. 

  5095. <a name="l05031"></a>05031   cheb12(12) = alam1-alam2
    5096. 

  5096. <a name="l05032"></a>05032   alam = x(1)*v(2)+x(3)*v(4)+x(5)*v(6)+x(7)*v(8) &amp;
    5097. 

  5097. <a name="l05033"></a>05033     +x(9)*v(10)+x(11)*v(12)
    5098. 

  5098. <a name="l05034"></a>05034   cheb24(2) = cheb12(2)+alam
    5099. 

  5099. <a name="l05035"></a>05035   cheb24(24) = cheb12(2)-alam
    5100. 

  5100. <a name="l05036"></a>05036   alam = x(11)*v(2)-x(9)*v(4)+x(7)*v(6)-x(5)*v(8) &amp;
    5101. 

  5101. <a name="l05037"></a>05037     +x(3)*v(10)-x(1)*v(12)
    5102. 

  5102. <a name="l05038"></a>05038   cheb24(12) = cheb12(12)+alam
    5103. 

  5103. <a name="l05039"></a>05039   cheb24(14) = cheb12(12)-alam
    5104. 

  5104. <a name="l05040"></a>05040   alam1 = v(1)-part1+part2
    5105. 

  5105. <a name="l05041"></a>05041   alam2 = x(10)*v(3)-part3+x(2)*v(11)
    5106. 

  5106. <a name="l05042"></a>05042   cheb12(6) = alam1+alam2
    5107. 

  5107. <a name="l05043"></a>05043   cheb12(8) = alam1-alam2
    5108. 

  5108. <a name="l05044"></a>05044   alam = x(5)*v(2)-x(9)*v(4)-x(1)*v(6) &amp;
    5109. 

  5109. <a name="l05045"></a>05045     -x(11)*v(8)+x(3)*v(10)+x(7)*v(12)
    5110. 

  5110. <a name="l05046"></a>05046   cheb24(6) = cheb12(6)+alam
    5111. 

  5111. <a name="l05047"></a>05047   cheb24(20) = cheb12(6)-alam
    5112. 

  5112. <a name="l05048"></a>05048   alam = x(7)*v(2)-x(3)*v(4)-x(11)*v(6)+x(1)*v(8) &amp;
    5113. 

  5113. <a name="l05049"></a>05049     -x(9)*v(10)-x(5)*v(12)
    5114. 

  5114. <a name="l05050"></a>05050   cheb24(8) = cheb12(8)+alam
    5115. 

  5115. <a name="l05051"></a>05051   cheb24(18) = cheb12(8)-alam
    5116. 

  5116. <a name="l05052"></a>05052 
    5117. 

  5117. <a name="l05053"></a>05053   <span class="keyword">do</span> i = 1, 6
    5118. 

  5118. <a name="l05054"></a>05054     j = 14-i
    5119. 

  5119. <a name="l05055"></a>05055     v(i) = fval(i)-fval(j)
    5120. 

  5120. <a name="l05056"></a>05056     fval(i) = fval(i)+fval(j)
    5121. 

  5121. <a name="l05057"></a>05057   <span class="keyword">end do</span>
    5122. 

  5122. <a name="l05058"></a>05058 
    5123. 

  5123. <a name="l05059"></a>05059   alam1 = v(1)+x(8)*v(5)
    5124. 

  5124. <a name="l05060"></a>05060   alam2 = x(4)*v(3)
    5125. 

  5125. <a name="l05061"></a>05061   cheb12(3) = alam1+alam2
    5126. 

  5126. <a name="l05062"></a>05062   cheb12(11) = alam1-alam2
    5127. 

  5127. <a name="l05063"></a>05063   cheb12(7) = v(1)-v(5)
    5128. 

  5128. <a name="l05064"></a>05064   alam = x(2)*v(2)+x(6)*v(4)+x(10)*v(6)
    5129. 

  5129. <a name="l05065"></a>05065   cheb24(3) = cheb12(3)+alam
    5130. 

  5130. <a name="l05066"></a>05066   cheb24(23) = cheb12(3)-alam
    5131. 

  5131. <a name="l05067"></a>05067   alam = x(6)*(v(2)-v(4)-v(6))
    5132. 

  5132. <a name="l05068"></a>05068   cheb24(7) = cheb12(7)+alam
    5133. 

  5133. <a name="l05069"></a>05069   cheb24(19) = cheb12(7)-alam
    5134. 

  5134. <a name="l05070"></a>05070   alam = x(10)*v(2)-x(6)*v(4)+x(2)*v(6)
    5135. 

  5135. <a name="l05071"></a>05071   cheb24(11) = cheb12(11)+alam
    5136. 

  5136. <a name="l05072"></a>05072   cheb24(15) = cheb12(11)-alam
    5137. 

  5137. <a name="l05073"></a>05073 
    5138. 

  5138. <a name="l05074"></a>05074   <span class="keyword">do</span> i = 1, 3
    5139. 

  5139. <a name="l05075"></a>05075     j = 8-i
    5140. 

  5140. <a name="l05076"></a>05076     v(i) = fval(i)-fval(j)
    5141. 

  5141. <a name="l05077"></a>05077     fval(i) = fval(i)+fval(j)
    5142. 

  5142. <a name="l05078"></a>05078   <span class="keyword">end do</span>
    5143. 

  5143. <a name="l05079"></a>05079 
    5144. 

  5144. <a name="l05080"></a>05080   cheb12(5) = v(1)+x(8)*v(3)
    5145. 

  5145. <a name="l05081"></a>05081   cheb12(9) = fval(1)-x(8)*fval(3)
    5146. 

  5146. <a name="l05082"></a>05082   alam = x(4)*v(2)
    5147. 

  5147. <a name="l05083"></a>05083   cheb24(5) = cheb12(5)+alam
    5148. 

  5148. <a name="l05084"></a>05084   cheb24(21) = cheb12(5)-alam
    5149. 

  5149. <a name="l05085"></a>05085   alam = x(8)*fval(2)-fval(4)
    5150. 

  5150. <a name="l05086"></a>05086   cheb24(9) = cheb12(9)+alam
    5151. 

  5151. <a name="l05087"></a>05087   cheb24(17) = cheb12(9)-alam
    5152. 

  5152. <a name="l05088"></a>05088   cheb12(1) = fval(1)+fval(3)
    5153. 

  5153. <a name="l05089"></a>05089   alam = fval(2)+fval(4)
    5154. 

  5154. <a name="l05090"></a>05090   cheb24(1) = cheb12(1)+alam
    5155. 

  5155. <a name="l05091"></a>05091   cheb24(25) = cheb12(1)-alam
    5156. 

  5156. <a name="l05092"></a>05092   cheb12(13) = v(1)-v(3)
    5157. 

  5157. <a name="l05093"></a>05093   cheb24(13) = cheb12(13)
    5158. 

  5158. <a name="l05094"></a>05094   alam = 1.0e+00/6.0e+00
    5159. 

  5159. <a name="l05095"></a>05095 
    5160. 

  5160. <a name="l05096"></a>05096   <span class="keyword">do</span> i = 2, 12
    5161. 

  5161. <a name="l05097"></a>05097     cheb12(i) = cheb12(i)*alam
    5162. 

  5162. <a name="l05098"></a>05098   <span class="keyword">end do</span>
    5163. 

  5163. <a name="l05099"></a>05099 
    5164. 

  5164. <a name="l05100"></a>05100   alam = 5.0e-01*alam
    5165. 

  5165. <a name="l05101"></a>05101   cheb12(1) = cheb12(1)*alam
    5166. 

  5166. <a name="l05102"></a>05102   cheb12(13) = cheb12(13)*alam
    5167. 

  5167. <a name="l05103"></a>05103 
    5168. 

  5168. <a name="l05104"></a>05104   <span class="keyword">do</span> i = 2, 24
    5169. 

  5169. <a name="l05105"></a>05105     cheb24(i) = cheb24(i)*alam
    5170. 

  5170. <a name="l05106"></a>05106   <span class="keyword">end do</span>
    5171. 

  5171. <a name="l05107"></a>05107 
    5172. 

  5172. <a name="l05108"></a>05108   cheb24(1) = 0.5E+00 * alam*cheb24(1)
    5173. 

  5173. <a name="l05109"></a>05109   cheb24(25) = 0.5E+00 * alam*cheb24(25)
    5174. 

  5174. <a name="l05110"></a>05110 
    5175. 

  5175. <a name="l05111"></a>05111   return
    5176. 

  5176. <a name="l05112"></a>05112 <span class="keyword">end</span>
    5177. 

  5177. <a name="l05113"></a><a class="code" href="quadpack_8f90.html#a5a75101d080f224c63adde98a0e64386">05113</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a5a75101d080f224c63adde98a0e64386">qextr</a> ( n, epstab, result, abserr, res3la, nres )
    5178. 

  5178. <a name="l05114"></a>05114 
    5179. 

  5179. <a name="l05115"></a>05115 <span class="comment">!*****************************************************************************80</span>
    5180. 

  5180. <a name="l05116"></a>05116 <span class="comment">!</span>
    5181. 

  5181. <a name="l05117"></a>05117 <span class="comment">!! QEXTR carries out the Epsilon extrapolation algorithm.</span>
    5182. 

  5182. <a name="l05118"></a>05118 <span class="comment">!</span>
    5183. 

  5183. <a name="l05119"></a>05119 <span class="comment">!  Discussion:</span>
    5184. 

  5184. <a name="l05120"></a>05120 <span class="comment">!</span>
    5185. 

  5185. <a name="l05121"></a>05121 <span class="comment">!    The routine determines the limit of a given sequence of approximations, </span>
    5186. 

  5186. <a name="l05122"></a>05122 <span class="comment">!    by means of the epsilon algorithm of P. Wynn.  An estimate of the </span>
    5187. 

  5187. <a name="l05123"></a>05123 <span class="comment">!    absolute error is also given.  The condensed epsilon table is computed.</span>
    5188. 

  5188. <a name="l05124"></a>05124 <span class="comment">!    Only those elements needed for the computation of the next diagonal</span>
    5189. 

  5189. <a name="l05125"></a>05125 <span class="comment">!    are preserved.</span>
    5190. 

  5190. <a name="l05126"></a>05126 <span class="comment">!</span>
    5191. 

  5191. <a name="l05127"></a>05127 <span class="comment">!  Author:</span>
    5192. 

  5192. <a name="l05128"></a>05128 <span class="comment">!</span>
    5193. 

  5193. <a name="l05129"></a>05129 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    5194. 

  5194. <a name="l05130"></a>05130 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    5195. 

  5195. <a name="l05131"></a>05131 <span class="comment">!</span>
    5196. 

  5196. <a name="l05132"></a>05132 <span class="comment">!  Reference:</span>
    5197. 

  5197. <a name="l05133"></a>05133 <span class="comment">!</span>
    5198. 

  5198. <a name="l05134"></a>05134 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    5199. 

  5199. <a name="l05135"></a>05135 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    5200. 

  5200. <a name="l05136"></a>05136 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    5201. 

  5201. <a name="l05137"></a>05137 <span class="comment">!    Springer Verlag, 1983</span>
    5202. 

  5202. <a name="l05138"></a>05138 <span class="comment">!</span>
    5203. 

  5203. <a name="l05139"></a>05139 <span class="comment">!  Parameters:</span>
    5204. 

  5204. <a name="l05140"></a>05140 <span class="comment">!</span>
    5205. 

  5205. <a name="l05141"></a>05141 <span class="comment">!    Input, integer N, indicates the entry of EPSTAB which contains</span>
    5206. 

  5206. <a name="l05142"></a>05142 <span class="comment">!    the new element in the first column of the epsilon table.</span>
    5207. 

  5207. <a name="l05143"></a>05143 <span class="comment">!</span>
    5208. 

  5208. <a name="l05144"></a>05144 <span class="comment">!    Input/output, real EPSTAB(52), the two lower diagonals of the triangular</span>
    5209. 

  5209. <a name="l05145"></a>05145 <span class="comment">!    epsilon table.  The elements are numbered starting at the right-hand </span>
    5210. 

  5210. <a name="l05146"></a>05146 <span class="comment">!    corner of the triangle.</span>
    5211. 

  5211. <a name="l05147"></a>05147 <span class="comment">!</span>
    5212. 

  5212. <a name="l05148"></a>05148 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    5213. 

  5213. <a name="l05149"></a>05149 <span class="comment">!</span>
    5214. 

  5214. <a name="l05150"></a>05150 <span class="comment">!    Output, real ABSERR, estimate of the absolute error computed from</span>
    5215. 

  5215. <a name="l05151"></a>05151 <span class="comment">!    RESULT and the 3 previous results.</span>
    5216. 

  5216. <a name="l05152"></a>05152 <span class="comment">!</span>
    5217. 

  5217. <a name="l05153"></a>05153 <span class="comment">!    ?, real RES3LA(3), the last 3 results.</span>
    5218. 

  5218. <a name="l05154"></a>05154 <span class="comment">!</span>
    5219. 

  5219. <a name="l05155"></a>05155 <span class="comment">!    Input/output, integer NRES, the number of calls to the routine.  This</span>
    5220. 

  5220. <a name="l05156"></a>05156 <span class="comment">!    should be zero on the first call, and is automatically updated</span>
    5221. 

  5221. <a name="l05157"></a>05157 <span class="comment">!    before return.</span>
    5222. 

  5222. <a name="l05158"></a>05158 <span class="comment">!</span>
    5223. 

  5223. <a name="l05159"></a>05159 <span class="comment">!  Local Parameters:</span>
    5224. 

  5224. <a name="l05160"></a>05160 <span class="comment">!</span>
    5225. 

  5225. <a name="l05161"></a>05161 <span class="comment">!           e0     - the 4 elements on which the</span>
    5226. 

  5226. <a name="l05162"></a>05162 <span class="comment">!           e1       computation of a new element in</span>
    5227. 

  5227. <a name="l05163"></a>05163 <span class="comment">!           e2       the epsilon table is based</span>
    5228. 

  5228. <a name="l05164"></a>05164 <span class="comment">!           e3                 e0</span>
    5229. 

  5229. <a name="l05165"></a>05165 <span class="comment">!                        e3    e1    new</span>
    5230. 

  5230. <a name="l05166"></a>05166 <span class="comment">!                              e2</span>
    5231. 

  5231. <a name="l05167"></a>05167 <span class="comment">!           newelm - number of elements to be computed in the new</span>
    5232. 

  5232. <a name="l05168"></a>05168 <span class="comment">!                    diagonal</span>
    5233. 

  5233. <a name="l05169"></a>05169 <span class="comment">!           error  - error = abs(e1-e0)+abs(e2-e1)+abs(new-e2)</span>
    5234. 

  5234. <a name="l05170"></a>05170 <span class="comment">!           result - the element in the new diagonal with least value</span>
    5235. 

  5235. <a name="l05171"></a>05171 <span class="comment">!                    of error</span>
    5236. 

  5236. <a name="l05172"></a>05172 <span class="comment">!           limexp is the maximum number of elements the epsilon table</span>
    5237. 

  5237. <a name="l05173"></a>05173 <span class="comment">!           can contain. if this number is reached, the upper diagonal</span>
    5238. 

  5238. <a name="l05174"></a>05174 <span class="comment">!           of the epsilon table is deleted.</span>
    5239. 

  5239. <a name="l05175"></a>05175 <span class="comment">!</span>
    5240. 

  5240. <a name="l05176"></a>05176   <span class="keyword">implicit none</span>
    5241. 

  5241. <a name="l05177"></a>05177 
    5242. 

  5242. <a name="l05178"></a>05178   <span class="keywordtype">real</span> abserr
    5243. 

  5243. <a name="l05179"></a>05179   <span class="keywordtype">real</span> delta1
    5244. 

  5244. <a name="l05180"></a>05180   <span class="keywordtype">real</span> delta2
    5245. 

  5245. <a name="l05181"></a>05181   <span class="keywordtype">real</span> delta3
    5246. 

  5246. <a name="l05182"></a>05182   <span class="keywordtype">real</span> epsinf
    5247. 

  5247. <a name="l05183"></a>05183   <span class="keywordtype">real</span> epstab(52)
    5248. 

  5248. <a name="l05184"></a>05184   <span class="keywordtype">real</span> error
    5249. 

  5249. <a name="l05185"></a>05185   <span class="keywordtype">real</span> err1
    5250. 

  5250. <a name="l05186"></a>05186   <span class="keywordtype">real</span> err2
    5251. 

  5251. <a name="l05187"></a>05187   <span class="keywordtype">real</span> err3
    5252. 

  5252. <a name="l05188"></a>05188   <span class="keywordtype">real</span> e0
    5253. 

  5253. <a name="l05189"></a>05189   <span class="keywordtype">real</span> e1
    5254. 

  5254. <a name="l05190"></a>05190   <span class="keywordtype">real</span> e1abs
    5255. 

  5255. <a name="l05191"></a>05191   <span class="keywordtype">real</span> e2
    5256. 

  5256. <a name="l05192"></a>05192   <span class="keywordtype">real</span> e3
    5257. 

  5257. <a name="l05193"></a>05193   <span class="keywordtype">integer</span> i
    5258. 

  5258. <a name="l05194"></a>05194   <span class="keywordtype">integer</span> ib
    5259. 

  5259. <a name="l05195"></a>05195   <span class="keywordtype">integer</span> ib2
    5260. 

  5260. <a name="l05196"></a>05196   <span class="keywordtype">integer</span> ie
    5261. 

  5261. <a name="l05197"></a>05197   <span class="keywordtype">integer</span> indx
    5262. 

  5262. <a name="l05198"></a>05198   <span class="keywordtype">integer</span> k1
    5263. 

  5263. <a name="l05199"></a>05199   <span class="keywordtype">integer</span> k2
    5264. 

  5264. <a name="l05200"></a>05200   <span class="keywordtype">integer</span> k3
    5265. 

  5265. <a name="l05201"></a>05201   <span class="keywordtype">integer</span> limexp
    5266. 

  5266. <a name="l05202"></a>05202   <span class="keywordtype">integer</span> n
    5267. 

  5267. <a name="l05203"></a>05203   <span class="keywordtype">integer</span> newelm
    5268. 

  5268. <a name="l05204"></a>05204   <span class="keywordtype">integer</span> nres
    5269. 

  5269. <a name="l05205"></a>05205   <span class="keywordtype">integer</span> num
    5270. 

  5270. <a name="l05206"></a>05206   <span class="keywordtype">real</span> res
    5271. 

  5271. <a name="l05207"></a>05207   <span class="keywordtype">real</span> result
    5272. 

  5272. <a name="l05208"></a>05208   <span class="keywordtype">real</span> res3la(3)
    5273. 

  5273. <a name="l05209"></a>05209   <span class="keywordtype">real</span> ss
    5274. 

  5274. <a name="l05210"></a>05210   <span class="keywordtype">real</span> tol1
    5275. 

  5275. <a name="l05211"></a>05211   <span class="keywordtype">real</span> tol2
    5276. 

  5276. <a name="l05212"></a>05212   <span class="keywordtype">real</span> tol3
    5277. 

  5277. <a name="l05213"></a>05213 
    5278. 

  5278. <a name="l05214"></a>05214   nres = nres+1
    5279. 

  5279. <a name="l05215"></a>05215   abserr = huge ( abserr )
    5280. 

  5280. <a name="l05216"></a>05216   result = epstab(n)
    5281. 

  5281. <a name="l05217"></a>05217 
    5282. 

  5282. <a name="l05218"></a>05218   <span class="keyword">if</span> ( n &lt; 3 ) <span class="keyword">then</span>
    5283. 

  5283. <a name="l05219"></a>05219     abserr = max ( abserr,0.5e+00* epsilon ( result ) *abs(result))
    5284. 

  5284. <a name="l05220"></a>05220     return
    5285. 

  5285. <a name="l05221"></a>05221   <span class="keyword">end if</span>
    5286. 

  5286. <a name="l05222"></a>05222 
    5287. 

  5287. <a name="l05223"></a>05223   limexp = 50
    5288. 

  5288. <a name="l05224"></a>05224   epstab(n+2) = epstab(n)
    5289. 

  5289. <a name="l05225"></a>05225   newelm = (n-1)/2
    5290. 

  5290. <a name="l05226"></a>05226   epstab(n) = huge ( epstab(n) )
    5291. 

  5291. <a name="l05227"></a>05227   num = n
    5292. 

  5292. <a name="l05228"></a>05228   k1 = n
    5293. 

  5293. <a name="l05229"></a>05229 
    5294. 

  5294. <a name="l05230"></a>05230   <span class="keyword">do</span> i = 1, newelm
    5295. 

  5295. <a name="l05231"></a>05231 
    5296. 

  5296. <a name="l05232"></a>05232     k2 = k1-1
    5297. 

  5297. <a name="l05233"></a>05233     k3 = k1-2
    5298. 

  5298. <a name="l05234"></a>05234     res = epstab(k1+2)
    5299. 

  5299. <a name="l05235"></a>05235     e0 = epstab(k3)
    5300. 

  5300. <a name="l05236"></a>05236     e1 = epstab(k2)
    5301. 

  5301. <a name="l05237"></a>05237     e2 = res
    5302. 

  5302. <a name="l05238"></a>05238     e1abs = abs(e1)
    5303. 

  5303. <a name="l05239"></a>05239     delta2 = e2-e1
    5304. 

  5304. <a name="l05240"></a>05240     err2 = abs(delta2)
    5305. 

  5305. <a name="l05241"></a>05241     tol2 = max ( abs(e2),e1abs)* epsilon ( e2 )
    5306. 

  5306. <a name="l05242"></a>05242     delta3 = e1-e0
    5307. 

  5307. <a name="l05243"></a>05243     err3 = abs(delta3)
    5308. 

  5308. <a name="l05244"></a>05244     tol3 = max ( e1abs,abs(e0))* epsilon ( e0 )
    5309. 

  5309. <a name="l05245"></a>05245 <span class="comment">!</span>
    5310. 

  5310. <a name="l05246"></a>05246 <span class="comment">!  If e0, e1 and e2 are equal to within machine accuracy, convergence </span>
    5311. 

  5311. <a name="l05247"></a>05247 <span class="comment">!  is assumed.</span>
    5312. 

  5312. <a name="l05248"></a>05248 <span class="comment">!</span>
    5313. 

  5313. <a name="l05249"></a>05249     <span class="keyword">if</span> ( err2 &lt;= tol2 .and. err3 &lt;= tol3 ) <span class="keyword">then</span>
    5314. 

  5314. <a name="l05250"></a>05250       result = res
    5315. 

  5315. <a name="l05251"></a>05251       abserr = err2+err3
    5316. 

  5316. <a name="l05252"></a>05252       abserr = max ( abserr,0.5e+00* epsilon ( result ) *abs(result))
    5317. 

  5317. <a name="l05253"></a>05253       return
    5318. 

  5318. <a name="l05254"></a>05254     <span class="keyword">end if</span>
    5319. 

  5319. <a name="l05255"></a>05255 
    5320. 

  5320. <a name="l05256"></a>05256     e3 = epstab(k1)
    5321. 

  5321. <a name="l05257"></a>05257     epstab(k1) = e1
    5322. 

  5322. <a name="l05258"></a>05258     delta1 = e1-e3
    5323. 

  5323. <a name="l05259"></a>05259     err1 = abs(delta1)
    5324. 

  5324. <a name="l05260"></a>05260     tol1 = max ( e1abs,abs(e3))* epsilon ( e3 )
    5325. 

  5325. <a name="l05261"></a>05261 <span class="comment">!</span>
    5326. 

  5326. <a name="l05262"></a>05262 <span class="comment">!  If two elements are very close to each other, omit a part</span>
    5327. 

  5327. <a name="l05263"></a>05263 <span class="comment">!  of the table by adjusting the value of N.</span>
    5328. 

  5328. <a name="l05264"></a>05264 <span class="comment">!</span>
    5329. 

  5329. <a name="l05265"></a>05265     <span class="keyword">if</span> ( err1 &lt;= tol1 .or. err2 &lt;= tol2 .or. err3 &lt;= tol3 ) go to 20
    5330. 

  5330. <a name="l05266"></a>05266 
    5331. 

  5331. <a name="l05267"></a>05267     ss = 1.0e+00/delta1+1.0e+00/delta2-1.0e+00/delta3
    5332. 

  5332. <a name="l05268"></a>05268     epsinf = abs ( ss*e1 )
    5333. 

  5333. <a name="l05269"></a>05269 <span class="comment">!</span>
    5334. 

  5334. <a name="l05270"></a>05270 <span class="comment">!  Test to detect irregular behavior in the table, and</span>
    5335. 

  5335. <a name="l05271"></a>05271 <span class="comment">!  eventually omit a part of the table adjusting the value of N.</span>
    5336. 

  5336. <a name="l05272"></a>05272 <span class="comment">!</span>
    5337. 

  5337. <a name="l05273"></a>05273     <span class="keyword">if</span> ( epsinf &gt; 1.0e-04 ) go to 30
    5338. 

  5338. <a name="l05274"></a>05274 
    5339. 

  5339. <a name="l05275"></a>05275 20  continue
    5340. 

  5340. <a name="l05276"></a>05276 
    5341. 

  5341. <a name="l05277"></a>05277     n = i+i-1
    5342. 

  5342. <a name="l05278"></a>05278     exit
    5343. 

  5343. <a name="l05279"></a>05279 <span class="comment">!</span>
    5344. 

  5344. <a name="l05280"></a>05280 <span class="comment">!  Compute a new element and eventually adjust the value of RESULT.</span>
    5345. 

  5345. <a name="l05281"></a>05281 <span class="comment">!</span>
    5346. 

  5346. <a name="l05282"></a>05282 30  continue
    5347. 

  5347. <a name="l05283"></a>05283 
    5348. 

  5348. <a name="l05284"></a>05284     res = e1+1.0e+00/ss
    5349. 

  5349. <a name="l05285"></a>05285     epstab(k1) = res
    5350. 

  5350. <a name="l05286"></a>05286     k1 = k1-2
    5351. 

  5351. <a name="l05287"></a>05287     error = err2+abs(res-e2)+err3
    5352. 

  5352. <a name="l05288"></a>05288 
    5353. 

  5353. <a name="l05289"></a>05289     <span class="keyword">if</span> ( error &lt;= abserr ) <span class="keyword">then</span>
    5354. 

  5354. <a name="l05290"></a>05290       abserr = error
    5355. 

  5355. <a name="l05291"></a>05291       result = res
    5356. 

  5356. <a name="l05292"></a>05292     <span class="keyword">end if</span>
    5357. 

  5357. <a name="l05293"></a>05293 
    5358. 

  5358. <a name="l05294"></a>05294   <span class="keyword">end do</span>
    5359. 

  5359. <a name="l05295"></a>05295 <span class="comment">!</span>
    5360. 

  5360. <a name="l05296"></a>05296 <span class="comment">!  Shift the table.</span>
    5361. 

  5361. <a name="l05297"></a>05297 <span class="comment">!</span>
    5362. 

  5362. <a name="l05298"></a>05298   <span class="keyword">if</span> ( n == limexp ) <span class="keyword">then</span>
    5363. 

  5363. <a name="l05299"></a>05299     n = 2*(limexp/2)-1
    5364. 

  5364. <a name="l05300"></a>05300   <span class="keyword">end if</span>
    5365. 

  5365. <a name="l05301"></a>05301 
    5366. 

  5366. <a name="l05302"></a>05302   <span class="keyword">if</span> ( (num/2)*2 == num ) <span class="keyword">then</span>
    5367. 

  5367. <a name="l05303"></a>05303     ib = 2
    5368. 

  5368. <a name="l05304"></a>05304   <span class="keyword">else</span>
    5369. 

  5369. <a name="l05305"></a>05305     ib = 1
    5370. 

  5370. <a name="l05306"></a>05306   <span class="keyword">end if</span>
    5371. 

  5371. <a name="l05307"></a>05307 
    5372. 

  5372. <a name="l05308"></a>05308   ie = newelm+1
    5373. 

  5373. <a name="l05309"></a>05309 
    5374. 

  5374. <a name="l05310"></a>05310   <span class="keyword">do</span> i = 1, ie
    5375. 

  5375. <a name="l05311"></a>05311     ib2 = ib+2
    5376. 

  5376. <a name="l05312"></a>05312     epstab(ib) = epstab(ib2)
    5377. 

  5377. <a name="l05313"></a>05313     ib = ib2
    5378. 

  5378. <a name="l05314"></a>05314   <span class="keyword">end do</span>
    5379. 

  5379. <a name="l05315"></a>05315 
    5380. 

  5380. <a name="l05316"></a>05316   <span class="keyword">if</span> ( num /= n ) <span class="keyword">then</span>
    5381. 

  5381. <a name="l05317"></a>05317 
    5382. 

  5382. <a name="l05318"></a>05318     indx = num-n+1
    5383. 

  5383. <a name="l05319"></a>05319 
    5384. 

  5384. <a name="l05320"></a>05320     <span class="keyword">do</span> i = 1, n
    5385. 

  5385. <a name="l05321"></a>05321       epstab(i)= epstab(indx)
    5386. 

  5386. <a name="l05322"></a>05322       indx = indx+1
    5387. 

  5387. <a name="l05323"></a>05323     <span class="keyword">end do</span>
    5388. 

  5388. <a name="l05324"></a>05324 
    5389. 

  5389. <a name="l05325"></a>05325   <span class="keyword">end if</span>
    5390. 

  5390. <a name="l05326"></a>05326 
    5391. 

  5391. <a name="l05327"></a>05327   <span class="keyword">if</span> ( nres &lt; 4 ) <span class="keyword">then</span>
    5392. 

  5392. <a name="l05328"></a>05328     res3la(nres) = result
    5393. 

  5393. <a name="l05329"></a>05329     abserr = huge ( abserr )
    5394. 

  5394. <a name="l05330"></a>05330   <span class="keyword">else</span>
    5395. 

  5395. <a name="l05331"></a>05331     abserr = abs(result-res3la(3))+abs(result-res3la(2)) &amp;
    5396. 

  5396. <a name="l05332"></a>05332       +abs(result-res3la(1))
    5397. 

  5397. <a name="l05333"></a>05333     res3la(1) = res3la(2)
    5398. 

  5398. <a name="l05334"></a>05334     res3la(2) = res3la(3)
    5399. 

  5399. <a name="l05335"></a>05335     res3la(3) = result
    5400. 

  5400. <a name="l05336"></a>05336   <span class="keyword">end if</span>
    5401. 

  5401. <a name="l05337"></a>05337 
    5402. 

  5402. <a name="l05338"></a>05338   abserr = max ( abserr,0.5e+00* epsilon ( result ) *abs(result))
    5403. 

  5403. <a name="l05339"></a>05339 
    5404. 

  5404. <a name="l05340"></a>05340   return
    5405. 

  5405. <a name="l05341"></a>05341 <span class="keyword">end</span>
    5406. 

  5406. <a name="l05342"></a><a class="code" href="quadpack_8f90.html#abbba06307e0e8c4daa2651945570ba1c">05342</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#abbba06307e0e8c4daa2651945570ba1c">qfour</a> ( f, a, b, omega, integr, epsabs, epsrel, limit, icall, &amp;
    5407. 

  5407. <a name="l05343"></a>05343   maxp1, result, abserr, neval, ier, alist, blist, rlist, elist, iord, &amp;
    5408. 

  5408. <a name="l05344"></a>05344   nnlog, momcom, chebmo )
    5409. 

  5409. <a name="l05345"></a>05345 
    5410. 

  5410. <a name="l05346"></a>05346 <span class="comment">!*****************************************************************************80</span>
    5411. 

  5411. <a name="l05347"></a>05347 <span class="comment">!</span>
    5412. 

  5412. <a name="l05348"></a>05348 <span class="comment">!! QFOUR estimates the integrals of oscillatory functions.</span>
    5413. 

  5413. <a name="l05349"></a>05349 <span class="comment">!</span>
    5414. 

  5414. <a name="l05350"></a>05350 <span class="comment">!  Discussion:</span>
    5415. 

  5415. <a name="l05351"></a>05351 <span class="comment">!</span>
    5416. 

  5416. <a name="l05352"></a>05352 <span class="comment">!    This routine calculates an approximation RESULT to a definite integral</span>
    5417. 

  5417. <a name="l05353"></a>05353 <span class="comment">!      I = integral of F(X) * COS(OMEGA*X) </span>
    5418. 

  5418. <a name="l05354"></a>05354 <span class="comment">!    or</span>
    5419. 

  5419. <a name="l05355"></a>05355 <span class="comment">!      I = integral of F(X) * SIN(OMEGA*X) </span>
    5420. 

  5420. <a name="l05356"></a>05356 <span class="comment">!    over (A,B), hopefully satisfying:</span>
    5421. 

  5421. <a name="l05357"></a>05357 <span class="comment">!      | I - RESULT | &lt;= max ( epsabs, epsrel * |I| ) ).</span>
    5422. 

  5422. <a name="l05358"></a>05358 <span class="comment">!</span>
    5423. 

  5423. <a name="l05359"></a>05359 <span class="comment">!    QFOUR is called by QAWO and QAWF.  It can also be called directly in </span>
    5424. 

  5424. <a name="l05360"></a>05360 <span class="comment">!    a user-written program.  In the latter case it is possible for the </span>
    5425. 

  5425. <a name="l05361"></a>05361 <span class="comment">!    user to determine the first dimension of array CHEBMO(MAXP1,25).</span>
    5426. 

  5426. <a name="l05362"></a>05362 <span class="comment">!    See also parameter description of MAXP1.  Additionally see</span>
    5427. 

  5427. <a name="l05363"></a>05363 <span class="comment">!    parameter description of ICALL for eventually re-using</span>
    5428. 

  5428. <a name="l05364"></a>05364 <span class="comment">!    Chebyshev moments computed during former call on subinterval</span>
    5429. 

  5429. <a name="l05365"></a>05365 <span class="comment">!    of equal length abs(B-A).</span>
    5430. 

  5430. <a name="l05366"></a>05366 <span class="comment">!</span>
    5431. 

  5431. <a name="l05367"></a>05367 <span class="comment">!  Author:</span>
    5432. 

  5432. <a name="l05368"></a>05368 <span class="comment">!</span>
    5433. 

  5433. <a name="l05369"></a>05369 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    5434. 

  5434. <a name="l05370"></a>05370 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    5435. 

  5435. <a name="l05371"></a>05371 <span class="comment">!</span>
    5436. 

  5436. <a name="l05372"></a>05372 <span class="comment">!  Reference:</span>
    5437. 

  5437. <a name="l05373"></a>05373 <span class="comment">!</span>
    5438. 

  5438. <a name="l05374"></a>05374 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    5439. 

  5439. <a name="l05375"></a>05375 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    5440. 

  5440. <a name="l05376"></a>05376 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    5441. 

  5441. <a name="l05377"></a>05377 <span class="comment">!    Springer Verlag, 1983</span>
    5442. 

  5442. <a name="l05378"></a>05378 <span class="comment">!</span>
    5443. 

  5443. <a name="l05379"></a>05379 <span class="comment">!  Parameters:</span>
    5444. 

  5444. <a name="l05380"></a>05380 <span class="comment">!</span>
    5445. 

  5445. <a name="l05381"></a>05381 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    5446. 

  5446. <a name="l05382"></a>05382 <span class="comment">!      function f ( x )</span>
    5447. 

  5447. <a name="l05383"></a>05383 <span class="comment">!      real f</span>
    5448. 

  5448. <a name="l05384"></a>05384 <span class="comment">!      real x</span>
    5449. 

  5449. <a name="l05385"></a>05385 <span class="comment">!    which evaluates the integrand function.</span>
    5450. 

  5450. <a name="l05386"></a>05386 <span class="comment">!</span>
    5451. 

  5451. <a name="l05387"></a>05387 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    5452. 

  5452. <a name="l05388"></a>05388 <span class="comment">!</span>
    5453. 

  5453. <a name="l05389"></a>05389 <span class="comment">!    Input, real OMEGA, the multiplier of X in the weight function.</span>
    5454. 

  5454. <a name="l05390"></a>05390 <span class="comment">!</span>
    5455. 

  5455. <a name="l05391"></a>05391 <span class="comment">!    Input, integer INTEGR, indicates the weight functions to be used.</span>
    5456. 

  5456. <a name="l05392"></a>05392 <span class="comment">!    = 1, w(x) = cos(omega*x)</span>
    5457. 

  5457. <a name="l05393"></a>05393 <span class="comment">!    = 2, w(x) = sin(omega*x)</span>
    5458. 

  5458. <a name="l05394"></a>05394 <span class="comment">!</span>
    5459. 

  5459. <a name="l05395"></a>05395 <span class="comment">!    Input, real EPSABS, EPSREL, the absolute and relative accuracy requested.</span>
    5460. 

  5460. <a name="l05396"></a>05396 <span class="comment">!</span>
    5461. 

  5461. <a name="l05397"></a>05397 <span class="comment">!    Input, integer LIMIT, the maximum number of subintervals of [A,B]</span>
    5462. 

  5462. <a name="l05398"></a>05398 <span class="comment">!    that can be generated.</span>
    5463. 

  5463. <a name="l05399"></a>05399 <span class="comment">!</span>
    5464. 

  5464. <a name="l05400"></a>05400 <span class="comment">!    icall  - integer</span>
    5465. 

  5465. <a name="l05401"></a>05401 <span class="comment">!                     if qfour is to be used only once, ICALL must</span>
    5466. 

  5466. <a name="l05402"></a>05402 <span class="comment">!                     be set to 1.  assume that during this call, the</span>
    5467. 

  5467. <a name="l05403"></a>05403 <span class="comment">!                     Chebyshev moments (for clenshaw-curtis integration</span>
    5468. 

  5468. <a name="l05404"></a>05404 <span class="comment">!                     of degree 24) have been computed for intervals of</span>
    5469. 

  5469. <a name="l05405"></a>05405 <span class="comment">!                     lenghts (abs(b-a))*2**(-l), l=0,1,2,...momcom-1.</span>
    5470. 

  5470. <a name="l05406"></a>05406 <span class="comment">!                     the Chebyshev moments already computed can be</span>
    5471. 

  5471. <a name="l05407"></a>05407 <span class="comment">!                     re-used in subsequent calls, if qfour must be</span>
    5472. 

  5472. <a name="l05408"></a>05408 <span class="comment">!                     called twice or more times on intervals of the</span>
    5473. 

  5473. <a name="l05409"></a>05409 <span class="comment">!                     same length abs(b-a). from the second call on, one</span>
    5474. 

  5474. <a name="l05410"></a>05410 <span class="comment">!                     has to put then ICALL &gt; 1.</span>
    5475. 

  5475. <a name="l05411"></a>05411 <span class="comment">!                     if ICALL &lt; 1, the routine will end with ier = 6.</span>
    5476. 

  5476. <a name="l05412"></a>05412 <span class="comment">!</span>
    5477. 

  5477. <a name="l05413"></a>05413 <span class="comment">!            maxp1  - integer</span>
    5478. 

  5478. <a name="l05414"></a>05414 <span class="comment">!                     gives an upper bound on the number of</span>
    5479. 

  5479. <a name="l05415"></a>05415 <span class="comment">!                     Chebyshev moments which can be stored, i.e.</span>
    5480. 

  5480. <a name="l05416"></a>05416 <span class="comment">!                     for the intervals of lenghts abs(b-a)*2**(-l),</span>
    5481. 

  5481. <a name="l05417"></a>05417 <span class="comment">!                     l=0,1, ..., maxp1-2, maxp1 &gt;= 1.</span>
    5482. 

  5482. <a name="l05418"></a>05418 <span class="comment">!                     if maxp1 &lt; 1, the routine will end with ier = 6.</span>
    5483. 

  5483. <a name="l05419"></a>05419 <span class="comment">!                     increasing (decreasing) the value of maxp1</span>
    5484. 

  5484. <a name="l05420"></a>05420 <span class="comment">!                     decreases (increases) the computational time but</span>
    5485. 

  5485. <a name="l05421"></a>05421 <span class="comment">!                     increases (decreases) the required memory space.</span>
    5486. 

  5486. <a name="l05422"></a>05422 <span class="comment">!</span>
    5487. 

  5487. <a name="l05423"></a>05423 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    5488. 

  5488. <a name="l05424"></a>05424 <span class="comment">!</span>
    5489. 

  5489. <a name="l05425"></a>05425 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    5490. 

  5490. <a name="l05426"></a>05426 <span class="comment">!</span>
    5491. 

  5491. <a name="l05427"></a>05427 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    5492. 

  5492. <a name="l05428"></a>05428 <span class="comment">!</span>
    5493. 

  5493. <a name="l05429"></a>05429 <span class="comment">!            ier    - integer</span>
    5494. 

  5494. <a name="l05430"></a>05430 <span class="comment">!                     ier = 0 normal and reliable termination of the</span>
    5495. 

  5495. <a name="l05431"></a>05431 <span class="comment">!                             routine. it is assumed that the</span>
    5496. 

  5496. <a name="l05432"></a>05432 <span class="comment">!                             requested accuracy has been achieved.</span>
    5497. 

  5497. <a name="l05433"></a>05433 <span class="comment">!                   - ier &gt; 0 abnormal termination of the routine.</span>
    5498. 

  5498. <a name="l05434"></a>05434 <span class="comment">!                             the estimates for integral and error are</span>
    5499. 

  5499. <a name="l05435"></a>05435 <span class="comment">!                             less reliable. it is assumed that the</span>
    5500. 

  5500. <a name="l05436"></a>05436 <span class="comment">!                             requested accuracy has not been achieved.</span>
    5501. 

  5501. <a name="l05437"></a>05437 <span class="comment">!                     ier = 1 maximum number of subdivisions allowed</span>
    5502. 

  5502. <a name="l05438"></a>05438 <span class="comment">!                             has been achieved. one can allow more</span>
    5503. 

  5503. <a name="l05439"></a>05439 <span class="comment">!                             subdivisions by increasing the value of</span>
    5504. 

  5504. <a name="l05440"></a>05440 <span class="comment">!                             limit (and taking according dimension</span>
    5505. 

  5505. <a name="l05441"></a>05441 <span class="comment">!                             adjustments into account). however, if</span>
    5506. 

  5506. <a name="l05442"></a>05442 <span class="comment">!                             this yields no improvement it is advised</span>
    5507. 

  5507. <a name="l05443"></a>05443 <span class="comment">!                             to analyze the integrand, in order to</span>
    5508. 

  5508. <a name="l05444"></a>05444 <span class="comment">!                             determine the integration difficulties.</span>
    5509. 

  5509. <a name="l05445"></a>05445 <span class="comment">!                             if the position of a local difficulty can</span>
    5510. 

  5510. <a name="l05446"></a>05446 <span class="comment">!                             be determined (e.g. singularity,</span>
    5511. 

  5511. <a name="l05447"></a>05447 <span class="comment">!                             discontinuity within the interval) one</span>
    5512. 

  5512. <a name="l05448"></a>05448 <span class="comment">!                             will probably gain from splitting up the</span>
    5513. 

  5513. <a name="l05449"></a>05449 <span class="comment">!                             interval at this point and calling the</span>
    5514. 

  5514. <a name="l05450"></a>05450 <span class="comment">!                             integrator on the subranges. if possible,</span>
    5515. 

  5515. <a name="l05451"></a>05451 <span class="comment">!                             an appropriate special-purpose integrator</span>
    5516. 

  5516. <a name="l05452"></a>05452 <span class="comment">!                             should be used which is designed for</span>
    5517. 

  5517. <a name="l05453"></a>05453 <span class="comment">!                             handling the type of difficulty involved.</span>
    5518. 

  5518. <a name="l05454"></a>05454 <span class="comment">!                         = 2 the occurrence of roundoff error is</span>
    5519. 

  5519. <a name="l05455"></a>05455 <span class="comment">!                             detected, which prevents the requested</span>
    5520. 

  5520. <a name="l05456"></a>05456 <span class="comment">!                             tolerance from being achieved.</span>
    5521. 

  5521. <a name="l05457"></a>05457 <span class="comment">!                             the error may be under-estimated.</span>
    5522. 

  5522. <a name="l05458"></a>05458 <span class="comment">!                         = 3 extremely bad integrand behavior occurs</span>
    5523. 

  5523. <a name="l05459"></a>05459 <span class="comment">!                             at some points of the integration</span>
    5524. 

  5524. <a name="l05460"></a>05460 <span class="comment">!                             interval.</span>
    5525. 

  5525. <a name="l05461"></a>05461 <span class="comment">!                         = 4 the algorithm does not converge. roundoff</span>
    5526. 

  5526. <a name="l05462"></a>05462 <span class="comment">!                             error is detected in the extrapolation</span>
    5527. 

  5527. <a name="l05463"></a>05463 <span class="comment">!                             table. it is presumed that the requested</span>
    5528. 

  5528. <a name="l05464"></a>05464 <span class="comment">!                             tolerance cannot be achieved due to</span>
    5529. 

  5529. <a name="l05465"></a>05465 <span class="comment">!                             roundoff in the extrapolation table, and</span>
    5530. 

  5530. <a name="l05466"></a>05466 <span class="comment">!                             that the returned result is the best which</span>
    5531. 

  5531. <a name="l05467"></a>05467 <span class="comment">!                             can be obtained.</span>
    5532. 

  5532. <a name="l05468"></a>05468 <span class="comment">!                         = 5 the integral is probably divergent, or</span>
    5533. 

  5533. <a name="l05469"></a>05469 <span class="comment">!                             slowly convergent. it must be noted that</span>
    5534. 

  5534. <a name="l05470"></a>05470 <span class="comment">!                             divergence can occur with any other value</span>
    5535. 

  5535. <a name="l05471"></a>05471 <span class="comment">!                             of ier &gt; 0.</span>
    5536. 

  5536. <a name="l05472"></a>05472 <span class="comment">!                         = 6 the input is invalid, because</span>
    5537. 

  5537. <a name="l05473"></a>05473 <span class="comment">!                             epsabs &lt; 0 and epsrel &lt; 0,</span>
    5538. 

  5538. <a name="l05474"></a>05474 <span class="comment">!                             or (integr /= 1 and integr /= 2) or</span>
    5539. 

  5539. <a name="l05475"></a>05475 <span class="comment">!                             ICALL &lt; 1 or maxp1 &lt; 1.</span>
    5540. 

  5540. <a name="l05476"></a>05476 <span class="comment">!                             result, abserr, neval, last, rlist(1),</span>
    5541. 

  5541. <a name="l05477"></a>05477 <span class="comment">!                             elist(1), iord(1) and nnlog(1) are set to</span>
    5542. 

  5542. <a name="l05478"></a>05478 <span class="comment">!                             zero. alist(1) and blist(1) are set to a</span>
    5543. 

  5543. <a name="l05479"></a>05479 <span class="comment">!                             and b respectively.</span>
    5544. 

  5544. <a name="l05480"></a>05480 <span class="comment">!</span>
    5545. 

  5545. <a name="l05481"></a>05481 <span class="comment">!    Workspace, real ALIST(LIMIT), BLIST(LIMIT), contains in entries 1 </span>
    5546. 

  5546. <a name="l05482"></a>05482 <span class="comment">!    through LAST the left and right ends of the partition subintervals.</span>
    5547. 

  5547. <a name="l05483"></a>05483 <span class="comment">!</span>
    5548. 

  5548. <a name="l05484"></a>05484 <span class="comment">!    Workspace, real RLIST(LIMIT), contains in entries 1 through LAST</span>
    5549. 

  5549. <a name="l05485"></a>05485 <span class="comment">!    the integral approximations on the subintervals.</span>
    5550. 

  5550. <a name="l05486"></a>05486 <span class="comment">!</span>
    5551. 

  5551. <a name="l05487"></a>05487 <span class="comment">!    Workspace, real ELIST(LIMIT), contains in entries 1 through LAST</span>
    5552. 

  5552. <a name="l05488"></a>05488 <span class="comment">!    the absolute error estimates on the subintervals.</span>
    5553. 

  5553. <a name="l05489"></a>05489 <span class="comment">!</span>
    5554. 

  5554. <a name="l05490"></a>05490 <span class="comment">!            iord   - integer</span>
    5555. 

  5555. <a name="l05491"></a>05491 <span class="comment">!                     vector of dimension at least limit, the first k</span>
    5556. 

  5556. <a name="l05492"></a>05492 <span class="comment">!                     elements of which are pointers to the error</span>
    5557. 

  5557. <a name="l05493"></a>05493 <span class="comment">!                     estimates over the subintervals, such that</span>
    5558. 

  5558. <a name="l05494"></a>05494 <span class="comment">!                     elist(iord(1)), ..., elist(iord(k)), form</span>
    5559. 

  5559. <a name="l05495"></a>05495 <span class="comment">!                     a decreasing sequence, with k = last</span>
    5560. 

  5560. <a name="l05496"></a>05496 <span class="comment">!                     if last &lt;= (limit/2+2), and</span>
    5561. 

  5561. <a name="l05497"></a>05497 <span class="comment">!                     k = limit+1-last otherwise.</span>
    5562. 

  5562. <a name="l05498"></a>05498 <span class="comment">!</span>
    5563. 

  5563. <a name="l05499"></a>05499 <span class="comment">!            nnlog  - integer</span>
    5564. 

  5564. <a name="l05500"></a>05500 <span class="comment">!                     vector of dimension at least limit, indicating the</span>
    5565. 

  5565. <a name="l05501"></a>05501 <span class="comment">!                     subdivision levels of the subintervals, i.e.</span>
    5566. 

  5566. <a name="l05502"></a>05502 <span class="comment">!                     iwork(i) = l means that the subinterval numbered</span>
    5567. 

  5567. <a name="l05503"></a>05503 <span class="comment">!                     i is of length abs(b-a)*2**(1-l)</span>
    5568. 

  5568. <a name="l05504"></a>05504 <span class="comment">!</span>
    5569. 

  5569. <a name="l05505"></a>05505 <span class="comment">!         on entry and return</span>
    5570. 

  5570. <a name="l05506"></a>05506 <span class="comment">!            momcom - integer</span>
    5571. 

  5571. <a name="l05507"></a>05507 <span class="comment">!                     indicating that the Chebyshev moments have been</span>
    5572. 

  5572. <a name="l05508"></a>05508 <span class="comment">!                     computed for intervals of lengths</span>
    5573. 

  5573. <a name="l05509"></a>05509 <span class="comment">!                     (abs(b-a))*2**(-l), l=0,1,2, ..., momcom-1,</span>
    5574. 

  5574. <a name="l05510"></a>05510 <span class="comment">!                     momcom &lt; maxp1</span>
    5575. 

  5575. <a name="l05511"></a>05511 <span class="comment">!</span>
    5576. 

  5576. <a name="l05512"></a>05512 <span class="comment">!            chebmo - real</span>
    5577. 

  5577. <a name="l05513"></a>05513 <span class="comment">!                     array of dimension (maxp1,25) containing the</span>
    5578. 

  5578. <a name="l05514"></a>05514 <span class="comment">!                     Chebyshev moments</span>
    5579. 

  5579. <a name="l05515"></a>05515 <span class="comment">!</span>
    5580. 

  5580. <a name="l05516"></a>05516 <span class="comment">!  Local Parameters:</span>
    5581. 

  5581. <a name="l05517"></a>05517 <span class="comment">!</span>
    5582. 

  5582. <a name="l05518"></a>05518 <span class="comment">!           alist     - list of left end points of all subintervals</span>
    5583. 

  5583. <a name="l05519"></a>05519 <span class="comment">!                       considered up to now</span>
    5584. 

  5584. <a name="l05520"></a>05520 <span class="comment">!           blist     - list of right end points of all subintervals</span>
    5585. 

  5585. <a name="l05521"></a>05521 <span class="comment">!                       considered up to now</span>
    5586. 

  5586. <a name="l05522"></a>05522 <span class="comment">!           rlist(i)  - approximation to the integral over</span>
    5587. 

  5587. <a name="l05523"></a>05523 <span class="comment">!                       (alist(i),blist(i))</span>
    5588. 

  5588. <a name="l05524"></a>05524 <span class="comment">!           rlist2    - array of dimension at least limexp+2 containing</span>
    5589. 

  5589. <a name="l05525"></a>05525 <span class="comment">!                       the part of the epsilon table which is still</span>
    5590. 

  5590. <a name="l05526"></a>05526 <span class="comment">!                       needed for further computations</span>
    5591. 

  5591. <a name="l05527"></a>05527 <span class="comment">!           elist(i)  - error estimate applying to rlist(i)</span>
    5592. 

  5592. <a name="l05528"></a>05528 <span class="comment">!           maxerr    - pointer to the interval with largest error</span>
    5593. 

  5593. <a name="l05529"></a>05529 <span class="comment">!                       estimate</span>
    5594. 

  5594. <a name="l05530"></a>05530 <span class="comment">!           errmax    - elist(maxerr)</span>
    5595. 

  5595. <a name="l05531"></a>05531 <span class="comment">!           erlast    - error on the interval currently subdivided</span>
    5596. 

  5596. <a name="l05532"></a>05532 <span class="comment">!           area      - sum of the integrals over the subintervals</span>
    5597. 

  5597. <a name="l05533"></a>05533 <span class="comment">!           errsum    - sum of the errors over the subintervals</span>
    5598. 

  5598. <a name="l05534"></a>05534 <span class="comment">!           errbnd    - requested accuracy max(epsabs,epsrel*</span>
    5599. 

  5599. <a name="l05535"></a>05535 <span class="comment">!                       abs(result))</span>
    5600. 

  5600. <a name="l05536"></a>05536 <span class="comment">!           *****1    - variable for the left subinterval</span>
    5601. 

  5601. <a name="l05537"></a>05537 <span class="comment">!           *****2    - variable for the right subinterval</span>
    5602. 

  5602. <a name="l05538"></a>05538 <span class="comment">!           last      - index for subdivision</span>
    5603. 

  5603. <a name="l05539"></a>05539 <span class="comment">!           nres      - number of calls to the extrapolation routine</span>
    5604. 

  5604. <a name="l05540"></a>05540 <span class="comment">!           numrl2    - number of elements in rlist2. if an appropriate</span>
    5605. 

  5605. <a name="l05541"></a>05541 <span class="comment">!                       approximation to the compounded integral has</span>
    5606. 

  5606. <a name="l05542"></a>05542 <span class="comment">!                       been obtained it is put in rlist2(numrl2) after</span>
    5607. 

  5607. <a name="l05543"></a>05543 <span class="comment">!                       numrl2 has been increased by one</span>
    5608. 

  5608. <a name="l05544"></a>05544 <span class="comment">!           small     - length of the smallest interval considered</span>
    5609. 

  5609. <a name="l05545"></a>05545 <span class="comment">!                       up to now, multiplied by 1.5</span>
    5610. 

  5610. <a name="l05546"></a>05546 <span class="comment">!           erlarg    - sum of the errors over the intervals larger</span>
    5611. 

  5611. <a name="l05547"></a>05547 <span class="comment">!                       than the smallest interval considered up to now</span>
    5612. 

  5612. <a name="l05548"></a>05548 <span class="comment">!           extrap    - logical variable denoting that the routine is</span>
    5613. 

  5613. <a name="l05549"></a>05549 <span class="comment">!                       attempting to perform extrapolation, i.e. before</span>
    5614. 

  5614. <a name="l05550"></a>05550 <span class="comment">!                       subdividing the smallest interval we try to</span>
    5615. 

  5615. <a name="l05551"></a>05551 <span class="comment">!                       decrease the value of erlarg</span>
    5616. 

  5616. <a name="l05552"></a>05552 <span class="comment">!           noext     - logical variable denoting that extrapolation</span>
    5617. 

  5617. <a name="l05553"></a>05553 <span class="comment">!                       is no longer allowed (true value)</span>
    5618. 

  5618. <a name="l05554"></a>05554 <span class="comment">!</span>
    5619. 

  5619. <a name="l05555"></a>05555   <span class="keyword">implicit none</span>
    5620. 

  5620. <a name="l05556"></a>05556 
    5621. 

  5621. <a name="l05557"></a>05557   <span class="keywordtype">integer</span> limit
    5622. 

  5622. <a name="l05558"></a>05558   <span class="keywordtype">integer</span> maxp1
    5623. 

  5623. <a name="l05559"></a>05559 
    5624. 

  5624. <a name="l05560"></a>05560   <span class="keywordtype">real</span> a
    5625. 

  5625. <a name="l05561"></a>05561   <span class="keywordtype">real</span> abseps
    5626. 

  5626. <a name="l05562"></a>05562   <span class="keywordtype">real</span> abserr
    5627. 

  5627. <a name="l05563"></a>05563   <span class="keywordtype">real</span> alist(limit)
    5628. 

  5628. <a name="l05564"></a>05564   <span class="keywordtype">real</span> area
    5629. 

  5629. <a name="l05565"></a>05565   <span class="keywordtype">real</span> area1
    5630. 

  5630. <a name="l05566"></a>05566   <span class="keywordtype">real</span> area12
    5631. 

  5631. <a name="l05567"></a>05567   <span class="keywordtype">real</span> area2
    5632. 

  5632. <a name="l05568"></a>05568   <span class="keywordtype">real</span> a1
    5633. 

  5633. <a name="l05569"></a>05569   <span class="keywordtype">real</span> a2
    5634. 

  5634. <a name="l05570"></a>05570   <span class="keywordtype">real</span> b
    5635. 

  5635. <a name="l05571"></a>05571   <span class="keywordtype">real</span> blist(limit)
    5636. 

  5636. <a name="l05572"></a>05572   <span class="keywordtype">real</span> b1
    5637. 

  5637. <a name="l05573"></a>05573   <span class="keywordtype">real</span> b2
    5638. 

  5638. <a name="l05574"></a>05574   <span class="keywordtype">real</span> chebmo(maxp1,25)
    5639. 

  5639. <a name="l05575"></a>05575   <span class="keywordtype">real</span> correc
    5640. 

  5640. <a name="l05576"></a>05576   <span class="keywordtype">real</span> defab1
    5641. 

  5641. <a name="l05577"></a>05577   <span class="keywordtype">real</span> defab2
    5642. 

  5642. <a name="l05578"></a>05578   <span class="keywordtype">real</span> defabs
    5643. 

  5643. <a name="l05579"></a>05579   <span class="keywordtype">real</span> domega
    5644. 

  5644. <a name="l05580"></a>05580   <span class="keywordtype">real</span> dres
    5645. 

  5645. <a name="l05581"></a>05581   <span class="keywordtype">real</span> elist(limit)
    5646. 

  5646. <a name="l05582"></a>05582   <span class="keywordtype">real</span> epsabs
    5647. 

  5647. <a name="l05583"></a>05583   <span class="keywordtype">real</span> epsrel
    5648. 

  5648. <a name="l05584"></a>05584   <span class="keywordtype">real</span> erlarg
    5649. 

  5649. <a name="l05585"></a>05585   <span class="keywordtype">real</span> erlast
    5650. 

  5650. <a name="l05586"></a>05586   <span class="keywordtype">real</span> errbnd
    5651. 

  5651. <a name="l05587"></a>05587   <span class="keywordtype">real</span> errmax
    5652. 

  5652. <a name="l05588"></a>05588   <span class="keywordtype">real</span> error1
    5653. 

  5653. <a name="l05589"></a>05589   <span class="keywordtype">real</span> erro12
    5654. 

  5654. <a name="l05590"></a>05590   <span class="keywordtype">real</span> error2
    5655. 

  5655. <a name="l05591"></a>05591   <span class="keywordtype">real</span> errsum
    5656. 

  5656. <a name="l05592"></a>05592   <span class="keywordtype">real</span> ertest
    5657. 

  5657. <a name="l05593"></a>05593   <span class="keywordtype">logical</span> extall
    5658. 

  5658. <a name="l05594"></a>05594   <span class="keywordtype">logical</span> extrap
    5659. 

  5659. <a name="l05595"></a>05595   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    5660. 

  5660. <a name="l05596"></a>05596   <span class="keywordtype">integer</span> icall
    5661. 

  5661. <a name="l05597"></a>05597   <span class="keywordtype">integer</span> id
    5662. 

  5662. <a name="l05598"></a>05598   <span class="keywordtype">integer</span> ier
    5663. 

  5663. <a name="l05599"></a>05599   <span class="keywordtype">integer</span> ierro
    5664. 

  5664. <a name="l05600"></a>05600   <span class="keywordtype">integer</span> integr
    5665. 

  5665. <a name="l05601"></a>05601   <span class="keywordtype">integer</span> iord(limit)
    5666. 

  5666. <a name="l05602"></a>05602   <span class="keywordtype">integer</span> iroff1
    5667. 

  5667. <a name="l05603"></a>05603   <span class="keywordtype">integer</span> iroff2
    5668. 

  5668. <a name="l05604"></a>05604   <span class="keywordtype">integer</span> iroff3
    5669. 

  5669. <a name="l05605"></a>05605   <span class="keywordtype">integer</span> jupbnd
    5670. 

  5670. <a name="l05606"></a>05606   <span class="keywordtype">integer</span> k
    5671. 

  5671. <a name="l05607"></a>05607   <span class="keywordtype">integer</span> ksgn
    5672. 

  5672. <a name="l05608"></a>05608   <span class="keywordtype">integer</span> ktmin
    5673. 

  5673. <a name="l05609"></a>05609   <span class="keywordtype">integer</span> last
    5674. 

  5674. <a name="l05610"></a>05610   <span class="keywordtype">integer</span> maxerr
    5675. 

  5675. <a name="l05611"></a>05611   <span class="keywordtype">integer</span> momcom
    5676. 

  5676. <a name="l05612"></a>05612   <span class="keywordtype">integer</span> nev
    5677. 

  5677. <a name="l05613"></a>05613   <span class="keywordtype">integer</span> neval
    5678. 

  5678. <a name="l05614"></a>05614   <span class="keywordtype">integer</span> nnlog(limit)
    5679. 

  5679. <a name="l05615"></a>05615   <span class="keywordtype">logical</span> noext
    5680. 

  5680. <a name="l05616"></a>05616   <span class="keywordtype">integer</span> nres
    5681. 

  5681. <a name="l05617"></a>05617   <span class="keywordtype">integer</span> nrmax
    5682. 

  5682. <a name="l05618"></a>05618   <span class="keywordtype">integer</span> nrmom
    5683. 

  5683. <a name="l05619"></a>05619   <span class="keywordtype">integer</span> numrl2
    5684. 

  5684. <a name="l05620"></a>05620   <span class="keywordtype">real</span> omega
    5685. 

  5685. <a name="l05621"></a>05621   <span class="keywordtype">real</span> resabs
    5686. 

  5686. <a name="l05622"></a>05622   <span class="keywordtype">real</span> reseps
    5687. 

  5687. <a name="l05623"></a>05623   <span class="keywordtype">real</span> result
    5688. 

  5688. <a name="l05624"></a>05624   <span class="keywordtype">real</span> res3la(3)
    5689. 

  5689. <a name="l05625"></a>05625   <span class="keywordtype">real</span> rlist(limit)
    5690. 

  5690. <a name="l05626"></a>05626   <span class="keywordtype">real</span> rlist2(52)
    5691. 

  5691. <a name="l05627"></a>05627   <span class="keywordtype">real</span> small
    5692. 

  5692. <a name="l05628"></a>05628   <span class="keywordtype">real</span> width
    5693. 

  5693. <a name="l05629"></a>05629 <span class="comment">!</span>
    5694. 

  5694. <a name="l05630"></a>05630 <span class="comment">!  the dimension of rlist2 is determined by  the value of</span>
    5695. 

  5695. <a name="l05631"></a>05631 <span class="comment">!  limexp in QEXTR (rlist2 should be of dimension</span>
    5696. 

  5696. <a name="l05632"></a>05632 <span class="comment">!  (limexp+2) at least).</span>
    5697. 

  5697. <a name="l05633"></a>05633 <span class="comment">!</span>
    5698. 

  5698. <a name="l05634"></a>05634 <span class="comment">!  Test on validity of parameters.</span>
    5699. 

  5699. <a name="l05635"></a>05635 <span class="comment">!</span>
    5700. 

  5700. <a name="l05636"></a>05636   ier = 0
    5701. 

  5701. <a name="l05637"></a>05637   neval = 0
    5702. 

  5702. <a name="l05638"></a>05638   last = 0
    5703. 

  5703. <a name="l05639"></a>05639   result = 0.0e+00
    5704. 

  5704. <a name="l05640"></a>05640   abserr = 0.0e+00
    5705. 

  5705. <a name="l05641"></a>05641   alist(1) = a
    5706. 

  5706. <a name="l05642"></a>05642   blist(1) = b
    5707. 

  5707. <a name="l05643"></a>05643   rlist(1) = 0.0e+00
    5708. 

  5708. <a name="l05644"></a>05644   elist(1) = 0.0e+00
    5709. 

  5709. <a name="l05645"></a>05645   iord(1) = 0
    5710. 

  5710. <a name="l05646"></a>05646   nnlog(1) = 0
    5711. 

  5711. <a name="l05647"></a>05647 
    5712. 

  5712. <a name="l05648"></a>05648   <span class="keyword">if</span> ( (integr /= 1.and.integr /= 2) .or. (epsabs &lt; 0.0e+00.and. &amp;
    5713. 

  5713. <a name="l05649"></a>05649     epsrel &lt; 0.0e+00) .or. icall &lt; 1 .or. maxp1 &lt; 1 ) <span class="keyword">then</span>
    5714. 

  5714. <a name="l05650"></a>05650     ier = 6
    5715. 

  5715. <a name="l05651"></a>05651     return
    5716. 

  5716. <a name="l05652"></a>05652   <span class="keyword">end if</span>
    5717. 

  5717. <a name="l05653"></a>05653 <span class="comment">!</span>
    5718. 

  5718. <a name="l05654"></a>05654 <span class="comment">!  First approximation to the integral.</span>
    5719. 

  5719. <a name="l05655"></a>05655 <span class="comment">!</span>
    5720. 

  5720. <a name="l05656"></a>05656   domega = abs ( omega )
    5721. 

  5721. <a name="l05657"></a>05657   nrmom = 0
    5722. 

  5722. <a name="l05658"></a>05658 
    5723. 

  5723. <a name="l05659"></a>05659   <span class="keyword">if</span> ( icall &lt;= 1 ) <span class="keyword">then</span>
    5724. 

  5724. <a name="l05660"></a>05660     momcom = 0
    5725. 

  5725. <a name="l05661"></a>05661   <span class="keyword">end if</span>
    5726. 

  5726. <a name="l05662"></a>05662 
    5727. 

  5727. <a name="l05663"></a>05663   call <a class="code" href="quadpack_8f90.html#ab0843f4831942d2c9bf3430cb71aca06">qc25o </a>( f, a, b, domega, integr, nrmom, maxp1, 0, result, abserr, &amp;
    5728. 

  5728. <a name="l05664"></a>05664     neval, defabs, resabs, momcom, chebmo )
    5729. 

  5729. <a name="l05665"></a>05665 <span class="comment">!</span>
    5730. 

  5730. <a name="l05666"></a>05666 <span class="comment">!  Test on accuracy.</span>
    5731. 

  5731. <a name="l05667"></a>05667 <span class="comment">!</span>
    5732. 

  5732. <a name="l05668"></a>05668   dres = abs(result)
    5733. 

  5733. <a name="l05669"></a>05669   errbnd = max ( epsabs,epsrel*dres)
    5734. 

  5734. <a name="l05670"></a>05670   rlist(1) = result
    5735. 

  5735. <a name="l05671"></a>05671   elist(1) = abserr
    5736. 

  5736. <a name="l05672"></a>05672   iord(1) = 1
    5737. 

  5737. <a name="l05673"></a>05673   <span class="keyword">if</span> ( abserr &lt;= 1.0e+02* epsilon ( defabs ) *defabs .and. &amp;
    5738. 

  5738. <a name="l05674"></a>05674     abserr &gt; errbnd ) ier = 2
    5739. 

  5739. <a name="l05675"></a>05675 
    5740. 

  5740. <a name="l05676"></a>05676   <span class="keyword">if</span> ( limit == 1 ) <span class="keyword">then</span>
    5741. 

  5741. <a name="l05677"></a>05677     ier = 1
    5742. 

  5742. <a name="l05678"></a>05678   <span class="keyword">end if</span>
    5743. 

  5743. <a name="l05679"></a>05679 
    5744. 

  5744. <a name="l05680"></a>05680   <span class="keyword">if</span> ( ier /= 0 .or. abserr &lt;= errbnd ) <span class="keyword">then</span>
    5745. 

  5745. <a name="l05681"></a>05681     go to 200
    5746. 

  5746. <a name="l05682"></a>05682   <span class="keyword">end if</span>
    5747. 

  5747. <a name="l05683"></a>05683 <span class="comment">!</span>
    5748. 

  5748. <a name="l05684"></a>05684 <span class="comment">!  Initializations</span>
    5749. 

  5749. <a name="l05685"></a>05685 <span class="comment">!</span>
    5750. 

  5750. <a name="l05686"></a>05686   errmax = abserr
    5751. 

  5751. <a name="l05687"></a>05687   maxerr = 1
    5752. 

  5752. <a name="l05688"></a>05688   area = result
    5753. 

  5753. <a name="l05689"></a>05689   errsum = abserr
    5754. 

  5754. <a name="l05690"></a>05690   abserr = huge ( abserr )
    5755. 

  5755. <a name="l05691"></a>05691   nrmax = 1
    5756. 

  5756. <a name="l05692"></a>05692   extrap = .false.
    5757. 

  5757. <a name="l05693"></a>05693   noext = .false.
    5758. 

  5758. <a name="l05694"></a>05694   ierro = 0
    5759. 

  5759. <a name="l05695"></a>05695   iroff1 = 0
    5760. 

  5760. <a name="l05696"></a>05696   iroff2 = 0
    5761. 

  5761. <a name="l05697"></a>05697   iroff3 = 0
    5762. 

  5762. <a name="l05698"></a>05698   ktmin = 0
    5763. 

  5763. <a name="l05699"></a>05699   small = abs(b-a)*7.5e-01
    5764. 

  5764. <a name="l05700"></a>05700   nres = 0
    5765. 

  5765. <a name="l05701"></a>05701   numrl2 = 0
    5766. 

  5766. <a name="l05702"></a>05702   extall = .false.
    5767. 

  5767. <a name="l05703"></a>05703 
    5768. 

  5768. <a name="l05704"></a>05704   <span class="keyword">if</span> ( 5.0e-01*abs(b-a)*domega &lt;= 2.0e+00) <span class="keyword">then</span>
    5769. 

  5769. <a name="l05705"></a>05705     numrl2 = 1
    5770. 

  5770. <a name="l05706"></a>05706     extall = .true.
    5771. 

  5771. <a name="l05707"></a>05707     rlist2(1) = result
    5772. 

  5772. <a name="l05708"></a>05708   <span class="keyword">end if</span>
    5773. 

  5773. <a name="l05709"></a>05709 
    5774. 

  5774. <a name="l05710"></a>05710   <span class="keyword">if</span> ( 2.5e-01 * abs(b-a) * domega &lt;= 2.0e+00 ) <span class="keyword">then</span>
    5775. 

  5775. <a name="l05711"></a>05711     extall = .true.
    5776. 

  5776. <a name="l05712"></a>05712   <span class="keyword">end if</span>
    5777. 

  5777. <a name="l05713"></a>05713 
    5778. 

  5778. <a name="l05714"></a>05714   <span class="keyword">if</span> ( dres &gt;= (1.0e+00-5.0e+01* epsilon ( defabs ) )*defabs ) <span class="keyword">then</span>
    5779. 

  5779. <a name="l05715"></a>05715     ksgn = 1
    5780. 

  5780. <a name="l05716"></a>05716   <span class="keyword">else</span>
    5781. 

  5781. <a name="l05717"></a>05717     ksgn = -1
    5782. 

  5782. <a name="l05718"></a>05718   <span class="keyword">end if</span>
    5783. 

  5783. <a name="l05719"></a>05719 <span class="comment">!</span>
    5784. 

  5784. <a name="l05720"></a>05720 <span class="comment">!  main do-loop</span>
    5785. 

  5785. <a name="l05721"></a>05721 <span class="comment">!</span>
    5786. 

  5786. <a name="l05722"></a>05722   <span class="keyword">do</span> last = 2, limit
    5787. 

  5787. <a name="l05723"></a>05723 <span class="comment">!</span>
    5788. 

  5788. <a name="l05724"></a>05724 <span class="comment">!  Bisect the subinterval with the nrmax-th largest error estimate.</span>
    5789. 

  5789. <a name="l05725"></a>05725 <span class="comment">!</span>
    5790. 

  5790. <a name="l05726"></a>05726     nrmom = nnlog(maxerr)+1
    5791. 

  5791. <a name="l05727"></a>05727     a1 = alist(maxerr)
    5792. 

  5792. <a name="l05728"></a>05728     b1 = 5.0e-01*(alist(maxerr)+blist(maxerr))
    5793. 

  5793. <a name="l05729"></a>05729     a2 = b1
    5794. 

  5794. <a name="l05730"></a>05730     b2 = blist(maxerr)
    5795. 

  5795. <a name="l05731"></a>05731     erlast = errmax
    5796. 

  5796. <a name="l05732"></a>05732 
    5797. 

  5797. <a name="l05733"></a>05733     call <a class="code" href="quadpack_8f90.html#ab0843f4831942d2c9bf3430cb71aca06">qc25o </a>( f, a1, b1, domega, integr, nrmom, maxp1, 0, area1, &amp;
    5798. 

  5798. <a name="l05734"></a>05734       error1, nev, resabs, defab1, momcom, chebmo )
    5799. 

  5799. <a name="l05735"></a>05735 
    5800. 

  5800. <a name="l05736"></a>05736     neval = neval+nev
    5801. 

  5801. <a name="l05737"></a>05737 
    5802. 

  5802. <a name="l05738"></a>05738     call <a class="code" href="quadpack_8f90.html#ab0843f4831942d2c9bf3430cb71aca06">qc25o </a>( f, a2, b2, domega, integr, nrmom, maxp1, 1, area2, &amp;
    5803. 

  5803. <a name="l05739"></a>05739       error2, nev, resabs, defab2, momcom, chebmo )
    5804. 

  5804. <a name="l05740"></a>05740 
    5805. 

  5805. <a name="l05741"></a>05741     neval = neval+nev
    5806. 

  5806. <a name="l05742"></a>05742 <span class="comment">!</span>
    5807. 

  5807. <a name="l05743"></a>05743 <span class="comment">!  Improve previous approximations to integral and error and</span>
    5808. 

  5808. <a name="l05744"></a>05744 <span class="comment">!  test for accuracy.</span>
    5809. 

  5809. <a name="l05745"></a>05745 <span class="comment">!</span>
    5810. 

  5810. <a name="l05746"></a>05746     area12 = area1+area2
    5811. 

  5811. <a name="l05747"></a>05747     erro12 = error1+error2
    5812. 

  5812. <a name="l05748"></a>05748     errsum = errsum+erro12-errmax
    5813. 

  5813. <a name="l05749"></a>05749     area = area+area12-rlist(maxerr)
    5814. 

  5814. <a name="l05750"></a>05750     <span class="keyword">if</span> ( defab1 == error1 .or. defab2 == error2 ) go to 25
    5815. 

  5815. <a name="l05751"></a>05751     <span class="keyword">if</span> ( abs(rlist(maxerr)-area12) &gt; 1.0e-05*abs(area12) &amp;
    5816. 

  5816. <a name="l05752"></a>05752     .or. erro12 &lt; 9.9e-01*errmax ) go to 20
    5817. 

  5817. <a name="l05753"></a>05753     <span class="keyword">if</span> ( extrap ) iroff2 = iroff2+1
    5818. 

  5818. <a name="l05754"></a>05754 
    5819. 

  5819. <a name="l05755"></a>05755     <span class="keyword">if</span> ( .not.extrap ) <span class="keyword">then</span>
    5820. 

  5820. <a name="l05756"></a>05756       iroff1 = iroff1+1
    5821. 

  5821. <a name="l05757"></a>05757     <span class="keyword">end if</span>
    5822. 

  5822. <a name="l05758"></a>05758 
    5823. 

  5823. <a name="l05759"></a>05759 20  continue
    5824. 

  5824. <a name="l05760"></a>05760 
    5825. 

  5825. <a name="l05761"></a>05761     <span class="keyword">if</span> ( last &gt; 10.and.erro12 &gt; errmax ) iroff3 = iroff3+1
    5826. 

  5826. <a name="l05762"></a>05762 
    5827. 

  5827. <a name="l05763"></a>05763 25  continue
    5828. 

  5828. <a name="l05764"></a>05764 
    5829. 

  5829. <a name="l05765"></a>05765     rlist(maxerr) = area1
    5830. 

  5830. <a name="l05766"></a>05766     rlist(last) = area2
    5831. 

  5831. <a name="l05767"></a>05767     nnlog(maxerr) = nrmom
    5832. 

  5832. <a name="l05768"></a>05768     nnlog(last) = nrmom
    5833. 

  5833. <a name="l05769"></a>05769     errbnd = max ( epsabs,epsrel*abs(area))
    5834. 

  5834. <a name="l05770"></a>05770 <span class="comment">!</span>
    5835. 

  5835. <a name="l05771"></a>05771 <span class="comment">!  Test for roundoff error and eventually set error flag</span>
    5836. 

  5836. <a name="l05772"></a>05772 <span class="comment">!</span>
    5837. 

  5837. <a name="l05773"></a>05773     <span class="keyword">if</span> ( iroff1+iroff2 &gt;= 10 .or. iroff3 &gt;= 20 ) ier = 2
    5838. 

  5838. <a name="l05774"></a>05774 
    5839. 

  5839. <a name="l05775"></a>05775     <span class="keyword">if</span> ( iroff2 &gt;= 5) ierro = 3
    5840. 

  5840. <a name="l05776"></a>05776 <span class="comment">!</span>
    5841. 

  5841. <a name="l05777"></a>05777 <span class="comment">!  Set error flag in the case that the number of subintervals</span>
    5842. 

  5842. <a name="l05778"></a>05778 <span class="comment">!  equals limit.</span>
    5843. 

  5843. <a name="l05779"></a>05779 <span class="comment">!</span>
    5844. 

  5844. <a name="l05780"></a>05780     <span class="keyword">if</span> ( last == limit ) <span class="keyword">then</span>
    5845. 

  5845. <a name="l05781"></a>05781       ier = 1
    5846. 

  5846. <a name="l05782"></a>05782     <span class="keyword">end if</span>
    5847. 

  5847. <a name="l05783"></a>05783 <span class="comment">!</span>
    5848. 

  5848. <a name="l05784"></a>05784 <span class="comment">!  Set error flag in the case of bad integrand behavior at</span>
    5849. 

  5849. <a name="l05785"></a>05785 <span class="comment">!  a point of the integration range.</span>
    5850. 

  5850. <a name="l05786"></a>05786 <span class="comment">!</span>
    5851. 

  5851. <a name="l05787"></a>05787     <span class="keyword">if</span> ( max ( abs(a1),abs(b2)) &lt;= (1.0e+00+1.0e+03* epsilon ( a1 ) ) &amp;
    5852. 

  5852. <a name="l05788"></a>05788     *(abs(a2)+1.0e+03* tiny ( a2 ) )) <span class="keyword">then</span>
    5853. 

  5853. <a name="l05789"></a>05789       ier = 4
    5854. 

  5854. <a name="l05790"></a>05790     <span class="keyword">end if</span>
    5855. 

  5855. <a name="l05791"></a>05791 <span class="comment">!</span>
    5856. 

  5856. <a name="l05792"></a>05792 <span class="comment">!  Append the newly-created intervals to the list.</span>
    5857. 

  5857. <a name="l05793"></a>05793 <span class="comment">!</span>
    5858. 

  5858. <a name="l05794"></a>05794     <span class="keyword">if</span> ( error2 &lt;= error1 ) <span class="keyword">then</span>
    5859. 

  5859. <a name="l05795"></a>05795       alist(last) = a2
    5860. 

  5860. <a name="l05796"></a>05796       blist(maxerr) = b1
    5861. 

  5861. <a name="l05797"></a>05797       blist(last) = b2
    5862. 

  5862. <a name="l05798"></a>05798       elist(maxerr) = error1
    5863. 

  5863. <a name="l05799"></a>05799       elist(last) = error2
    5864. 

  5864. <a name="l05800"></a>05800     <span class="keyword">else</span>
    5865. 

  5865. <a name="l05801"></a>05801       alist(maxerr) = a2
    5866. 

  5866. <a name="l05802"></a>05802       alist(last) = a1
    5867. 

  5867. <a name="l05803"></a>05803       blist(last) = b1
    5868. 

  5868. <a name="l05804"></a>05804       rlist(maxerr) = area2
    5869. 

  5869. <a name="l05805"></a>05805       rlist(last) = area1
    5870. 

  5870. <a name="l05806"></a>05806       elist(maxerr) = error2
    5871. 

  5871. <a name="l05807"></a>05807       elist(last) = error1
    5872. 

  5872. <a name="l05808"></a>05808     <span class="keyword">end if</span>
    5873. 

  5873. <a name="l05809"></a>05809 <span class="comment">!</span>
    5874. 

  5874. <a name="l05810"></a>05810 <span class="comment">!  Call QSORT to maintain the descending ordering</span>
    5875. 

  5875. <a name="l05811"></a>05811 <span class="comment">!  in the list of error estimates and select the subinterval</span>
    5876. 

  5876. <a name="l05812"></a>05812 <span class="comment">!  with nrmax-th largest error estimate (to be bisected next).</span>
    5877. 

  5877. <a name="l05813"></a>05813 <span class="comment">!</span>
    5878. 

  5878. <a name="l05814"></a>05814 
    5879. 

  5879. <a name="l05815"></a>05815     call <a class="code" href="quadpack_8f90.html#a55e08a684c5a6315fb37dd0fdc66d8e6">qsort </a>( limit, last, maxerr, errmax, elist, iord, nrmax )
    5880. 

  5880. <a name="l05816"></a>05816 
    5881. 

  5881. <a name="l05817"></a>05817     <span class="keyword">if</span> ( errsum &lt;= errbnd ) <span class="keyword">then</span>
    5882. 

  5882. <a name="l05818"></a>05818       go to 170
    5883. 

  5883. <a name="l05819"></a>05819     <span class="keyword">end if</span>
    5884. 

  5884. <a name="l05820"></a>05820 
    5885. 

  5885. <a name="l05821"></a>05821     <span class="keyword">if</span> ( ier /= 0 ) <span class="keyword">then</span>
    5886. 

  5886. <a name="l05822"></a>05822       exit
    5887. 

  5887. <a name="l05823"></a>05823     <span class="keyword">end if</span>
    5888. 

  5888. <a name="l05824"></a>05824 
    5889. 

  5889. <a name="l05825"></a>05825     <span class="keyword">if</span> ( last == 2 .and. extall ) go to 120
    5890. 

  5890. <a name="l05826"></a>05826 
    5891. 

  5891. <a name="l05827"></a>05827     <span class="keyword">if</span> ( noext ) <span class="keyword">then</span>
    5892. 

  5892. <a name="l05828"></a>05828       cycle
    5893. 

  5893. <a name="l05829"></a>05829     <span class="keyword">end if</span>
    5894. 

  5894. <a name="l05830"></a>05830 
    5895. 

  5895. <a name="l05831"></a>05831     <span class="keyword">if</span> ( .not. extall ) go to 50
    5896. 

  5896. <a name="l05832"></a>05832     erlarg = erlarg-erlast
    5897. 

  5897. <a name="l05833"></a>05833     <span class="keyword">if</span> ( abs(b1-a1) &gt; small ) erlarg = erlarg+erro12
    5898. 

  5898. <a name="l05834"></a>05834     <span class="keyword">if</span> ( extrap ) go to 70
    5899. 

  5899. <a name="l05835"></a>05835 <span class="comment">!</span>
    5900. 

  5900. <a name="l05836"></a>05836 <span class="comment">!  Test whether the interval to be bisected next is the</span>
    5901. 

  5901. <a name="l05837"></a>05837 <span class="comment">!  smallest interval.</span>
    5902. 

  5902. <a name="l05838"></a>05838 <span class="comment">!</span>
    5903. 

  5903. <a name="l05839"></a>05839 50  continue
    5904. 

  5904. <a name="l05840"></a>05840 
    5905. 

  5905. <a name="l05841"></a>05841     width = abs(blist(maxerr)-alist(maxerr))
    5906. 

  5906. <a name="l05842"></a>05842 
    5907. 

  5907. <a name="l05843"></a>05843     <span class="keyword">if</span> ( width &gt; small ) <span class="keyword">then</span>
    5908. 

  5908. <a name="l05844"></a>05844       cycle
    5909. 

  5909. <a name="l05845"></a>05845     <span class="keyword">end if</span>
    5910. 

  5910. <a name="l05846"></a>05846 
    5911. 

  5911. <a name="l05847"></a>05847     <span class="keyword">if</span> ( extall ) go to 60
    5912. 

  5912. <a name="l05848"></a>05848 <span class="comment">!</span>
    5913. 

  5913. <a name="l05849"></a>05849 <span class="comment">!  Test whether we can start with the extrapolation procedure</span>
    5914. 

  5914. <a name="l05850"></a>05850 <span class="comment">!  (we do this if we integrate over the next interval with</span>
    5915. 

  5915. <a name="l05851"></a>05851 <span class="comment">!  use of a Gauss-Kronrod rule - see QC25O).</span>
    5916. 

  5916. <a name="l05852"></a>05852 <span class="comment">!</span>
    5917. 

  5917. <a name="l05853"></a>05853     small = small*5.0e-01
    5918. 

  5918. <a name="l05854"></a>05854 
    5919. 

  5919. <a name="l05855"></a>05855     <span class="keyword">if</span> ( 2.5e-01*width*domega &gt; 2.0e+00 ) <span class="keyword">then</span>
    5920. 

  5920. <a name="l05856"></a>05856       cycle
    5921. 

  5921. <a name="l05857"></a>05857     <span class="keyword">end if</span>
    5922. 

  5922. <a name="l05858"></a>05858 
    5923. 

  5923. <a name="l05859"></a>05859     extall = .true.
    5924. 

  5924. <a name="l05860"></a>05860     go to 130
    5925. 

  5925. <a name="l05861"></a>05861 
    5926. 

  5926. <a name="l05862"></a>05862 60  continue
    5927. 

  5927. <a name="l05863"></a>05863 
    5928. 

  5928. <a name="l05864"></a>05864     extrap = .true.
    5929. 

  5929. <a name="l05865"></a>05865     nrmax = 2
    5930. 

  5930. <a name="l05866"></a>05866 
    5931. 

  5931. <a name="l05867"></a>05867 70  continue
    5932. 

  5932. <a name="l05868"></a>05868 
    5933. 

  5933. <a name="l05869"></a>05869     <span class="keyword">if</span> ( ierro == 3 .or. erlarg &lt;= ertest ) go to 90
    5934. 

  5934. <a name="l05870"></a>05870 <span class="comment">!</span>
    5935. 

  5935. <a name="l05871"></a>05871 <span class="comment">!  The smallest interval has the largest error.</span>
    5936. 

  5936. <a name="l05872"></a>05872 <span class="comment">!  Before bisecting decrease the sum of the errors over the</span>
    5937. 

  5937. <a name="l05873"></a>05873 <span class="comment">!  larger intervals (ERLARG) and perform extrapolation.</span>
    5938. 

  5938. <a name="l05874"></a>05874 <span class="comment">!</span>
    5939. 

  5939. <a name="l05875"></a>05875     jupbnd = last
    5940. 

  5940. <a name="l05876"></a>05876 
    5941. 

  5941. <a name="l05877"></a>05877     <span class="keyword">if</span> ( last &gt; (limit/2+2) ) <span class="keyword">then</span>
    5942. 

  5942. <a name="l05878"></a>05878       jupbnd = limit+3-last
    5943. 

  5943. <a name="l05879"></a>05879     <span class="keyword">end if</span>
    5944. 

  5944. <a name="l05880"></a>05880 
    5945. 

  5945. <a name="l05881"></a>05881     id = nrmax
    5946. 

  5946. <a name="l05882"></a>05882 
    5947. 

  5947. <a name="l05883"></a>05883     <span class="keyword">do</span> k = id, jupbnd
    5948. 

  5948. <a name="l05884"></a>05884       maxerr = iord(nrmax)
    5949. 

  5949. <a name="l05885"></a>05885       errmax = elist(maxerr)
    5950. 

  5950. <a name="l05886"></a>05886       <span class="keyword">if</span> ( abs(blist(maxerr)-alist(maxerr)) &gt; small ) go to 140
    5951. 

  5951. <a name="l05887"></a>05887       nrmax = nrmax+1
    5952. 

  5952. <a name="l05888"></a>05888     <span class="keyword">end do</span>
    5953. 

  5953. <a name="l05889"></a>05889 <span class="comment">!</span>
    5954. 

  5954. <a name="l05890"></a>05890 <span class="comment">!  Perform extrapolation.</span>
    5955. 

  5955. <a name="l05891"></a>05891 <span class="comment">!</span>
    5956. 

  5956. <a name="l05892"></a>05892 90  continue
    5957. 

  5957. <a name="l05893"></a>05893 
    5958. 

  5958. <a name="l05894"></a>05894     numrl2 = numrl2+1
    5959. 

  5959. <a name="l05895"></a>05895     rlist2(numrl2) = area
    5960. 

  5960. <a name="l05896"></a>05896 
    5961. 

  5961. <a name="l05897"></a>05897     <span class="keyword">if</span> ( numrl2 &lt; 3 ) go to 110
    5962. 

  5962. <a name="l05898"></a>05898 
    5963. 

  5963. <a name="l05899"></a>05899     call <a class="code" href="quadpack_8f90.html#a5a75101d080f224c63adde98a0e64386">qextr </a>( numrl2, rlist2, reseps, abseps, res3la, nres )
    5964. 

  5964. <a name="l05900"></a>05900     ktmin = ktmin+1
    5965. 

  5965. <a name="l05901"></a>05901 
    5966. 

  5966. <a name="l05902"></a>05902     <span class="keyword">if</span> ( ktmin &gt; 5.and.abserr &lt; 1.0e-03*errsum ) <span class="keyword">then</span>
    5967. 

  5967. <a name="l05903"></a>05903       ier = 5
    5968. 

  5968. <a name="l05904"></a>05904     <span class="keyword">end if</span>
    5969. 

  5969. <a name="l05905"></a>05905 
    5970. 

  5970. <a name="l05906"></a>05906     <span class="keyword">if</span> ( abseps &gt;= abserr ) go to 100
    5971. 

  5971. <a name="l05907"></a>05907 
    5972. 

  5972. <a name="l05908"></a>05908     ktmin = 0
    5973. 

  5973. <a name="l05909"></a>05909     abserr = abseps
    5974. 

  5974. <a name="l05910"></a>05910     result = reseps
    5975. 

  5975. <a name="l05911"></a>05911     correc = erlarg
    5976. 

  5976. <a name="l05912"></a>05912     ertest = max ( epsabs, epsrel*abs(reseps))
    5977. 

  5977. <a name="l05913"></a>05913 
    5978. 

  5978. <a name="l05914"></a>05914     <span class="keyword">if</span> ( abserr &lt;= ertest ) <span class="keyword">then</span>
    5979. 

  5979. <a name="l05915"></a>05915       exit
    5980. 

  5980. <a name="l05916"></a>05916     <span class="keyword">end if</span>
    5981. 

  5981. <a name="l05917"></a>05917 <span class="comment">!</span>
    5982. 

  5982. <a name="l05918"></a>05918 <span class="comment">!  Prepare bisection of the smallest interval.</span>
    5983. 

  5983. <a name="l05919"></a>05919 <span class="comment">!</span>
    5984. 

  5984. <a name="l05920"></a>05920 100 continue
    5985. 

  5985. <a name="l05921"></a>05921 
    5986. 

  5986. <a name="l05922"></a>05922     <span class="keyword">if</span> ( numrl2 == 1 ) <span class="keyword">then</span>
    5987. 

  5987. <a name="l05923"></a>05923       noext = .true.
    5988. 

  5988. <a name="l05924"></a>05924     <span class="keyword">end if</span>
    5989. 

  5989. <a name="l05925"></a>05925 
    5990. 

  5990. <a name="l05926"></a>05926     <span class="keyword">if</span> ( ier == 5 ) <span class="keyword">then</span>
    5991. 

  5991. <a name="l05927"></a>05927       exit
    5992. 

  5992. <a name="l05928"></a>05928     <span class="keyword">end if</span>
    5993. 

  5993. <a name="l05929"></a>05929 
    5994. 

  5994. <a name="l05930"></a>05930 110 continue
    5995. 

  5995. <a name="l05931"></a>05931 
    5996. 

  5996. <a name="l05932"></a>05932     maxerr = iord(1)
    5997. 

  5997. <a name="l05933"></a>05933     errmax = elist(maxerr)
    5998. 

  5998. <a name="l05934"></a>05934     nrmax = 1
    5999. 

  5999. <a name="l05935"></a>05935     extrap = .false.
    6000. 

  6000. <a name="l05936"></a>05936     small = small*5.0e-01
    6001. 

  6001. <a name="l05937"></a>05937     erlarg = errsum
    6002. 

  6002. <a name="l05938"></a>05938     cycle
    6003. 

  6003. <a name="l05939"></a>05939 
    6004. 

  6004. <a name="l05940"></a>05940 120 continue
    6005. 

  6005. <a name="l05941"></a>05941 
    6006. 

  6006. <a name="l05942"></a>05942     small = small * 5.0e-01
    6007. 

  6007. <a name="l05943"></a>05943     numrl2 = numrl2 + 1
    6008. 

  6008. <a name="l05944"></a>05944     rlist2(numrl2) = area
    6009. 

  6009. <a name="l05945"></a>05945 
    6010. 

  6010. <a name="l05946"></a>05946 130 continue
    6011. 

  6011. <a name="l05947"></a>05947 
    6012. 

  6012. <a name="l05948"></a>05948     ertest = errbnd
    6013. 

  6013. <a name="l05949"></a>05949     erlarg = errsum
    6014. 

  6014. <a name="l05950"></a>05950 
    6015. 

  6015. <a name="l05951"></a>05951 140 continue
    6016. 

  6016. <a name="l05952"></a>05952 
    6017. 

  6017. <a name="l05953"></a>05953   <span class="keyword">end do</span>
    6018. 

  6018. <a name="l05954"></a>05954 <span class="comment">!</span>
    6019. 

  6019. <a name="l05955"></a>05955 <span class="comment">!  set the final result.</span>
    6020. 

  6020. <a name="l05956"></a>05956 <span class="comment">!</span>
    6021. 

  6021. <a name="l05957"></a>05957   <span class="keyword">if</span> ( abserr == huge ( abserr ) .or. nres == 0 ) <span class="keyword">then</span>
    6022. 

  6022. <a name="l05958"></a>05958     go to 170
    6023. 

  6023. <a name="l05959"></a>05959   <span class="keyword">end if</span>
    6024. 

  6024. <a name="l05960"></a>05960 
    6025. 

  6025. <a name="l05961"></a>05961   <span class="keyword">if</span> ( ier+ierro == 0 ) go to 165
    6026. 

  6026. <a name="l05962"></a>05962   <span class="keyword">if</span> ( ierro == 3 ) abserr = abserr+correc
    6027. 

  6027. <a name="l05963"></a>05963   <span class="keyword">if</span> ( ier == 0 ) ier = 3
    6028. 

  6028. <a name="l05964"></a>05964   <span class="keyword">if</span> ( result /= 0.0e+00.and.area /= 0.0e+00 ) go to 160
    6029. 

  6029. <a name="l05965"></a>05965   <span class="keyword">if</span> ( abserr &gt; errsum ) go to 170
    6030. 

  6030. <a name="l05966"></a>05966   <span class="keyword">if</span> ( area == 0.0e+00 ) go to 190
    6031. 

  6031. <a name="l05967"></a>05967   go to 165
    6032. 

  6032. <a name="l05968"></a>05968 
    6033. 

  6033. <a name="l05969"></a>05969 160 continue
    6034. 

  6034. <a name="l05970"></a>05970 
    6035. 

  6035. <a name="l05971"></a>05971   <span class="keyword">if</span> ( abserr/abs(result) &gt; errsum/abs(area) ) go to 170
    6036. 

  6036. <a name="l05972"></a>05972 <span class="comment">!</span>
    6037. 

  6037. <a name="l05973"></a>05973 <span class="comment">!  Test on divergence.</span>
    6038. 

  6038. <a name="l05974"></a>05974 <span class="comment">!</span>
    6039. 

  6039. <a name="l05975"></a>05975   165 continue
    6040. 

  6040. <a name="l05976"></a>05976 
    6041. 

  6041. <a name="l05977"></a>05977   <span class="keyword">if</span> ( ksgn == (-1) .and. max ( abs(result),abs(area)) &lt;=  &amp;
    6042. 

  6042. <a name="l05978"></a>05978    defabs*1.0e-02 ) go to 190
    6043. 

  6043. <a name="l05979"></a>05979 
    6044. 

  6044. <a name="l05980"></a>05980   <span class="keyword">if</span> ( 1.0e-02 &gt; (result/area) .or. (result/area) &gt; 1.0e+02 &amp;
    6045. 

  6045. <a name="l05981"></a>05981    .or. errsum &gt;= abs(area) ) ier = 6
    6046. 

  6046. <a name="l05982"></a>05982 
    6047. 

  6047. <a name="l05983"></a>05983   go to 190
    6048. 

  6048. <a name="l05984"></a>05984 <span class="comment">!</span>
    6049. 

  6049. <a name="l05985"></a>05985 <span class="comment">!  Compute global integral sum.</span>
    6050. 

  6050. <a name="l05986"></a>05986 <span class="comment">!</span>
    6051. 

  6051. <a name="l05987"></a>05987 170 continue
    6052. 

  6052. <a name="l05988"></a>05988 
    6053. 

  6053. <a name="l05989"></a>05989   result = sum ( rlist(1:last) )
    6054. 

  6054. <a name="l05990"></a>05990 
    6055. 

  6055. <a name="l05991"></a>05991   abserr = errsum
    6056. 

  6056. <a name="l05992"></a>05992 
    6057. 

  6057. <a name="l05993"></a>05993 190 continue
    6058. 

  6058. <a name="l05994"></a>05994 
    6059. 

  6059. <a name="l05995"></a>05995   <span class="keyword">if</span> (ier &gt; 2) ier=ier-1
    6060. 

  6060. <a name="l05996"></a>05996 
    6061. 

  6061. <a name="l05997"></a>05997 200 continue
    6062. 

  6062. <a name="l05998"></a>05998 
    6063. 

  6063. <a name="l05999"></a>05999   <span class="keyword">if</span> ( integr == 2 .and. omega &lt; 0.0e+00 ) <span class="keyword">then</span>
    6064. 

  6064. <a name="l06000"></a>06000     result = -result
    6065. 

  6065. <a name="l06001"></a>06001   <span class="keyword">end if</span>
    6066. 

  6066. <a name="l06002"></a>06002 
    6067. 

  6067. <a name="l06003"></a>06003   return
    6068. 

  6068. <a name="l06004"></a>06004 <span class="keyword">end</span>
    6069. 

  6069. <a name="l06005"></a><a class="code" href="quadpack_8f90.html#a1722ad5ba07cec52d38c9ebf9df80a2d">06005</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a1722ad5ba07cec52d38c9ebf9df80a2d">qk15</a> ( f, a, b, result, abserr, resabs, resasc )
    6070. 

  6070. <a name="l06006"></a>06006 
    6071. 

  6071. <a name="l06007"></a>06007 <span class="comment">!*****************************************************************************80</span>
    6072. 

  6072. <a name="l06008"></a>06008 <span class="comment">!</span>
    6073. 

  6073. <a name="l06009"></a>06009 <span class="comment">!! QK15 carries out a 15 point Gauss-Kronrod quadrature rule.</span>
    6074. 

  6074. <a name="l06010"></a>06010 <span class="comment">!</span>
    6075. 

  6075. <a name="l06011"></a>06011 <span class="comment">!  Discussion:</span>
    6076. 

  6076. <a name="l06012"></a>06012 <span class="comment">!</span>
    6077. 

  6077. <a name="l06013"></a>06013 <span class="comment">!    This routine approximates</span>
    6078. 

  6078. <a name="l06014"></a>06014 <span class="comment">!      I = integral ( A &lt;= X &lt;= B ) F(X) dx</span>
    6079. 

  6079. <a name="l06015"></a>06015 <span class="comment">!    with an error estimate, and</span>
    6080. 

  6080. <a name="l06016"></a>06016 <span class="comment">!      J = integral ( A &lt;= X &lt;= B ) | F(X) | dx</span>
    6081. 

  6081. <a name="l06017"></a>06017 <span class="comment">!</span>
    6082. 

  6082. <a name="l06018"></a>06018 <span class="comment">!  Author:</span>
    6083. 

  6083. <a name="l06019"></a>06019 <span class="comment">!</span>
    6084. 

  6084. <a name="l06020"></a>06020 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    6085. 

  6085. <a name="l06021"></a>06021 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    6086. 

  6086. <a name="l06022"></a>06022 <span class="comment">!</span>
    6087. 

  6087. <a name="l06023"></a>06023 <span class="comment">!  Reference:</span>
    6088. 

  6088. <a name="l06024"></a>06024 <span class="comment">!</span>
    6089. 

  6089. <a name="l06025"></a>06025 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    6090. 

  6090. <a name="l06026"></a>06026 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    6091. 

  6091. <a name="l06027"></a>06027 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    6092. 

  6092. <a name="l06028"></a>06028 <span class="comment">!    Springer Verlag, 1983</span>
    6093. 

  6093. <a name="l06029"></a>06029 <span class="comment">!</span>
    6094. 

  6094. <a name="l06030"></a>06030 <span class="comment">!  Parameters:</span>
    6095. 

  6095. <a name="l06031"></a>06031 <span class="comment">!</span>
    6096. 

  6096. <a name="l06032"></a>06032 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    6097. 

  6097. <a name="l06033"></a>06033 <span class="comment">!      function f ( x )</span>
    6098. 

  6098. <a name="l06034"></a>06034 <span class="comment">!      real f</span>
    6099. 

  6099. <a name="l06035"></a>06035 <span class="comment">!      real x</span>
    6100. 

  6100. <a name="l06036"></a>06036 <span class="comment">!    which evaluates the integrand function.</span>
    6101. 

  6101. <a name="l06037"></a>06037 <span class="comment">!</span>
    6102. 

  6102. <a name="l06038"></a>06038 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    6103. 

  6103. <a name="l06039"></a>06039 <span class="comment">!</span>
    6104. 

  6104. <a name="l06040"></a>06040 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    6105. 

  6105. <a name="l06041"></a>06041 <span class="comment">!    RESULT is computed by applying the 15-point Kronrod rule (RESK) </span>
    6106. 

  6106. <a name="l06042"></a>06042 <span class="comment">!    obtained by optimal addition of abscissae to the 7-point Gauss rule </span>
    6107. 

  6107. <a name="l06043"></a>06043 <span class="comment">!    (RESG).</span>
    6108. 

  6108. <a name="l06044"></a>06044 <span class="comment">!</span>
    6109. 

  6109. <a name="l06045"></a>06045 <span class="comment">!    Output, real ABSERR, an estimate of | I - RESULT |.</span>
    6110. 

  6110. <a name="l06046"></a>06046 <span class="comment">!</span>
    6111. 

  6111. <a name="l06047"></a>06047 <span class="comment">!    Output, real RESABS, approximation to the integral of the absolute</span>
    6112. 

  6112. <a name="l06048"></a>06048 <span class="comment">!    value of F.</span>
    6113. 

  6113. <a name="l06049"></a>06049 <span class="comment">!</span>
    6114. 

  6114. <a name="l06050"></a>06050 <span class="comment">!    Output, real RESASC, approximation to the integral | F-I/(B-A) | </span>
    6115. 

  6115. <a name="l06051"></a>06051 <span class="comment">!    over [A,B].</span>
    6116. 

  6116. <a name="l06052"></a>06052 <span class="comment">!</span>
    6117. 

  6117. <a name="l06053"></a>06053 <span class="comment">!  Local Parameters:</span>
    6118. 

  6118. <a name="l06054"></a>06054 <span class="comment">!</span>
    6119. 

  6119. <a name="l06055"></a>06055 <span class="comment">!           the abscissae and weights are given for the interval (-1,1).</span>
    6120. 

  6120. <a name="l06056"></a>06056 <span class="comment">!           because of symmetry only the positive abscissae and their</span>
    6121. 

  6121. <a name="l06057"></a>06057 <span class="comment">!           corresponding weights are given.</span>
    6122. 

  6122. <a name="l06058"></a>06058 <span class="comment">!</span>
    6123. 

  6123. <a name="l06059"></a>06059 <span class="comment">!           xgk    - abscissae of the 15-point Kronrod rule</span>
    6124. 

  6124. <a name="l06060"></a>06060 <span class="comment">!                    xgk(2), xgk(4), ...  abscissae of the 7-point</span>
    6125. 

  6125. <a name="l06061"></a>06061 <span class="comment">!                    Gauss rule</span>
    6126. 

  6126. <a name="l06062"></a>06062 <span class="comment">!                    xgk(1), xgk(3), ...  abscissae which are optimally</span>
    6127. 

  6127. <a name="l06063"></a>06063 <span class="comment">!                    added to the 7-point Gauss rule</span>
    6128. 

  6128. <a name="l06064"></a>06064 <span class="comment">!</span>
    6129. 

  6129. <a name="l06065"></a>06065 <span class="comment">!           wgk    - weights of the 15-point Kronrod rule</span>
    6130. 

  6130. <a name="l06066"></a>06066 <span class="comment">!</span>
    6131. 

  6131. <a name="l06067"></a>06067 <span class="comment">!           wg     - weights of the 7-point Gauss rule</span>
    6132. 

  6132. <a name="l06068"></a>06068 <span class="comment">!</span>
    6133. 

  6133. <a name="l06069"></a>06069 <span class="comment">!           centr  - mid point of the interval</span>
    6134. 

  6134. <a name="l06070"></a>06070 <span class="comment">!           hlgth  - half-length of the interval</span>
    6135. 

  6135. <a name="l06071"></a>06071 <span class="comment">!           absc   - abscissa</span>
    6136. 

  6136. <a name="l06072"></a>06072 <span class="comment">!           fval*  - function value</span>
    6137. 

  6137. <a name="l06073"></a>06073 <span class="comment">!           resg   - result of the 7-point Gauss formula</span>
    6138. 

  6138. <a name="l06074"></a>06074 <span class="comment">!           resk   - result of the 15-point Kronrod formula</span>
    6139. 

  6139. <a name="l06075"></a>06075 <span class="comment">!           reskh  - approximation to the mean value of f over (a,b),</span>
    6140. 

  6140. <a name="l06076"></a>06076 <span class="comment">!                    i.e. to i/(b-a)</span>
    6141. 

  6141. <a name="l06077"></a>06077 <span class="comment">!</span>
    6142. 

  6142. <a name="l06078"></a>06078   <span class="keyword">implicit none</span>
    6143. 

  6143. <a name="l06079"></a>06079 
    6144. 

  6144. <a name="l06080"></a>06080   <span class="keywordtype">real</span> a
    6145. 

  6145. <a name="l06081"></a>06081   <span class="keywordtype">real</span> absc
    6146. 

  6146. <a name="l06082"></a>06082   <span class="keywordtype">real</span> abserr
    6147. 

  6147. <a name="l06083"></a>06083   <span class="keywordtype">real</span> b
    6148. 

  6148. <a name="l06084"></a>06084   <span class="keywordtype">real</span> centr
    6149. 

  6149. <a name="l06085"></a>06085   <span class="keywordtype">real</span> dhlgth
    6150. 

  6150. <a name="l06086"></a>06086   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    6151. 

  6151. <a name="l06087"></a>06087   <span class="keywordtype">real</span> fc
    6152. 

  6152. <a name="l06088"></a>06088   <span class="keywordtype">real</span> fsum
    6153. 

  6153. <a name="l06089"></a>06089   <span class="keywordtype">real</span> fval1
    6154. 

  6154. <a name="l06090"></a>06090   <span class="keywordtype">real</span> fval2
    6155. 

  6155. <a name="l06091"></a>06091   <span class="keywordtype">real</span> fv1(7)
    6156. 

  6156. <a name="l06092"></a>06092   <span class="keywordtype">real</span> fv2(7)
    6157. 

  6157. <a name="l06093"></a>06093   <span class="keywordtype">real</span> hlgth
    6158. 

  6158. <a name="l06094"></a>06094   <span class="keywordtype">integer</span> j
    6159. 

  6159. <a name="l06095"></a>06095   <span class="keywordtype">integer</span> jtw
    6160. 

  6160. <a name="l06096"></a>06096   <span class="keywordtype">integer</span> jtwm1
    6161. 

  6161. <a name="l06097"></a>06097   <span class="keywordtype">real</span> resabs
    6162. 

  6162. <a name="l06098"></a>06098   <span class="keywordtype">real</span> resasc
    6163. 

  6163. <a name="l06099"></a>06099   <span class="keywordtype">real</span> resg
    6164. 

  6164. <a name="l06100"></a>06100   <span class="keywordtype">real</span> resk
    6165. 

  6165. <a name="l06101"></a>06101   <span class="keywordtype">real</span> reskh
    6166. 

  6166. <a name="l06102"></a>06102   <span class="keywordtype">real</span> result
    6167. 

  6167. <a name="l06103"></a>06103   <span class="keywordtype">real</span> wg(4)
    6168. 

  6168. <a name="l06104"></a>06104   <span class="keywordtype">real</span> wgk(8)
    6169. 

  6169. <a name="l06105"></a>06105   <span class="keywordtype">real</span> xgk(8)
    6170. 

  6170. <a name="l06106"></a>06106 
    6171. 

  6171. <a name="l06107"></a>06107   <span class="keyword">data</span> xgk(1),xgk(2),xgk(3),xgk(4),xgk(5),xgk(6),xgk(7),xgk(8)/ &amp;
    6172. 

  6172. <a name="l06108"></a>06108        9.914553711208126e-01,   9.491079123427585e-01, &amp;
    6173. 

  6173. <a name="l06109"></a>06109        8.648644233597691e-01,   7.415311855993944e-01, &amp;
    6174. 

  6174. <a name="l06110"></a>06110        5.860872354676911e-01,   4.058451513773972e-01, &amp;
    6175. 

  6175. <a name="l06111"></a>06111        2.077849550078985e-01,   0.0e+00              /
    6176. 

  6176. <a name="l06112"></a>06112   <span class="keyword">data</span> wgk(1),wgk(2),wgk(3),wgk(4),wgk(5),wgk(6),wgk(7),wgk(8)/ &amp;
    6177. 

  6177. <a name="l06113"></a>06113        2.293532201052922e-02,   6.309209262997855e-02, &amp;
    6178. 

  6178. <a name="l06114"></a>06114        1.047900103222502e-01,   1.406532597155259e-01, &amp;
    6179. 

  6179. <a name="l06115"></a>06115        1.690047266392679e-01,   1.903505780647854e-01, &amp;
    6180. 

  6180. <a name="l06116"></a>06116        2.044329400752989e-01,   2.094821410847278e-01/
    6181. 

  6181. <a name="l06117"></a>06117   <span class="keyword">data</span> wg(1),wg(2),wg(3),wg(4)/ &amp;
    6182. 

  6182. <a name="l06118"></a>06118        1.294849661688697e-01,   2.797053914892767e-01, &amp;
    6183. 

  6183. <a name="l06119"></a>06119        3.818300505051189e-01,   4.179591836734694e-01/
    6184. 

  6184. <a name="l06120"></a>06120 <span class="comment">!</span>
    6185. 

  6185. <a name="l06121"></a>06121   centr = 5.0e-01*(a+b)
    6186. 

  6186. <a name="l06122"></a>06122   hlgth = 5.0e-01*(b-a)
    6187. 

  6187. <a name="l06123"></a>06123   dhlgth = abs(hlgth)
    6188. 

  6188. <a name="l06124"></a>06124 <span class="comment">!</span>
    6189. 

  6189. <a name="l06125"></a>06125 <span class="comment">!  Compute the 15-point Kronrod approximation to the integral,</span>
    6190. 

  6190. <a name="l06126"></a>06126 <span class="comment">!  and estimate the absolute error.</span>
    6191. 

  6191. <a name="l06127"></a>06127 <span class="comment">!</span>
    6192. 

  6192. <a name="l06128"></a>06128   fc = f(centr)
    6193. 

  6193. <a name="l06129"></a>06129   resg = fc*wg(4)
    6194. 

  6194. <a name="l06130"></a>06130   resk = fc*wgk(8)
    6195. 

  6195. <a name="l06131"></a>06131   resabs = abs(resk)
    6196. 

  6196. <a name="l06132"></a>06132 
    6197. 

  6197. <a name="l06133"></a>06133   <span class="keyword">do</span> j = 1, 3
    6198. 

  6198. <a name="l06134"></a>06134     jtw = j*2
    6199. 

  6199. <a name="l06135"></a>06135     absc = hlgth*xgk(jtw)
    6200. 

  6200. <a name="l06136"></a>06136     fval1 = f(centr-absc)
    6201. 

  6201. <a name="l06137"></a>06137     fval2 = f(centr+absc)
    6202. 

  6202. <a name="l06138"></a>06138     fv1(jtw) = fval1
    6203. 

  6203. <a name="l06139"></a>06139     fv2(jtw) = fval2
    6204. 

  6204. <a name="l06140"></a>06140     fsum = fval1+fval2
    6205. 

  6205. <a name="l06141"></a>06141     resg = resg+wg(j)*fsum
    6206. 

  6206. <a name="l06142"></a>06142     resk = resk+wgk(jtw)*fsum
    6207. 

  6207. <a name="l06143"></a>06143     resabs = resabs+wgk(jtw)*(abs(fval1)+abs(fval2))
    6208. 

  6208. <a name="l06144"></a>06144   <span class="keyword">end do</span>
    6209. 

  6209. <a name="l06145"></a>06145 
    6210. 

  6210. <a name="l06146"></a>06146   <span class="keyword">do</span> j = 1, 4
    6211. 

  6211. <a name="l06147"></a>06147     jtwm1 = j*2-1
    6212. 

  6212. <a name="l06148"></a>06148     absc = hlgth*xgk(jtwm1)
    6213. 

  6213. <a name="l06149"></a>06149     fval1 = f(centr-absc)
    6214. 

  6214. <a name="l06150"></a>06150     fval2 = f(centr+absc)
    6215. 

  6215. <a name="l06151"></a>06151     fv1(jtwm1) = fval1
    6216. 

  6216. <a name="l06152"></a>06152     fv2(jtwm1) = fval2
    6217. 

  6217. <a name="l06153"></a>06153     fsum = fval1+fval2
    6218. 

  6218. <a name="l06154"></a>06154     resk = resk+wgk(jtwm1)*fsum
    6219. 

  6219. <a name="l06155"></a>06155     resabs = resabs+wgk(jtwm1)*(abs(fval1)+abs(fval2))
    6220. 

  6220. <a name="l06156"></a>06156   <span class="keyword">end do</span>
    6221. 

  6221. <a name="l06157"></a>06157 
    6222. 

  6222. <a name="l06158"></a>06158   reskh = resk * 5.0e-01
    6223. 

  6223. <a name="l06159"></a>06159   resasc = wgk(8)*abs(fc-reskh)
    6224. 

  6224. <a name="l06160"></a>06160 
    6225. 

  6225. <a name="l06161"></a>06161   <span class="keyword">do</span> j = 1, 7
    6226. 

  6226. <a name="l06162"></a>06162     resasc = resasc+wgk(j)*(abs(fv1(j)-reskh)+abs(fv2(j)-reskh))
    6227. 

  6227. <a name="l06163"></a>06163   <span class="keyword">end do</span>
    6228. 

  6228. <a name="l06164"></a>06164 
    6229. 

  6229. <a name="l06165"></a>06165   result = resk*hlgth
    6230. 

  6230. <a name="l06166"></a>06166   resabs = resabs*dhlgth
    6231. 

  6231. <a name="l06167"></a>06167   resasc = resasc*dhlgth
    6232. 

  6232. <a name="l06168"></a>06168   abserr = abs((resk-resg)*hlgth)
    6233. 

  6233. <a name="l06169"></a>06169 
    6234. 

  6234. <a name="l06170"></a>06170   <span class="keyword">if</span> ( resasc /= 0.0e+00.and.abserr /= 0.0e+00 ) <span class="keyword">then</span>
    6235. 

  6235. <a name="l06171"></a>06171     abserr = resasc*min ( 1.0e+00,(2.0e+02*abserr/resasc)**1.5e+00)
    6236. 

  6236. <a name="l06172"></a>06172   <span class="keyword">end if</span>
    6237. 

  6237. <a name="l06173"></a>06173 
    6238. 

  6238. <a name="l06174"></a>06174   <span class="keyword">if</span> ( resabs &gt; tiny ( resabs ) / (5.0e+01* epsilon ( resabs ) ) ) <span class="keyword">then</span>
    6239. 

  6239. <a name="l06175"></a>06175     abserr = max (( epsilon ( resabs ) *5.0e+01)*resabs,abserr)
    6240. 

  6240. <a name="l06176"></a>06176   <span class="keyword">end if</span>
    6241. 

  6241. <a name="l06177"></a>06177 
    6242. 

  6242. <a name="l06178"></a>06178   return
    6243. 

  6243. <a name="l06179"></a>06179 <span class="keyword">end</span>
    6244. 

  6244. <a name="l06180"></a><a class="code" href="quadpack_8f90.html#a59164415fc33f2f3bf4ebc4ee2220f7e">06180</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a59164415fc33f2f3bf4ebc4ee2220f7e">qk15i</a> ( f, boun, inf, a, b, result, abserr, resabs, resasc )
    6245. 

  6245. <a name="l06181"></a>06181 
    6246. 

  6246. <a name="l06182"></a>06182 <span class="comment">!*****************************************************************************80</span>
    6247. 

  6247. <a name="l06183"></a>06183 <span class="comment">!</span>
    6248. 

  6248. <a name="l06184"></a>06184 <span class="comment">!! QK15I applies a 15 point Gauss-Kronrod quadrature on an infinite interval.</span>
    6249. 

  6249. <a name="l06185"></a>06185 <span class="comment">!</span>
    6250. 

  6250. <a name="l06186"></a>06186 <span class="comment">!  Discussion:</span>
    6251. 

  6251. <a name="l06187"></a>06187 <span class="comment">!</span>
    6252. 

  6252. <a name="l06188"></a>06188 <span class="comment">!    The original infinite integration range is mapped onto the interval </span>
    6253. 

  6253. <a name="l06189"></a>06189 <span class="comment">!    (0,1) and (a,b) is a part of (0,1).  The routine then computes:</span>
    6254. 

  6254. <a name="l06190"></a>06190 <span class="comment">!</span>
    6255. 

  6255. <a name="l06191"></a>06191 <span class="comment">!    i = integral of transformed integrand over (a,b),</span>
    6256. 

  6256. <a name="l06192"></a>06192 <span class="comment">!    j = integral of abs(transformed integrand) over (a,b).</span>
    6257. 

  6257. <a name="l06193"></a>06193 <span class="comment">!</span>
    6258. 

  6258. <a name="l06194"></a>06194 <span class="comment">!  Author:</span>
    6259. 

  6259. <a name="l06195"></a>06195 <span class="comment">!</span>
    6260. 

  6260. <a name="l06196"></a>06196 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    6261. 

  6261. <a name="l06197"></a>06197 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    6262. 

  6262. <a name="l06198"></a>06198 <span class="comment">!</span>
    6263. 

  6263. <a name="l06199"></a>06199 <span class="comment">!  Reference:</span>
    6264. 

  6264. <a name="l06200"></a>06200 <span class="comment">!</span>
    6265. 

  6265. <a name="l06201"></a>06201 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    6266. 

  6266. <a name="l06202"></a>06202 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    6267. 

  6267. <a name="l06203"></a>06203 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    6268. 

  6268. <a name="l06204"></a>06204 <span class="comment">!    Springer Verlag, 1983</span>
    6269. 

  6269. <a name="l06205"></a>06205 <span class="comment">!</span>
    6270. 

  6270. <a name="l06206"></a>06206 <span class="comment">!  Parameters:</span>
    6271. 

  6271. <a name="l06207"></a>06207 <span class="comment">!</span>
    6272. 

  6272. <a name="l06208"></a>06208 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    6273. 

  6273. <a name="l06209"></a>06209 <span class="comment">!      function f ( x )</span>
    6274. 

  6274. <a name="l06210"></a>06210 <span class="comment">!      real f</span>
    6275. 

  6275. <a name="l06211"></a>06211 <span class="comment">!      real x</span>
    6276. 

  6276. <a name="l06212"></a>06212 <span class="comment">!    which evaluates the integrand function.</span>
    6277. 

  6277. <a name="l06213"></a>06213 <span class="comment">!</span>
    6278. 

  6278. <a name="l06214"></a>06214 <span class="comment">!    Input, real BOUN, the finite bound of the original integration range,</span>
    6279. 

  6279. <a name="l06215"></a>06215 <span class="comment">!    or zero if INF is 2.</span>
    6280. 

  6280. <a name="l06216"></a>06216 <span class="comment">!</span>
    6281. 

  6281. <a name="l06217"></a>06217 <span class="comment">!    Input, integer INF, indicates the type of the interval.</span>
    6282. 

  6282. <a name="l06218"></a>06218 <span class="comment">!    -1: the original interval is (-infinity,BOUN),</span>
    6283. 

  6283. <a name="l06219"></a>06219 <span class="comment">!    +1, the original interval is (BOUN,+infinity),</span>
    6284. 

  6284. <a name="l06220"></a>06220 <span class="comment">!    +2, the original interval is (-infinity,+infinity) and</span>
    6285. 

  6285. <a name="l06221"></a>06221 <span class="comment">!    the integral is computed as the sum of two integrals, one </span>
    6286. 

  6286. <a name="l06222"></a>06222 <span class="comment">!    over (-infinity,0) and one over (0,+infinity).</span>
    6287. 

  6287. <a name="l06223"></a>06223 <span class="comment">!</span>
    6288. 

  6288. <a name="l06224"></a>06224 <span class="comment">!    Input, real A, B, the limits of integration, over a subrange of [0,1].</span>
    6289. 

  6289. <a name="l06225"></a>06225 <span class="comment">!</span>
    6290. 

  6290. <a name="l06226"></a>06226 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    6291. 

  6291. <a name="l06227"></a>06227 <span class="comment">!    RESULT is computed by applying the 15-point Kronrod rule (RESK) obtained </span>
    6292. 

  6292. <a name="l06228"></a>06228 <span class="comment">!    by optimal addition of abscissae to the 7-point Gauss rule (RESG).</span>
    6293. 

  6293. <a name="l06229"></a>06229 <span class="comment">!</span>
    6294. 

  6294. <a name="l06230"></a>06230 <span class="comment">!    Output, real ABSERR, an estimate of | I - RESULT |.</span>
    6295. 

  6295. <a name="l06231"></a>06231 <span class="comment">!</span>
    6296. 

  6296. <a name="l06232"></a>06232 <span class="comment">!    Output, real RESABS, approximation to the integral of the absolute</span>
    6297. 

  6297. <a name="l06233"></a>06233 <span class="comment">!    value of F.</span>
    6298. 

  6298. <a name="l06234"></a>06234 <span class="comment">!</span>
    6299. 

  6299. <a name="l06235"></a>06235 <span class="comment">!    Output, real RESASC, approximation to the integral of the</span>
    6300. 

  6300. <a name="l06236"></a>06236 <span class="comment">!    transformated integrand | F-I/(B-A) | over [A,B].</span>
    6301. 

  6301. <a name="l06237"></a>06237 <span class="comment">!</span>
    6302. 

  6302. <a name="l06238"></a>06238 <span class="comment">!  Local Parameters:</span>
    6303. 

  6303. <a name="l06239"></a>06239 <span class="comment">!</span>
    6304. 

  6304. <a name="l06240"></a>06240 <span class="comment">!           centr  - mid point of the interval</span>
    6305. 

  6305. <a name="l06241"></a>06241 <span class="comment">!           hlgth  - half-length of the interval</span>
    6306. 

  6306. <a name="l06242"></a>06242 <span class="comment">!           absc*  - abscissa</span>
    6307. 

  6307. <a name="l06243"></a>06243 <span class="comment">!           tabsc* - transformed abscissa</span>
    6308. 

  6308. <a name="l06244"></a>06244 <span class="comment">!           fval*  - function value</span>
    6309. 

  6309. <a name="l06245"></a>06245 <span class="comment">!           resg   - result of the 7-point Gauss formula</span>
    6310. 

  6310. <a name="l06246"></a>06246 <span class="comment">!           resk   - result of the 15-point Kronrod formula</span>
    6311. 

  6311. <a name="l06247"></a>06247 <span class="comment">!           reskh  - approximation to the mean value of the transformed</span>
    6312. 

  6312. <a name="l06248"></a>06248 <span class="comment">!                    integrand over (a,b), i.e. to i/(b-a)</span>
    6313. 

  6313. <a name="l06249"></a>06249 <span class="comment">!</span>
    6314. 

  6314. <a name="l06250"></a>06250   <span class="keyword">implicit none</span>
    6315. 

  6315. <a name="l06251"></a>06251 
    6316. 

  6316. <a name="l06252"></a>06252   <span class="keywordtype">real</span> a
    6317. 

  6317. <a name="l06253"></a>06253   <span class="keywordtype">real</span> absc
    6318. 

  6318. <a name="l06254"></a>06254   <span class="keywordtype">real</span> absc1
    6319. 

  6319. <a name="l06255"></a>06255   <span class="keywordtype">real</span> absc2
    6320. 

  6320. <a name="l06256"></a>06256   <span class="keywordtype">real</span> abserr
    6321. 

  6321. <a name="l06257"></a>06257   <span class="keywordtype">real</span> b
    6322. 

  6322. <a name="l06258"></a>06258   <span class="keywordtype">real</span> boun
    6323. 

  6323. <a name="l06259"></a>06259   <span class="keywordtype">real</span> centr
    6324. 

  6324. <a name="l06260"></a>06260   <span class="keywordtype">real</span> dinf
    6325. 

  6325. <a name="l06261"></a>06261   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    6326. 

  6326. <a name="l06262"></a>06262   <span class="keywordtype">real</span> fc
    6327. 

  6327. <a name="l06263"></a>06263   <span class="keywordtype">real</span> fsum
    6328. 

  6328. <a name="l06264"></a>06264   <span class="keywordtype">real</span> fval1
    6329. 

  6329. <a name="l06265"></a>06265   <span class="keywordtype">real</span> fval2
    6330. 

  6330. <a name="l06266"></a>06266   <span class="keywordtype">real</span> fv1(7)
    6331. 

  6331. <a name="l06267"></a>06267   <span class="keywordtype">real</span> fv2(7)
    6332. 

  6332. <a name="l06268"></a>06268   <span class="keywordtype">real</span> hlgth
    6333. 

  6333. <a name="l06269"></a>06269   <span class="keywordtype">integer</span> inf
    6334. 

  6334. <a name="l06270"></a>06270   <span class="keywordtype">integer</span> j
    6335. 

  6335. <a name="l06271"></a>06271   <span class="keywordtype">real</span> resabs
    6336. 

  6336. <a name="l06272"></a>06272   <span class="keywordtype">real</span> resasc
    6337. 

  6337. <a name="l06273"></a>06273   <span class="keywordtype">real</span> resg
    6338. 

  6338. <a name="l06274"></a>06274   <span class="keywordtype">real</span> resk
    6339. 

  6339. <a name="l06275"></a>06275   <span class="keywordtype">real</span> reskh
    6340. 

  6340. <a name="l06276"></a>06276   <span class="keywordtype">real</span> result
    6341. 

  6341. <a name="l06277"></a>06277   <span class="keywordtype">real</span> tabsc1
    6342. 

  6342. <a name="l06278"></a>06278   <span class="keywordtype">real</span> tabsc2
    6343. 

  6343. <a name="l06279"></a>06279   <span class="keywordtype">real</span> wg(8)
    6344. 

  6344. <a name="l06280"></a>06280   <span class="keywordtype">real</span> wgk(8)
    6345. 

  6345. <a name="l06281"></a>06281   <span class="keywordtype">real</span> xgk(8)
    6346. 

  6346. <a name="l06282"></a>06282 <span class="comment">!</span>
    6347. 

  6347. <a name="l06283"></a>06283 <span class="comment">!  the abscissae and weights are supplied for the interval</span>
    6348. 

  6348. <a name="l06284"></a>06284 <span class="comment">!  (-1,1).  because of symmetry only the positive abscissae and</span>
    6349. 

  6349. <a name="l06285"></a>06285 <span class="comment">!  their corresponding weights are given.</span>
    6350. 

  6350. <a name="l06286"></a>06286 <span class="comment">!</span>
    6351. 

  6351. <a name="l06287"></a>06287 <span class="comment">!           xgk    - abscissae of the 15-point Kronrod rule</span>
    6352. 

  6352. <a name="l06288"></a>06288 <span class="comment">!                    xgk(2), xgk(4), ... abscissae of the 7-point Gauss</span>
    6353. 

  6353. <a name="l06289"></a>06289 <span class="comment">!                    rule</span>
    6354. 

  6354. <a name="l06290"></a>06290 <span class="comment">!                    xgk(1), xgk(3), ...  abscissae which are optimally</span>
    6355. 

  6355. <a name="l06291"></a>06291 <span class="comment">!                    added to the 7-point Gauss rule</span>
    6356. 

  6356. <a name="l06292"></a>06292 <span class="comment">!</span>
    6357. 

  6357. <a name="l06293"></a>06293 <span class="comment">!           wgk    - weights of the 15-point Kronrod rule</span>
    6358. 

  6358. <a name="l06294"></a>06294 <span class="comment">!</span>
    6359. 

  6359. <a name="l06295"></a>06295 <span class="comment">!           wg     - weights of the 7-point Gauss rule, corresponding</span>
    6360. 

  6360. <a name="l06296"></a>06296 <span class="comment">!                    to the abscissae xgk(2), xgk(4), ...</span>
    6361. 

  6361. <a name="l06297"></a>06297 <span class="comment">!                    wg(1), wg(3), ... are set to zero.</span>
    6362. 

  6362. <a name="l06298"></a>06298 <span class="comment">!</span>
    6363. 

  6363. <a name="l06299"></a>06299   <span class="keyword">data</span> xgk(1),xgk(2),xgk(3),xgk(4),xgk(5),xgk(6),xgk(7),xgk(8)/ &amp;
    6364. 

  6364. <a name="l06300"></a>06300        9.914553711208126e-01,     9.491079123427585e-01, &amp;
    6365. 

  6365. <a name="l06301"></a>06301        8.648644233597691e-01,     7.415311855993944e-01, &amp;
    6366. 

  6366. <a name="l06302"></a>06302        5.860872354676911e-01,     4.058451513773972e-01, &amp;
    6367. 

  6367. <a name="l06303"></a>06303        2.077849550078985e-01,     0.0000000000000000e+00/
    6368. 

  6368. <a name="l06304"></a>06304 
    6369. 

  6369. <a name="l06305"></a>06305   <span class="keyword">data</span> wgk(1),wgk(2),wgk(3),wgk(4),wgk(5),wgk(6),wgk(7),wgk(8)/ &amp;
    6370. 

  6370. <a name="l06306"></a>06306        2.293532201052922e-02,     6.309209262997855e-02, &amp;
    6371. 

  6371. <a name="l06307"></a>06307        1.047900103222502e-01,     1.406532597155259e-01, &amp;
    6372. 

  6372. <a name="l06308"></a>06308        1.690047266392679e-01,     1.903505780647854e-01, &amp;
    6373. 

  6373. <a name="l06309"></a>06309        2.044329400752989e-01,     2.094821410847278e-01/
    6374. 

  6374. <a name="l06310"></a>06310 
    6375. 

  6375. <a name="l06311"></a>06311   <span class="keyword">data</span> wg(1),wg(2),wg(3),wg(4),wg(5),wg(6),wg(7),wg(8)/ &amp;
    6376. 

  6376. <a name="l06312"></a>06312        0.0000000000000000e+00,     1.294849661688697e-01, &amp;
    6377. 

  6377. <a name="l06313"></a>06313        0.0000000000000000e+00,     2.797053914892767e-01, &amp;
    6378. 

  6378. <a name="l06314"></a>06314        0.0000000000000000e+00,     3.818300505051189e-01, &amp;
    6379. 

  6379. <a name="l06315"></a>06315        0.0000000000000000e+00,     4.179591836734694e-01/
    6380. 

  6380. <a name="l06316"></a>06316 
    6381. 

  6381. <a name="l06317"></a>06317   dinf = min ( 1, inf )
    6382. 

  6382. <a name="l06318"></a>06318 
    6383. 

  6383. <a name="l06319"></a>06319   centr = 5.0e-01*(a+b)
    6384. 

  6384. <a name="l06320"></a>06320   hlgth = 5.0e-01*(b-a)
    6385. 

  6385. <a name="l06321"></a>06321   tabsc1 = boun+dinf*(1.0e+00-centr)/centr
    6386. 

  6386. <a name="l06322"></a>06322   fval1 = f(tabsc1)
    6387. 

  6387. <a name="l06323"></a>06323   <span class="keyword">if</span> ( inf == 2 ) fval1 = fval1+f(-tabsc1)
    6388. 

  6388. <a name="l06324"></a>06324   fc = (fval1/centr)/centr
    6389. 

  6389. <a name="l06325"></a>06325 <span class="comment">!</span>
    6390. 

  6390. <a name="l06326"></a>06326 <span class="comment">!  Compute the 15-point Kronrod approximation to the integral,</span>
    6391. 

  6391. <a name="l06327"></a>06327 <span class="comment">!  and estimate the error.</span>
    6392. 

  6392. <a name="l06328"></a>06328 <span class="comment">!</span>
    6393. 

  6393. <a name="l06329"></a>06329   resg = wg(8)*fc
    6394. 

  6394. <a name="l06330"></a>06330   resk = wgk(8)*fc
    6395. 

  6395. <a name="l06331"></a>06331   resabs = abs(resk)
    6396. 

  6396. <a name="l06332"></a>06332 
    6397. 

  6397. <a name="l06333"></a>06333   <span class="keyword">do</span> j = 1, 7
    6398. 

  6398. <a name="l06334"></a>06334 
    6399. 

  6399. <a name="l06335"></a>06335     absc = hlgth*xgk(j)
    6400. 

  6400. <a name="l06336"></a>06336     absc1 = centr-absc
    6401. 

  6401. <a name="l06337"></a>06337     absc2 = centr+absc
    6402. 

  6402. <a name="l06338"></a>06338     tabsc1 = boun+dinf*(1.0e+00-absc1)/absc1
    6403. 

  6403. <a name="l06339"></a>06339     tabsc2 = boun+dinf*(1.0e+00-absc2)/absc2
    6404. 

  6404. <a name="l06340"></a>06340     fval1 = f(tabsc1)
    6405. 

  6405. <a name="l06341"></a>06341     fval2 = f(tabsc2)
    6406. 

  6406. <a name="l06342"></a>06342 
    6407. 

  6407. <a name="l06343"></a>06343     <span class="keyword">if</span> ( inf == 2 ) <span class="keyword">then</span>
    6408. 

  6408. <a name="l06344"></a>06344       fval1 = fval1+f(-tabsc1)
    6409. 

  6409. <a name="l06345"></a>06345       fval2 = fval2+f(-tabsc2)
    6410. 

  6410. <a name="l06346"></a>06346     <span class="keyword">end if</span>
    6411. 

  6411. <a name="l06347"></a>06347 
    6412. 

  6412. <a name="l06348"></a>06348     fval1 = (fval1/absc1)/absc1
    6413. 

  6413. <a name="l06349"></a>06349     fval2 = (fval2/absc2)/absc2
    6414. 

  6414. <a name="l06350"></a>06350     fv1(j) = fval1
    6415. 

  6415. <a name="l06351"></a>06351     fv2(j) = fval2
    6416. 

  6416. <a name="l06352"></a>06352     fsum = fval1+fval2
    6417. 

  6417. <a name="l06353"></a>06353     resg = resg+wg(j)*fsum
    6418. 

  6418. <a name="l06354"></a>06354     resk = resk+wgk(j)*fsum
    6419. 

  6419. <a name="l06355"></a>06355     resabs = resabs+wgk(j)*(abs(fval1)+abs(fval2))
    6420. 

  6420. <a name="l06356"></a>06356   <span class="keyword">end do</span>
    6421. 

  6421. <a name="l06357"></a>06357 
    6422. 

  6422. <a name="l06358"></a>06358   reskh = resk * 5.0e-01
    6423. 

  6423. <a name="l06359"></a>06359   resasc = wgk(8) * abs(fc-reskh)
    6424. 

  6424. <a name="l06360"></a>06360 
    6425. 

  6425. <a name="l06361"></a>06361   <span class="keyword">do</span> j = 1, 7
    6426. 

  6426. <a name="l06362"></a>06362     resasc = resasc + wgk(j)*(abs(fv1(j)-reskh)+abs(fv2(j)-reskh))
    6427. 

  6427. <a name="l06363"></a>06363   <span class="keyword">end do</span>
    6428. 

  6428. <a name="l06364"></a>06364 
    6429. 

  6429. <a name="l06365"></a>06365   result = resk * hlgth
    6430. 

  6430. <a name="l06366"></a>06366   resasc = resasc * hlgth
    6431. 

  6431. <a name="l06367"></a>06367   resabs = resabs * hlgth
    6432. 

  6432. <a name="l06368"></a>06368   abserr = abs ( ( resk - resg ) * hlgth )
    6433. 

  6433. <a name="l06369"></a>06369 
    6434. 

  6434. <a name="l06370"></a>06370   <span class="keyword">if</span> ( resasc /= 0.0e+00.and.abserr /= 0.0e+00) <span class="keyword">then</span>
    6435. 

  6435. <a name="l06371"></a>06371     abserr = resasc* min ( 1.0e+00,(2.0e+02*abserr/resasc)**1.5e+00)
    6436. 

  6436. <a name="l06372"></a>06372   <span class="keyword">end if</span>
    6437. 

  6437. <a name="l06373"></a>06373 
    6438. 

  6438. <a name="l06374"></a>06374   <span class="keyword">if</span> ( resabs &gt; tiny ( resabs ) / ( 5.0e+01 * epsilon ( resabs ) ) ) <span class="keyword">then</span>
    6439. 

  6439. <a name="l06375"></a>06375     abserr = max (( epsilon ( resabs ) *5.0e+01)*resabs,abserr)
    6440. 

  6440. <a name="l06376"></a>06376   <span class="keyword">end if</span>
    6441. 

  6441. <a name="l06377"></a>06377 
    6442. 

  6442. <a name="l06378"></a>06378   return
    6443. 

  6443. <a name="l06379"></a>06379 <span class="keyword">end</span>
    6444. 

  6444. <a name="l06380"></a><a class="code" href="quadpack_8f90.html#a0c083838940925726abd5bc85fa29587">06380</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a0c083838940925726abd5bc85fa29587">qk15w</a> ( f, w, p1, p2, p3, p4, kp, a, b, result, abserr, resabs, &amp;
    6445. 

  6445. <a name="l06381"></a>06381   resasc )
    6446. 

  6446. <a name="l06382"></a>06382 
    6447. 

  6447. <a name="l06383"></a>06383 <span class="comment">!*****************************************************************************80</span>
    6448. 

  6448. <a name="l06384"></a>06384 <span class="comment">!</span>
    6449. 

  6449. <a name="l06385"></a>06385 <span class="comment">!! QK15W applies a 15 point Gauss-Kronrod rule for a weighted integrand.</span>
    6450. 

  6450. <a name="l06386"></a>06386 <span class="comment">!</span>
    6451. 

  6451. <a name="l06387"></a>06387 <span class="comment">!  Discussion:</span>
    6452. 

  6452. <a name="l06388"></a>06388 <span class="comment">!</span>
    6453. 

  6453. <a name="l06389"></a>06389 <span class="comment">!    This routine approximates </span>
    6454. 

  6454. <a name="l06390"></a>06390 <span class="comment">!      i = integral of f*w over (a,b), </span>
    6455. 

  6455. <a name="l06391"></a>06391 <span class="comment">!    with error estimate, and</span>
    6456. 

  6456. <a name="l06392"></a>06392 <span class="comment">!      j = integral of abs(f*w) over (a,b)</span>
    6457. 

  6457. <a name="l06393"></a>06393 <span class="comment">!</span>
    6458. 

  6458. <a name="l06394"></a>06394 <span class="comment">!  Author:</span>
    6459. 

  6459. <a name="l06395"></a>06395 <span class="comment">!</span>
    6460. 

  6460. <a name="l06396"></a>06396 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    6461. 

  6461. <a name="l06397"></a>06397 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    6462. 

  6462. <a name="l06398"></a>06398 <span class="comment">!</span>
    6463. 

  6463. <a name="l06399"></a>06399 <span class="comment">!  Reference:</span>
    6464. 

  6464. <a name="l06400"></a>06400 <span class="comment">!</span>
    6465. 

  6465. <a name="l06401"></a>06401 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    6466. 

  6466. <a name="l06402"></a>06402 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    6467. 

  6467. <a name="l06403"></a>06403 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    6468. 

  6468. <a name="l06404"></a>06404 <span class="comment">!    Springer Verlag, 1983</span>
    6469. 

  6469. <a name="l06405"></a>06405 <span class="comment">!</span>
    6470. 

  6470. <a name="l06406"></a>06406 <span class="comment">!  Parameters:</span>
    6471. 

  6471. <a name="l06407"></a>06407 <span class="comment">!</span>
    6472. 

  6472. <a name="l06408"></a>06408 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    6473. 

  6473. <a name="l06409"></a>06409 <span class="comment">!      function f ( x )</span>
    6474. 

  6474. <a name="l06410"></a>06410 <span class="comment">!      real f</span>
    6475. 

  6475. <a name="l06411"></a>06411 <span class="comment">!      real x</span>
    6476. 

  6476. <a name="l06412"></a>06412 <span class="comment">!    which evaluates the integrand function.</span>
    6477. 

  6477. <a name="l06413"></a>06413 <span class="comment">!</span>
    6478. 

  6478. <a name="l06414"></a>06414 <span class="comment">!              w      - real</span>
    6479. 

  6479. <a name="l06415"></a>06415 <span class="comment">!                       function subprogram defining the integrand</span>
    6480. 

  6480. <a name="l06416"></a>06416 <span class="comment">!                       weight function w(x). the actual name for w</span>
    6481. 

  6481. <a name="l06417"></a>06417 <span class="comment">!                       needs to be declared e x t e r n a l in the</span>
    6482. 

  6482. <a name="l06418"></a>06418 <span class="comment">!                       calling program.</span>
    6483. 

  6483. <a name="l06419"></a>06419 <span class="comment">!</span>
    6484. 

  6484. <a name="l06420"></a>06420 <span class="comment">!    ?, real P1, P2, P3, P4, parameters in the weight function</span>
    6485. 

  6485. <a name="l06421"></a>06421 <span class="comment">!</span>
    6486. 

  6486. <a name="l06422"></a>06422 <span class="comment">!    Input, integer KP, key for indicating the type of weight function</span>
    6487. 

  6487. <a name="l06423"></a>06423 <span class="comment">!</span>
    6488. 

  6488. <a name="l06424"></a>06424 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    6489. 

  6489. <a name="l06425"></a>06425 <span class="comment">!</span>
    6490. 

  6490. <a name="l06426"></a>06426 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    6491. 

  6491. <a name="l06427"></a>06427 <span class="comment">!    RESULT is computed by applying the 15-point Kronrod rule (RESK) obtained by</span>
    6492. 

  6492. <a name="l06428"></a>06428 <span class="comment">!    optimal addition of abscissae to the 7-point Gauss rule (RESG).</span>
    6493. 

  6493. <a name="l06429"></a>06429 <span class="comment">!</span>
    6494. 

  6494. <a name="l06430"></a>06430 <span class="comment">!    Output, real ABSERR, an estimate of | I - RESULT |.</span>
    6495. 

  6495. <a name="l06431"></a>06431 <span class="comment">!</span>
    6496. 

  6496. <a name="l06432"></a>06432 <span class="comment">!    Output, real RESABS, approximation to the integral of the absolute</span>
    6497. 

  6497. <a name="l06433"></a>06433 <span class="comment">!    value of F.</span>
    6498. 

  6498. <a name="l06434"></a>06434 <span class="comment">!</span>
    6499. 

  6499. <a name="l06435"></a>06435 <span class="comment">!    Output, real RESASC, approximation to the integral | F-I/(B-A) | </span>
    6500. 

  6500. <a name="l06436"></a>06436 <span class="comment">!    over [A,B].</span>
    6501. 

  6501. <a name="l06437"></a>06437 <span class="comment">!</span>
    6502. 

  6502. <a name="l06438"></a>06438 <span class="comment">!  Local Parameters:</span>
    6503. 

  6503. <a name="l06439"></a>06439 <span class="comment">!</span>
    6504. 

  6504. <a name="l06440"></a>06440 <span class="comment">!           centr  - mid point of the interval</span>
    6505. 

  6505. <a name="l06441"></a>06441 <span class="comment">!           hlgth  - half-length of the interval</span>
    6506. 

  6506. <a name="l06442"></a>06442 <span class="comment">!           absc*  - abscissa</span>
    6507. 

  6507. <a name="l06443"></a>06443 <span class="comment">!           fval*  - function value</span>
    6508. 

  6508. <a name="l06444"></a>06444 <span class="comment">!           resg   - result of the 7-point Gauss formula</span>
    6509. 

  6509. <a name="l06445"></a>06445 <span class="comment">!           resk   - result of the 15-point Kronrod formula</span>
    6510. 

  6510. <a name="l06446"></a>06446 <span class="comment">!           reskh  - approximation to the mean value of f*w over (a,b),</span>
    6511. 

  6511. <a name="l06447"></a>06447 <span class="comment">!                    i.e. to i/(b-a)</span>
    6512. 

  6512. <a name="l06448"></a>06448 <span class="comment">!</span>
    6513. 

  6513. <a name="l06449"></a>06449   <span class="keyword">implicit none</span>
    6514. 

  6514. <a name="l06450"></a>06450 
    6515. 

  6515. <a name="l06451"></a>06451   <span class="keywordtype">real</span> a
    6516. 

  6516. <a name="l06452"></a>06452   <span class="keywordtype">real</span> absc
    6517. 

  6517. <a name="l06453"></a>06453   <span class="keywordtype">real</span> absc1
    6518. 

  6518. <a name="l06454"></a>06454   <span class="keywordtype">real</span> absc2
    6519. 

  6519. <a name="l06455"></a>06455   <span class="keywordtype">real</span> abserr
    6520. 

  6520. <a name="l06456"></a>06456   <span class="keywordtype">real</span> b
    6521. 

  6521. <a name="l06457"></a>06457   <span class="keywordtype">real</span> centr
    6522. 

  6522. <a name="l06458"></a>06458   <span class="keywordtype">real</span> dhlgth
    6523. 

  6523. <a name="l06459"></a>06459   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    6524. 

  6524. <a name="l06460"></a>06460   <span class="keywordtype">real</span> fc
    6525. 

  6525. <a name="l06461"></a>06461   <span class="keywordtype">real</span> fsum
    6526. 

  6526. <a name="l06462"></a>06462   <span class="keywordtype">real</span> fval1
    6527. 

  6527. <a name="l06463"></a>06463   <span class="keywordtype">real</span> fval2
    6528. 

  6528. <a name="l06464"></a>06464   <span class="keywordtype">real</span> fv1(7)
    6529. 

  6529. <a name="l06465"></a>06465   <span class="keywordtype">real</span> fv2(7)
    6530. 

  6530. <a name="l06466"></a>06466   <span class="keywordtype">real</span> hlgth
    6531. 

  6531. <a name="l06467"></a>06467   <span class="keywordtype">integer</span> j
    6532. 

  6532. <a name="l06468"></a>06468   <span class="keywordtype">integer</span> jtw
    6533. 

  6533. <a name="l06469"></a>06469   <span class="keywordtype">integer</span> jtwm1
    6534. 

  6534. <a name="l06470"></a>06470   <span class="keywordtype">integer</span> kp
    6535. 

  6535. <a name="l06471"></a>06471   <span class="keywordtype">real</span> p1
    6536. 

  6536. <a name="l06472"></a>06472   <span class="keywordtype">real</span> p2
    6537. 

  6537. <a name="l06473"></a>06473   <span class="keywordtype">real</span> p3
    6538. 

  6538. <a name="l06474"></a>06474   <span class="keywordtype">real</span> p4
    6539. 

  6539. <a name="l06475"></a>06475   <span class="keywordtype">real</span> resabs
    6540. 

  6540. <a name="l06476"></a>06476   <span class="keywordtype">real</span> resasc
    6541. 

  6541. <a name="l06477"></a>06477   <span class="keywordtype">real</span> resg
    6542. 

  6542. <a name="l06478"></a>06478   <span class="keywordtype">real</span> resk
    6543. 

  6543. <a name="l06479"></a>06479   <span class="keywordtype">real</span> reskh
    6544. 

  6544. <a name="l06480"></a>06480   <span class="keywordtype">real</span> result
    6545. 

  6545. <a name="l06481"></a>06481   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: w
    6546. 

  6546. <a name="l06482"></a>06482   <span class="keywordtype">real</span>, <span class="keywordtype">dimension ( 4 )</span> :: wg = (/ 
    6547. 

  6547. <a name="l06483"></a>06483     1.294849661688697e-01,     2.797053914892767e-01, 
    6548. 

  6548. <a name="l06484"></a>06484     3.818300505051889e-01,     4.179591836734694e-01 /)
    6549. 

  6549. <a name="l06485"></a>06485   <span class="keywordtype">real</span> wgk(8)
    6550. 

  6550. <a name="l06486"></a>06486   <span class="keywordtype">real</span> xgk(8)
    6551. 

  6551. <a name="l06487"></a>06487 <span class="comment">!</span>
    6552. 

  6552. <a name="l06488"></a>06488 <span class="comment">!  the abscissae and weights are given for the interval (-1,1).</span>
    6553. 

  6553. <a name="l06489"></a>06489 <span class="comment">!  because of symmetry only the positive abscissae and their</span>
    6554. 

  6554. <a name="l06490"></a>06490 <span class="comment">!  corresponding weights are given.</span>
    6555. 

  6555. <a name="l06491"></a>06491 <span class="comment">!</span>
    6556. 

  6556. <a name="l06492"></a>06492 <span class="comment">!           xgk    - abscissae of the 15-point Gauss-Kronrod rule</span>
    6557. 

  6557. <a name="l06493"></a>06493 <span class="comment">!                    xgk(2), xgk(4), ... abscissae of the 7-point Gauss</span>
    6558. 

  6558. <a name="l06494"></a>06494 <span class="comment">!                    rule</span>
    6559. 

  6559. <a name="l06495"></a>06495 <span class="comment">!                    xgk(1), xgk(3), ... abscissae which are optimally</span>
    6560. 

  6560. <a name="l06496"></a>06496 <span class="comment">!                    added to the 7-point Gauss rule</span>
    6561. 

  6561. <a name="l06497"></a>06497 <span class="comment">!</span>
    6562. 

  6562. <a name="l06498"></a>06498 <span class="comment">!           wgk    - weights of the 15-point Gauss-Kronrod rule</span>
    6563. 

  6563. <a name="l06499"></a>06499 <span class="comment">!</span>
    6564. 

  6564. <a name="l06500"></a>06500 <span class="comment">!           wg     - weights of the 7-point Gauss rule</span>
    6565. 

  6565. <a name="l06501"></a>06501 <span class="comment">!</span>
    6566. 

  6566. <a name="l06502"></a>06502   <span class="keyword">data</span> xgk(1),xgk(2),xgk(3),xgk(4),xgk(5),xgk(6),xgk(7),xgk(8)/ &amp;
    6567. 

  6567. <a name="l06503"></a>06503        9.914553711208126e-01,     9.491079123427585e-01, &amp;
    6568. 

  6568. <a name="l06504"></a>06504        8.648644233597691e-01,     7.415311855993944e-01, &amp;
    6569. 

  6569. <a name="l06505"></a>06505        5.860872354676911e-01,     4.058451513773972e-01, &amp;
    6570. 

  6570. <a name="l06506"></a>06506        2.077849550789850e-01,     0.000000000000000e+00/
    6571. 

  6571. <a name="l06507"></a>06507 
    6572. 

  6572. <a name="l06508"></a>06508   <span class="keyword">data</span> wgk(1),wgk(2),wgk(3),wgk(4),wgk(5),wgk(6),wgk(7),wgk(8)/ &amp;
    6573. 

  6573. <a name="l06509"></a>06509        2.293532201052922e-02,     6.309209262997855e-02, &amp;
    6574. 

  6574. <a name="l06510"></a>06510        1.047900103222502e-01,     1.406532597155259e-01, &amp;
    6575. 

  6575. <a name="l06511"></a>06511        1.690047266392679e-01,     1.903505780647854e-01, &amp;
    6576. 

  6576. <a name="l06512"></a>06512        2.044329400752989e-01,     2.094821410847278e-01/
    6577. 

  6577. <a name="l06513"></a>06513 <span class="comment">!</span>
    6578. 

  6578. <a name="l06514"></a>06514   centr = 5.0e-01*(a+b)
    6579. 

  6579. <a name="l06515"></a>06515   hlgth = 5.0e-01*(b-a)
    6580. 

  6580. <a name="l06516"></a>06516   dhlgth = abs(hlgth)
    6581. 

  6581. <a name="l06517"></a>06517 <span class="comment">!</span>
    6582. 

  6582. <a name="l06518"></a>06518 <span class="comment">!  Compute the 15-point Kronrod approximation to the integral,</span>
    6583. 

  6583. <a name="l06519"></a>06519 <span class="comment">!  and estimate the error.</span>
    6584. 

  6584. <a name="l06520"></a>06520 <span class="comment">!</span>
    6585. 

  6585. <a name="l06521"></a>06521   fc = f(centr)*w(centr,p1,p2,p3,p4,kp)
    6586. 

  6586. <a name="l06522"></a>06522   resg = wg(4)*fc
    6587. 

  6587. <a name="l06523"></a>06523   resk = wgk(8)*fc
    6588. 

  6588. <a name="l06524"></a>06524   resabs = abs(resk)
    6589. 

  6589. <a name="l06525"></a>06525 
    6590. 

  6590. <a name="l06526"></a>06526   <span class="keyword">do</span> j = 1, 3
    6591. 

  6591. <a name="l06527"></a>06527     jtw = j*2
    6592. 

  6592. <a name="l06528"></a>06528     absc = hlgth*xgk(jtw)
    6593. 

  6593. <a name="l06529"></a>06529     absc1 = centr-absc
    6594. 

  6594. <a name="l06530"></a>06530     absc2 = centr+absc
    6595. 

  6595. <a name="l06531"></a>06531     fval1 = f(absc1)*w(absc1,p1,p2,p3,p4,kp)
    6596. 

  6596. <a name="l06532"></a>06532     fval2 = f(absc2)*w(absc2,p1,p2,p3,p4,kp)
    6597. 

  6597. <a name="l06533"></a>06533     fv1(jtw) = fval1
    6598. 

  6598. <a name="l06534"></a>06534     fv2(jtw) = fval2
    6599. 

  6599. <a name="l06535"></a>06535     fsum = fval1+fval2
    6600. 

  6600. <a name="l06536"></a>06536     resg = resg+wg(j)*fsum
    6601. 

  6601. <a name="l06537"></a>06537     resk = resk+wgk(jtw)*fsum
    6602. 

  6602. <a name="l06538"></a>06538     resabs = resabs+wgk(jtw)*(abs(fval1)+abs(fval2))
    6603. 

  6603. <a name="l06539"></a>06539   <span class="keyword">end do</span>
    6604. 

  6604. <a name="l06540"></a>06540 
    6605. 

  6605. <a name="l06541"></a>06541   <span class="keyword">do</span> j = 1, 4
    6606. 

  6606. <a name="l06542"></a>06542     jtwm1 = j*2-1
    6607. 

  6607. <a name="l06543"></a>06543     absc = hlgth*xgk(jtwm1)
    6608. 

  6608. <a name="l06544"></a>06544     absc1 = centr-absc
    6609. 

  6609. <a name="l06545"></a>06545     absc2 = centr+absc
    6610. 

  6610. <a name="l06546"></a>06546     fval1 = f(absc1)*w(absc1,p1,p2,p3,p4,kp)
    6611. 

  6611. <a name="l06547"></a>06547     fval2 = f(absc2)*w(absc2,p1,p2,p3,p4,kp)
    6612. 

  6612. <a name="l06548"></a>06548     fv1(jtwm1) = fval1
    6613. 

  6613. <a name="l06549"></a>06549     fv2(jtwm1) = fval2
    6614. 

  6614. <a name="l06550"></a>06550     fsum = fval1+fval2
    6615. 

  6615. <a name="l06551"></a>06551     resk = resk+wgk(jtwm1)*fsum
    6616. 

  6616. <a name="l06552"></a>06552     resabs = resabs+wgk(jtwm1)*(abs(fval1)+abs(fval2))
    6617. 

  6617. <a name="l06553"></a>06553   <span class="keyword">end do</span>
    6618. 

  6618. <a name="l06554"></a>06554 
    6619. 

  6619. <a name="l06555"></a>06555   reskh = resk*5.0e-01
    6620. 

  6620. <a name="l06556"></a>06556   resasc = wgk(8)*abs(fc-reskh)
    6621. 

  6621. <a name="l06557"></a>06557 
    6622. 

  6622. <a name="l06558"></a>06558   <span class="keyword">do</span> j = 1, 7
    6623. 

  6623. <a name="l06559"></a>06559     resasc = resasc+wgk(j)*(abs(fv1(j)-reskh)+abs(fv2(j)-reskh))
    6624. 

  6624. <a name="l06560"></a>06560   <span class="keyword">end do</span>
    6625. 

  6625. <a name="l06561"></a>06561 
    6626. 

  6626. <a name="l06562"></a>06562   result = resk*hlgth
    6627. 

  6627. <a name="l06563"></a>06563   resabs = resabs*dhlgth
    6628. 

  6628. <a name="l06564"></a>06564   resasc = resasc*dhlgth
    6629. 

  6629. <a name="l06565"></a>06565   abserr = abs((resk-resg)*hlgth)
    6630. 

  6630. <a name="l06566"></a>06566 
    6631. 

  6631. <a name="l06567"></a>06567   <span class="keyword">if</span> ( resasc /= 0.0e+00.and.abserr /= 0.0e+00) <span class="keyword">then</span>
    6632. 

  6632. <a name="l06568"></a>06568     abserr = resasc*min ( 1.0e+00,(2.0e+02*abserr/resasc)**1.5e+00)
    6633. 

  6633. <a name="l06569"></a>06569   <span class="keyword">end if</span>
    6634. 

  6634. <a name="l06570"></a>06570 
    6635. 

  6635. <a name="l06571"></a>06571   <span class="keyword">if</span> ( resabs &gt; tiny ( resabs ) /(5.0e+01* epsilon ( resabs ) ) ) <span class="keyword">then</span>
    6636. 

  6636. <a name="l06572"></a>06572     abserr = max ( ( epsilon ( resabs ) * 5.0e+01)*resabs,abserr)
    6637. 

  6637. <a name="l06573"></a>06573   <span class="keyword">end if</span>
    6638. 

  6638. <a name="l06574"></a>06574 
    6639. 

  6639. <a name="l06575"></a>06575   return
    6640. 

  6640. <a name="l06576"></a>06576 <span class="keyword">end</span>
    6641. 

  6641. <a name="l06577"></a><a class="code" href="quadpack_8f90.html#a27241a527b249e9de59a5ed6bee5f805">06577</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a27241a527b249e9de59a5ed6bee5f805">qk21</a> ( f, a, b, result, abserr, resabs, resasc )
    6642. 

  6642. <a name="l06578"></a>06578 
    6643. 

  6643. <a name="l06579"></a>06579 <span class="comment">!*****************************************************************************80</span>
    6644. 

  6644. <a name="l06580"></a>06580 <span class="comment">!</span>
    6645. 

  6645. <a name="l06581"></a>06581 <span class="comment">!! QK21 carries out a 21 point Gauss-Kronrod quadrature rule.</span>
    6646. 

  6646. <a name="l06582"></a>06582 <span class="comment">!</span>
    6647. 

  6647. <a name="l06583"></a>06583 <span class="comment">!  Discussion:</span>
    6648. 

  6648. <a name="l06584"></a>06584 <span class="comment">!</span>
    6649. 

  6649. <a name="l06585"></a>06585 <span class="comment">!    This routine approximates</span>
    6650. 

  6650. <a name="l06586"></a>06586 <span class="comment">!      I = integral ( A &lt;= X &lt;= B ) F(X) dx</span>
    6651. 

  6651. <a name="l06587"></a>06587 <span class="comment">!    with an error estimate, and</span>
    6652. 

  6652. <a name="l06588"></a>06588 <span class="comment">!      J = integral ( A &lt;= X &lt;= B ) | F(X) | dx</span>
    6653. 

  6653. <a name="l06589"></a>06589 <span class="comment">!</span>
    6654. 

  6654. <a name="l06590"></a>06590 <span class="comment">!  Author:</span>
    6655. 

  6655. <a name="l06591"></a>06591 <span class="comment">!</span>
    6656. 

  6656. <a name="l06592"></a>06592 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    6657. 

  6657. <a name="l06593"></a>06593 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    6658. 

  6658. <a name="l06594"></a>06594 <span class="comment">!</span>
    6659. 

  6659. <a name="l06595"></a>06595 <span class="comment">!  Reference:</span>
    6660. 

  6660. <a name="l06596"></a>06596 <span class="comment">!</span>
    6661. 

  6661. <a name="l06597"></a>06597 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    6662. 

  6662. <a name="l06598"></a>06598 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    6663. 

  6663. <a name="l06599"></a>06599 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    6664. 

  6664. <a name="l06600"></a>06600 <span class="comment">!    Springer Verlag, 1983</span>
    6665. 

  6665. <a name="l06601"></a>06601 <span class="comment">!</span>
    6666. 

  6666. <a name="l06602"></a>06602 <span class="comment">!  Parameters:</span>
    6667. 

  6667. <a name="l06603"></a>06603 <span class="comment">!</span>
    6668. 

  6668. <a name="l06604"></a>06604 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    6669. 

  6669. <a name="l06605"></a>06605 <span class="comment">!      function f ( x )</span>
    6670. 

  6670. <a name="l06606"></a>06606 <span class="comment">!      real f</span>
    6671. 

  6671. <a name="l06607"></a>06607 <span class="comment">!      real x</span>
    6672. 

  6672. <a name="l06608"></a>06608 <span class="comment">!    which evaluates the integrand function.</span>
    6673. 

  6673. <a name="l06609"></a>06609 <span class="comment">!</span>
    6674. 

  6674. <a name="l06610"></a>06610 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    6675. 

  6675. <a name="l06611"></a>06611 <span class="comment">!</span>
    6676. 

  6676. <a name="l06612"></a>06612 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    6677. 

  6677. <a name="l06613"></a>06613 <span class="comment">!    RESULT is computed by applying the 21-point Kronrod rule (resk) </span>
    6678. 

  6678. <a name="l06614"></a>06614 <span class="comment">!    obtained by optimal addition of abscissae to the 10-point Gauss </span>
    6679. 

  6679. <a name="l06615"></a>06615 <span class="comment">!    rule (resg).</span>
    6680. 

  6680. <a name="l06616"></a>06616 <span class="comment">!</span>
    6681. 

  6681. <a name="l06617"></a>06617 <span class="comment">!    Output, real ABSERR, an estimate of | I - RESULT |.</span>
    6682. 

  6682. <a name="l06618"></a>06618 <span class="comment">!</span>
    6683. 

  6683. <a name="l06619"></a>06619 <span class="comment">!    Output, real RESABS, approximation to the integral of the absolute</span>
    6684. 

  6684. <a name="l06620"></a>06620 <span class="comment">!    value of F.</span>
    6685. 

  6685. <a name="l06621"></a>06621 <span class="comment">!</span>
    6686. 

  6686. <a name="l06622"></a>06622 <span class="comment">!    Output, real RESASC, approximation to the integral | F-I/(B-A) | </span>
    6687. 

  6687. <a name="l06623"></a>06623 <span class="comment">!    over [A,B].</span>
    6688. 

  6688. <a name="l06624"></a>06624 <span class="comment">!</span>
    6689. 

  6689. <a name="l06625"></a>06625   <span class="keyword">implicit none</span>
    6690. 

  6690. <a name="l06626"></a>06626 
    6691. 

  6691. <a name="l06627"></a>06627   <span class="keywordtype">real</span> a
    6692. 

  6692. <a name="l06628"></a>06628   <span class="keywordtype">real</span> absc
    6693. 

  6693. <a name="l06629"></a>06629   <span class="keywordtype">real</span> abserr
    6694. 

  6694. <a name="l06630"></a>06630   <span class="keywordtype">real</span> b
    6695. 

  6695. <a name="l06631"></a>06631   <span class="keywordtype">real</span> centr
    6696. 

  6696. <a name="l06632"></a>06632   <span class="keywordtype">real</span> dhlgth
    6697. 

  6697. <a name="l06633"></a>06633   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    6698. 

  6698. <a name="l06634"></a>06634   <span class="keywordtype">real</span> fc
    6699. 

  6699. <a name="l06635"></a>06635   <span class="keywordtype">real</span> fsum
    6700. 

  6700. <a name="l06636"></a>06636   <span class="keywordtype">real</span> fval1
    6701. 

  6701. <a name="l06637"></a>06637   <span class="keywordtype">real</span> fval2
    6702. 

  6702. <a name="l06638"></a>06638   <span class="keywordtype">real</span> fv1(10)
    6703. 

  6703. <a name="l06639"></a>06639   <span class="keywordtype">real</span> fv2(10)
    6704. 

  6704. <a name="l06640"></a>06640   <span class="keywordtype">real</span> hlgth
    6705. 

  6705. <a name="l06641"></a>06641   <span class="keywordtype">integer</span> j
    6706. 

  6706. <a name="l06642"></a>06642   <span class="keywordtype">integer</span> jtw
    6707. 

  6707. <a name="l06643"></a>06643   <span class="keywordtype">integer</span> jtwm1
    6708. 

  6708. <a name="l06644"></a>06644   <span class="keywordtype">real</span> resabs
    6709. 

  6709. <a name="l06645"></a>06645   <span class="keywordtype">real</span> resasc
    6710. 

  6710. <a name="l06646"></a>06646   <span class="keywordtype">real</span> resg
    6711. 

  6711. <a name="l06647"></a>06647   <span class="keywordtype">real</span> resk
    6712. 

  6712. <a name="l06648"></a>06648   <span class="keywordtype">real</span> reskh
    6713. 

  6713. <a name="l06649"></a>06649   <span class="keywordtype">real</span> result
    6714. 

  6714. <a name="l06650"></a>06650   <span class="keywordtype">real</span> wg(5)
    6715. 

  6715. <a name="l06651"></a>06651   <span class="keywordtype">real</span> wgk(11)
    6716. 

  6716. <a name="l06652"></a>06652   <span class="keywordtype">real</span> xgk(11)
    6717. 

  6717. <a name="l06653"></a>06653 <span class="comment">!</span>
    6718. 

  6718. <a name="l06654"></a>06654 <span class="comment">!           the abscissae and weights are given for the interval (-1,1).</span>
    6719. 

  6719. <a name="l06655"></a>06655 <span class="comment">!           because of symmetry only the positive abscissae and their</span>
    6720. 

  6720. <a name="l06656"></a>06656 <span class="comment">!           corresponding weights are given.</span>
    6721. 

  6721. <a name="l06657"></a>06657 <span class="comment">!</span>
    6722. 

  6722. <a name="l06658"></a>06658 <span class="comment">!           xgk    - abscissae of the 21-point Kronrod rule</span>
    6723. 

  6723. <a name="l06659"></a>06659 <span class="comment">!                    xgk(2), xgk(4), ...  abscissae of the 10-point</span>
    6724. 

  6724. <a name="l06660"></a>06660 <span class="comment">!                    Gauss rule</span>
    6725. 

  6725. <a name="l06661"></a>06661 <span class="comment">!                    xgk(1), xgk(3), ...  abscissae which are optimally</span>
    6726. 

  6726. <a name="l06662"></a>06662 <span class="comment">!                    added to the 10-point Gauss rule</span>
    6727. 

  6727. <a name="l06663"></a>06663 <span class="comment">!</span>
    6728. 

  6728. <a name="l06664"></a>06664 <span class="comment">!           wgk    - weights of the 21-point Kronrod rule</span>
    6729. 

  6729. <a name="l06665"></a>06665 <span class="comment">!</span>
    6730. 

  6730. <a name="l06666"></a>06666 <span class="comment">!           wg     - weights of the 10-point Gauss rule</span>
    6731. 

  6731. <a name="l06667"></a>06667 <span class="comment">!</span>
    6732. 

  6732. <a name="l06668"></a>06668   <span class="keyword">data</span> xgk(1),xgk(2),xgk(3),xgk(4),xgk(5),xgk(6),xgk(7),xgk(8), &amp;
    6733. 

  6733. <a name="l06669"></a>06669     xgk(9),xgk(10),xgk(11)/ &amp;
    6734. 

  6734. <a name="l06670"></a>06670        9.956571630258081e-01,     9.739065285171717e-01, &amp;
    6735. 

  6735. <a name="l06671"></a>06671        9.301574913557082e-01,     8.650633666889845e-01, &amp;
    6736. 

  6736. <a name="l06672"></a>06672        7.808177265864169e-01,     6.794095682990244e-01, &amp;
    6737. 

  6737. <a name="l06673"></a>06673        5.627571346686047e-01,     4.333953941292472e-01, &amp;
    6738. 

  6738. <a name="l06674"></a>06674        2.943928627014602e-01,     1.488743389816312e-01, &amp;
    6739. 

  6739. <a name="l06675"></a>06675        0.000000000000000e+00/
    6740. 

  6740. <a name="l06676"></a>06676 <span class="comment">!</span>
    6741. 

  6741. <a name="l06677"></a>06677   <span class="keyword">data</span> wgk(1),wgk(2),wgk(3),wgk(4),wgk(5),wgk(6),wgk(7),wgk(8), &amp;
    6742. 

  6742. <a name="l06678"></a>06678     wgk(9),wgk(10),wgk(11)/ &amp;
    6743. 

  6743. <a name="l06679"></a>06679        1.169463886737187e-02,     3.255816230796473e-02, &amp;
    6744. 

  6744. <a name="l06680"></a>06680        5.475589657435200e-02,     7.503967481091995e-02, &amp;
    6745. 

  6745. <a name="l06681"></a>06681        9.312545458369761e-02,     1.093871588022976e-01, &amp;
    6746. 

  6746. <a name="l06682"></a>06682        1.234919762620659e-01,     1.347092173114733e-01, &amp;
    6747. 

  6747. <a name="l06683"></a>06683        1.427759385770601e-01,     1.477391049013385e-01, &amp;
    6748. 

  6748. <a name="l06684"></a>06684        1.494455540029169e-01/
    6749. 

  6749. <a name="l06685"></a>06685 <span class="comment">!</span>
    6750. 

  6750. <a name="l06686"></a>06686   <span class="keyword">data</span> wg(1),wg(2),wg(3),wg(4),wg(5)/ &amp;
    6751. 

  6751. <a name="l06687"></a>06687        6.667134430868814e-02,     1.494513491505806e-01, &amp;
    6752. 

  6752. <a name="l06688"></a>06688        2.190863625159820e-01,     2.692667193099964e-01, &amp;
    6753. 

  6753. <a name="l06689"></a>06689        2.955242247147529e-01/
    6754. 

  6754. <a name="l06690"></a>06690 <span class="comment">!</span>
    6755. 

  6755. <a name="l06691"></a>06691 <span class="comment">!</span>
    6756. 

  6756. <a name="l06692"></a>06692 <span class="comment">!           list of major variables</span>
    6757. 

  6757. <a name="l06693"></a>06693 <span class="comment">!</span>
    6758. 

  6758. <a name="l06694"></a>06694 <span class="comment">!           centr  - mid point of the interval</span>
    6759. 

  6759. <a name="l06695"></a>06695 <span class="comment">!           hlgth  - half-length of the interval</span>
    6760. 

  6760. <a name="l06696"></a>06696 <span class="comment">!           absc   - abscissa</span>
    6761. 

  6761. <a name="l06697"></a>06697 <span class="comment">!           fval*  - function value</span>
    6762. 

  6762. <a name="l06698"></a>06698 <span class="comment">!           resg   - result of the 10-point Gauss formula</span>
    6763. 

  6763. <a name="l06699"></a>06699 <span class="comment">!           resk   - result of the 21-point Kronrod formula</span>
    6764. 

  6764. <a name="l06700"></a>06700 <span class="comment">!           reskh  - approximation to the mean value of f over (a,b),</span>
    6765. 

  6765. <a name="l06701"></a>06701 <span class="comment">!                    i.e. to i/(b-a)</span>
    6766. 

  6766. <a name="l06702"></a>06702 <span class="comment">!</span>
    6767. 

  6767. <a name="l06703"></a>06703   centr = 5.0e-01*(a+b)
    6768. 

  6768. <a name="l06704"></a>06704   hlgth = 5.0e-01*(b-a)
    6769. 

  6769. <a name="l06705"></a>06705   dhlgth = abs(hlgth)
    6770. 

  6770. <a name="l06706"></a>06706 <span class="comment">!</span>
    6771. 

  6771. <a name="l06707"></a>06707 <span class="comment">!  Compute the 21-point Kronrod approximation to the</span>
    6772. 

  6772. <a name="l06708"></a>06708 <span class="comment">!  integral, and estimate the absolute error.</span>
    6773. 

  6773. <a name="l06709"></a>06709 <span class="comment">!</span>
    6774. 

  6774. <a name="l06710"></a>06710   resg = 0.0e+00
    6775. 

  6775. <a name="l06711"></a>06711   fc = f(centr)
    6776. 

  6776. <a name="l06712"></a>06712   resk = wgk(11)*fc
    6777. 

  6777. <a name="l06713"></a>06713   resabs = abs(resk)
    6778. 

  6778. <a name="l06714"></a>06714 
    6779. 

  6779. <a name="l06715"></a>06715   <span class="keyword">do</span> j = 1, 5
    6780. 

  6780. <a name="l06716"></a>06716     jtw = 2*j
    6781. 

  6781. <a name="l06717"></a>06717     absc = hlgth*xgk(jtw)
    6782. 

  6782. <a name="l06718"></a>06718     fval1 = f(centr-absc)
    6783. 

  6783. <a name="l06719"></a>06719     fval2 = f(centr+absc)
    6784. 

  6784. <a name="l06720"></a>06720     fv1(jtw) = fval1
    6785. 

  6785. <a name="l06721"></a>06721     fv2(jtw) = fval2
    6786. 

  6786. <a name="l06722"></a>06722     fsum = fval1+fval2
    6787. 

  6787. <a name="l06723"></a>06723     resg = resg+wg(j)*fsum
    6788. 

  6788. <a name="l06724"></a>06724     resk = resk+wgk(jtw)*fsum
    6789. 

  6789. <a name="l06725"></a>06725     resabs = resabs+wgk(jtw)*(abs(fval1)+abs(fval2))
    6790. 

  6790. <a name="l06726"></a>06726   <span class="keyword">end do</span>
    6791. 

  6791. <a name="l06727"></a>06727 
    6792. 

  6792. <a name="l06728"></a>06728   <span class="keyword">do</span> j = 1, 5
    6793. 

  6793. <a name="l06729"></a>06729     jtwm1 = 2*j-1
    6794. 

  6794. <a name="l06730"></a>06730     absc = hlgth*xgk(jtwm1)
    6795. 

  6795. <a name="l06731"></a>06731     fval1 = f(centr-absc)
    6796. 

  6796. <a name="l06732"></a>06732     fval2 = f(centr+absc)
    6797. 

  6797. <a name="l06733"></a>06733     fv1(jtwm1) = fval1
    6798. 

  6798. <a name="l06734"></a>06734     fv2(jtwm1) = fval2
    6799. 

  6799. <a name="l06735"></a>06735     fsum = fval1+fval2
    6800. 

  6800. <a name="l06736"></a>06736     resk = resk+wgk(jtwm1)*fsum
    6801. 

  6801. <a name="l06737"></a>06737     resabs = resabs+wgk(jtwm1)*(abs(fval1)+abs(fval2))
    6802. 

  6802. <a name="l06738"></a>06738   <span class="keyword">end do</span>
    6803. 

  6803. <a name="l06739"></a>06739 
    6804. 

  6804. <a name="l06740"></a>06740   reskh = resk*5.0e-01
    6805. 

  6805. <a name="l06741"></a>06741   resasc = wgk(11)*abs(fc-reskh)
    6806. 

  6806. <a name="l06742"></a>06742 
    6807. 

  6807. <a name="l06743"></a>06743   <span class="keyword">do</span> j = 1, 10
    6808. 

  6808. <a name="l06744"></a>06744     resasc = resasc+wgk(j)*(abs(fv1(j)-reskh)+abs(fv2(j)-reskh))
    6809. 

  6809. <a name="l06745"></a>06745   <span class="keyword">end do</span>
    6810. 

  6810. <a name="l06746"></a>06746 
    6811. 

  6811. <a name="l06747"></a>06747   result = resk*hlgth
    6812. 

  6812. <a name="l06748"></a>06748   resabs = resabs*dhlgth
    6813. 

  6813. <a name="l06749"></a>06749   resasc = resasc*dhlgth
    6814. 

  6814. <a name="l06750"></a>06750   abserr = abs((resk-resg)*hlgth)
    6815. 

  6815. <a name="l06751"></a>06751 
    6816. 

  6816. <a name="l06752"></a>06752   <span class="keyword">if</span> ( resasc /= 0.0e+00.and.abserr /= 0.0e+00) <span class="keyword">then</span>
    6817. 

  6817. <a name="l06753"></a>06753     abserr = resasc*min ( 1.0e+00,(2.0e+02*abserr/resasc)**1.5e+00)
    6818. 

  6818. <a name="l06754"></a>06754   <span class="keyword">end if</span>
    6819. 

  6819. <a name="l06755"></a>06755 
    6820. 

  6820. <a name="l06756"></a>06756   <span class="keyword">if</span> ( resabs &gt; tiny ( resabs ) /(5.0e+01* epsilon ( resabs ) )) <span class="keyword">then</span>
    6821. 

  6821. <a name="l06757"></a>06757     abserr = max (( epsilon ( resabs ) *5.0e+01)*resabs,abserr)
    6822. 

  6822. <a name="l06758"></a>06758   <span class="keyword">end if</span>
    6823. 

  6823. <a name="l06759"></a>06759 
    6824. 

  6824. <a name="l06760"></a>06760   return
    6825. 

  6825. <a name="l06761"></a>06761 <span class="keyword">end</span>
    6826. 

  6826. <a name="l06762"></a><a class="code" href="quadpack_8f90.html#aded2e8dd2218fbd159b78c0e8975a4cd">06762</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#aded2e8dd2218fbd159b78c0e8975a4cd">qk31</a> ( f, a, b, result, abserr, resabs, resasc )
    6827. 

  6827. <a name="l06763"></a>06763 
    6828. 

  6828. <a name="l06764"></a>06764 <span class="comment">!*****************************************************************************80</span>
    6829. 

  6829. <a name="l06765"></a>06765 <span class="comment">!</span>
    6830. 

  6830. <a name="l06766"></a>06766 <span class="comment">!! QK31 carries out a 31 point Gauss-Kronrod quadrature rule.</span>
    6831. 

  6831. <a name="l06767"></a>06767 <span class="comment">!</span>
    6832. 

  6832. <a name="l06768"></a>06768 <span class="comment">!  Discussion:</span>
    6833. 

  6833. <a name="l06769"></a>06769 <span class="comment">!</span>
    6834. 

  6834. <a name="l06770"></a>06770 <span class="comment">!    This routine approximates</span>
    6835. 

  6835. <a name="l06771"></a>06771 <span class="comment">!      I = integral ( A &lt;= X &lt;= B ) F(X) dx</span>
    6836. 

  6836. <a name="l06772"></a>06772 <span class="comment">!    with an error estimate, and</span>
    6837. 

  6837. <a name="l06773"></a>06773 <span class="comment">!      J = integral ( A &lt;= X &lt;= B ) | F(X) | dx</span>
    6838. 

  6838. <a name="l06774"></a>06774 <span class="comment">!</span>
    6839. 

  6839. <a name="l06775"></a>06775 <span class="comment">!  Author:</span>
    6840. 

  6840. <a name="l06776"></a>06776 <span class="comment">!</span>
    6841. 

  6841. <a name="l06777"></a>06777 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    6842. 

  6842. <a name="l06778"></a>06778 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    6843. 

  6843. <a name="l06779"></a>06779 <span class="comment">!</span>
    6844. 

  6844. <a name="l06780"></a>06780 <span class="comment">!  Reference:</span>
    6845. 

  6845. <a name="l06781"></a>06781 <span class="comment">!</span>
    6846. 

  6846. <a name="l06782"></a>06782 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    6847. 

  6847. <a name="l06783"></a>06783 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    6848. 

  6848. <a name="l06784"></a>06784 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    6849. 

  6849. <a name="l06785"></a>06785 <span class="comment">!    Springer Verlag, 1983</span>
    6850. 

  6850. <a name="l06786"></a>06786 <span class="comment">!</span>
    6851. 

  6851. <a name="l06787"></a>06787 <span class="comment">!  Parameters:</span>
    6852. 

  6852. <a name="l06788"></a>06788 <span class="comment">!</span>
    6853. 

  6853. <a name="l06789"></a>06789 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    6854. 

  6854. <a name="l06790"></a>06790 <span class="comment">!      function f ( x )</span>
    6855. 

  6855. <a name="l06791"></a>06791 <span class="comment">!      real f</span>
    6856. 

  6856. <a name="l06792"></a>06792 <span class="comment">!      real x</span>
    6857. 

  6857. <a name="l06793"></a>06793 <span class="comment">!    which evaluates the integrand function.</span>
    6858. 

  6858. <a name="l06794"></a>06794 <span class="comment">!</span>
    6859. 

  6859. <a name="l06795"></a>06795 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    6860. 

  6860. <a name="l06796"></a>06796 <span class="comment">!</span>
    6861. 

  6861. <a name="l06797"></a>06797 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    6862. 

  6862. <a name="l06798"></a>06798 <span class="comment">!                       result is computed by applying the 31-point</span>
    6863. 

  6863. <a name="l06799"></a>06799 <span class="comment">!                       Gauss-Kronrod rule (resk), obtained by optimal</span>
    6864. 

  6864. <a name="l06800"></a>06800 <span class="comment">!                       addition of abscissae to the 15-point Gauss</span>
    6865. 

  6865. <a name="l06801"></a>06801 <span class="comment">!                       rule (resg).</span>
    6866. 

  6866. <a name="l06802"></a>06802 <span class="comment">!</span>
    6867. 

  6867. <a name="l06803"></a>06803 <span class="comment">!    Output, real ABSERR, an estimate of | I - RESULT |.</span>
    6868. 

  6868. <a name="l06804"></a>06804 <span class="comment">!</span>
    6869. 

  6869. <a name="l06805"></a>06805 <span class="comment">!    Output, real RESABS, approximation to the integral of the absolute</span>
    6870. 

  6870. <a name="l06806"></a>06806 <span class="comment">!    value of F.</span>
    6871. 

  6871. <a name="l06807"></a>06807 <span class="comment">!</span>
    6872. 

  6872. <a name="l06808"></a>06808 <span class="comment">!    Output, real RESASC, approximation to the integral | F-I/(B-A) | </span>
    6873. 

  6873. <a name="l06809"></a>06809 <span class="comment">!    over [A,B].</span>
    6874. 

  6874. <a name="l06810"></a>06810 <span class="comment">!</span>
    6875. 

  6875. <a name="l06811"></a>06811   <span class="keyword">implicit none</span>
    6876. 

  6876. <a name="l06812"></a>06812 
    6877. 

  6877. <a name="l06813"></a>06813   <span class="keywordtype">real</span> a
    6878. 

  6878. <a name="l06814"></a>06814   <span class="keywordtype">real</span> absc
    6879. 

  6879. <a name="l06815"></a>06815   <span class="keywordtype">real</span> abserr
    6880. 

  6880. <a name="l06816"></a>06816   <span class="keywordtype">real</span> b
    6881. 

  6881. <a name="l06817"></a>06817   <span class="keywordtype">real</span> centr
    6882. 

  6882. <a name="l06818"></a>06818   <span class="keywordtype">real</span> dhlgth
    6883. 

  6883. <a name="l06819"></a>06819   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    6884. 

  6884. <a name="l06820"></a>06820   <span class="keywordtype">real</span> fc
    6885. 

  6885. <a name="l06821"></a>06821   <span class="keywordtype">real</span> fsum
    6886. 

  6886. <a name="l06822"></a>06822   <span class="keywordtype">real</span> fval1
    6887. 

  6887. <a name="l06823"></a>06823   <span class="keywordtype">real</span> fval2
    6888. 

  6888. <a name="l06824"></a>06824   <span class="keywordtype">real</span> fv1(15)
    6889. 

  6889. <a name="l06825"></a>06825   <span class="keywordtype">real</span> fv2(15)
    6890. 

  6890. <a name="l06826"></a>06826   <span class="keywordtype">real</span> hlgth
    6891. 

  6891. <a name="l06827"></a>06827   <span class="keywordtype">integer</span> j
    6892. 

  6892. <a name="l06828"></a>06828   <span class="keywordtype">integer</span> jtw
    6893. 

  6893. <a name="l06829"></a>06829   <span class="keywordtype">integer</span> jtwm1
    6894. 

  6894. <a name="l06830"></a>06830   <span class="keywordtype">real</span> resabs
    6895. 

  6895. <a name="l06831"></a>06831   <span class="keywordtype">real</span> resasc
    6896. 

  6896. <a name="l06832"></a>06832   <span class="keywordtype">real</span> resg
    6897. 

  6897. <a name="l06833"></a>06833   <span class="keywordtype">real</span> resk
    6898. 

  6898. <a name="l06834"></a>06834   <span class="keywordtype">real</span> reskh
    6899. 

  6899. <a name="l06835"></a>06835   <span class="keywordtype">real</span> result
    6900. 

  6900. <a name="l06836"></a>06836   <span class="keywordtype">real</span> wg(8)
    6901. 

  6901. <a name="l06837"></a>06837   <span class="keywordtype">real</span> wgk(16)
    6902. 

  6902. <a name="l06838"></a>06838   <span class="keywordtype">real</span> xgk(16)
    6903. 

  6903. <a name="l06839"></a>06839 <span class="comment">!</span>
    6904. 

  6904. <a name="l06840"></a>06840 <span class="comment">!           the abscissae and weights are given for the interval (-1,1).</span>
    6905. 

  6905. <a name="l06841"></a>06841 <span class="comment">!           because of symmetry only the positive abscissae and their</span>
    6906. 

  6906. <a name="l06842"></a>06842 <span class="comment">!           corresponding weights are given.</span>
    6907. 

  6907. <a name="l06843"></a>06843 <span class="comment">!</span>
    6908. 

  6908. <a name="l06844"></a>06844 <span class="comment">!           xgk    - abscissae of the 31-point Kronrod rule</span>
    6909. 

  6909. <a name="l06845"></a>06845 <span class="comment">!                    xgk(2), xgk(4), ...  abscissae of the 15-point</span>
    6910. 

  6910. <a name="l06846"></a>06846 <span class="comment">!                    Gauss rule</span>
    6911. 

  6911. <a name="l06847"></a>06847 <span class="comment">!                    xgk(1), xgk(3), ...  abscissae which are optimally</span>
    6912. 

  6912. <a name="l06848"></a>06848 <span class="comment">!                    added to the 15-point Gauss rule</span>
    6913. 

  6913. <a name="l06849"></a>06849 <span class="comment">!</span>
    6914. 

  6914. <a name="l06850"></a>06850 <span class="comment">!           wgk    - weights of the 31-point Kronrod rule</span>
    6915. 

  6915. <a name="l06851"></a>06851 <span class="comment">!</span>
    6916. 

  6916. <a name="l06852"></a>06852 <span class="comment">!           wg     - weights of the 15-point Gauss rule</span>
    6917. 

  6917. <a name="l06853"></a>06853 <span class="comment">!</span>
    6918. 

  6918. <a name="l06854"></a>06854   <span class="keyword">data</span> xgk(1),xgk(2),xgk(3),xgk(4),xgk(5),xgk(6),xgk(7),xgk(8), &amp;
    6919. 

  6919. <a name="l06855"></a>06855     xgk(9),xgk(10),xgk(11),xgk(12),xgk(13),xgk(14),xgk(15),xgk(16)/ &amp;
    6920. 

  6920. <a name="l06856"></a>06856        9.980022986933971e-01,   9.879925180204854e-01, &amp;
    6921. 

  6921. <a name="l06857"></a>06857        9.677390756791391e-01,   9.372733924007059e-01, &amp;
    6922. 

  6922. <a name="l06858"></a>06858        8.972645323440819e-01,   8.482065834104272e-01, &amp;
    6923. 

  6923. <a name="l06859"></a>06859        7.904185014424659e-01,   7.244177313601700e-01, &amp;
    6924. 

  6924. <a name="l06860"></a>06860        6.509967412974170e-01,   5.709721726085388e-01, &amp;
    6925. 

  6925. <a name="l06861"></a>06861        4.850818636402397e-01,   3.941513470775634e-01, &amp;
    6926. 

  6926. <a name="l06862"></a>06862        2.991800071531688e-01,   2.011940939974345e-01, &amp;
    6927. 

  6927. <a name="l06863"></a>06863        1.011420669187175e-01,   0.0e+00               /
    6928. 

  6928. <a name="l06864"></a>06864   <span class="keyword">data</span> wgk(1),wgk(2),wgk(3),wgk(4),wgk(5),wgk(6),wgk(7),wgk(8), &amp;
    6929. 

  6929. <a name="l06865"></a>06865     wgk(9),wgk(10),wgk(11),wgk(12),wgk(13),wgk(14),wgk(15),wgk(16)/ &amp;
    6930. 

  6930. <a name="l06866"></a>06866        5.377479872923349e-03,   1.500794732931612e-02, &amp;
    6931. 

  6931. <a name="l06867"></a>06867        2.546084732671532e-02,   3.534636079137585e-02, &amp;
    6932. 

  6932. <a name="l06868"></a>06868        4.458975132476488e-02,   5.348152469092809e-02, &amp;
    6933. 

  6933. <a name="l06869"></a>06869        6.200956780067064e-02,   6.985412131872826e-02, &amp;
    6934. 

  6934. <a name="l06870"></a>06870        7.684968075772038e-02,   8.308050282313302e-02, &amp;
    6935. 

  6935. <a name="l06871"></a>06871        8.856444305621177e-02,   9.312659817082532e-02, &amp;
    6936. 

  6936. <a name="l06872"></a>06872        9.664272698362368e-02,   9.917359872179196e-02, &amp;
    6937. 

  6937. <a name="l06873"></a>06873        1.007698455238756e-01,   1.013300070147915e-01/
    6938. 

  6938. <a name="l06874"></a>06874   <span class="keyword">data</span> wg(1),wg(2),wg(3),wg(4),wg(5),wg(6),wg(7),wg(8)/ &amp;
    6939. 

  6939. <a name="l06875"></a>06875        3.075324199611727e-02,   7.036604748810812e-02, &amp;
    6940. 

  6940. <a name="l06876"></a>06876        1.071592204671719e-01,   1.395706779261543e-01, &amp;
    6941. 

  6941. <a name="l06877"></a>06877        1.662692058169939e-01,   1.861610000155622e-01, &amp;
    6942. 

  6942. <a name="l06878"></a>06878        1.984314853271116e-01,   2.025782419255613e-01/
    6943. 

  6943. <a name="l06879"></a>06879 <span class="comment">!</span>
    6944. 

  6944. <a name="l06880"></a>06880 <span class="comment">!</span>
    6945. 

  6945. <a name="l06881"></a>06881 <span class="comment">!           list of major variables</span>
    6946. 

  6946. <a name="l06882"></a>06882 <span class="comment">!</span>
    6947. 

  6947. <a name="l06883"></a>06883 <span class="comment">!           centr  - mid point of the interval</span>
    6948. 

  6948. <a name="l06884"></a>06884 <span class="comment">!           hlgth  - half-length of the interval</span>
    6949. 

  6949. <a name="l06885"></a>06885 <span class="comment">!           absc   - abscissa</span>
    6950. 

  6950. <a name="l06886"></a>06886 <span class="comment">!           fval*  - function value</span>
    6951. 

  6951. <a name="l06887"></a>06887 <span class="comment">!           resg   - result of the 15-point Gauss formula</span>
    6952. 

  6952. <a name="l06888"></a>06888 <span class="comment">!           resk   - result of the 31-point Kronrod formula</span>
    6953. 

  6953. <a name="l06889"></a>06889 <span class="comment">!           reskh  - approximation to the mean value of f over (a,b),</span>
    6954. 

  6954. <a name="l06890"></a>06890 <span class="comment">!                    i.e. to i/(b-a)</span>
    6955. 

  6955. <a name="l06891"></a>06891 <span class="comment">!</span>
    6956. 

  6956. <a name="l06892"></a>06892   centr = 5.0e-01*(a+b)
    6957. 

  6957. <a name="l06893"></a>06893   hlgth = 5.0e-01*(b-a)
    6958. 

  6958. <a name="l06894"></a>06894   dhlgth = abs(hlgth)
    6959. 

  6959. <a name="l06895"></a>06895 <span class="comment">!</span>
    6960. 

  6960. <a name="l06896"></a>06896 <span class="comment">!  Compute the 31-point Kronrod approximation to the integral,</span>
    6961. 

  6961. <a name="l06897"></a>06897 <span class="comment">!  and estimate the absolute error.</span>
    6962. 

  6962. <a name="l06898"></a>06898 <span class="comment">!</span>
    6963. 

  6963. <a name="l06899"></a>06899   fc = f(centr)
    6964. 

  6964. <a name="l06900"></a>06900   resg = wg(8)*fc
    6965. 

  6965. <a name="l06901"></a>06901   resk = wgk(16)*fc
    6966. 

  6966. <a name="l06902"></a>06902   resabs = abs(resk)
    6967. 

  6967. <a name="l06903"></a>06903 
    6968. 

  6968. <a name="l06904"></a>06904   <span class="keyword">do</span> j = 1, 7
    6969. 

  6969. <a name="l06905"></a>06905     jtw = j*2
    6970. 

  6970. <a name="l06906"></a>06906     absc = hlgth*xgk(jtw)
    6971. 

  6971. <a name="l06907"></a>06907     fval1 = f(centr-absc)
    6972. 

  6972. <a name="l06908"></a>06908     fval2 = f(centr+absc)
    6973. 

  6973. <a name="l06909"></a>06909     fv1(jtw) = fval1
    6974. 

  6974. <a name="l06910"></a>06910     fv2(jtw) = fval2
    6975. 

  6975. <a name="l06911"></a>06911     fsum = fval1+fval2
    6976. 

  6976. <a name="l06912"></a>06912     resg = resg+wg(j)*fsum
    6977. 

  6977. <a name="l06913"></a>06913     resk = resk+wgk(jtw)*fsum
    6978. 

  6978. <a name="l06914"></a>06914     resabs = resabs+wgk(jtw)*(abs(fval1)+abs(fval2))
    6979. 

  6979. <a name="l06915"></a>06915   <span class="keyword">end do</span>
    6980. 

  6980. <a name="l06916"></a>06916 
    6981. 

  6981. <a name="l06917"></a>06917   <span class="keyword">do</span> j = 1, 8
    6982. 

  6982. <a name="l06918"></a>06918     jtwm1 = j*2-1
    6983. 

  6983. <a name="l06919"></a>06919     absc = hlgth*xgk(jtwm1)
    6984. 

  6984. <a name="l06920"></a>06920     fval1 = f(centr-absc)
    6985. 

  6985. <a name="l06921"></a>06921     fval2 = f(centr+absc)
    6986. 

  6986. <a name="l06922"></a>06922     fv1(jtwm1) = fval1
    6987. 

  6987. <a name="l06923"></a>06923     fv2(jtwm1) = fval2
    6988. 

  6988. <a name="l06924"></a>06924     fsum = fval1+fval2
    6989. 

  6989. <a name="l06925"></a>06925     resk = resk+wgk(jtwm1)*fsum
    6990. 

  6990. <a name="l06926"></a>06926     resabs = resabs+wgk(jtwm1)*(abs(fval1)+abs(fval2))
    6991. 

  6991. <a name="l06927"></a>06927   <span class="keyword">end do</span>
    6992. 

  6992. <a name="l06928"></a>06928 
    6993. 

  6993. <a name="l06929"></a>06929   reskh = resk*5.0e-01
    6994. 

  6994. <a name="l06930"></a>06930   resasc = wgk(16)*abs(fc-reskh)
    6995. 

  6995. <a name="l06931"></a>06931 
    6996. 

  6996. <a name="l06932"></a>06932   <span class="keyword">do</span> j = 1, 15
    6997. 

  6997. <a name="l06933"></a>06933     resasc = resasc+wgk(j)*(abs(fv1(j)-reskh)+abs(fv2(j)-reskh))
    6998. 

  6998. <a name="l06934"></a>06934  <span class="keyword">end do</span>
    6999. 

  6999. <a name="l06935"></a>06935 
    7000. 

  7000. <a name="l06936"></a>06936   result = resk*hlgth
    7001. 

  7001. <a name="l06937"></a>06937   resabs = resabs*dhlgth
    7002. 

  7002. <a name="l06938"></a>06938   resasc = resasc*dhlgth
    7003. 

  7003. <a name="l06939"></a>06939   abserr = abs((resk-resg)*hlgth)
    7004. 

  7004. <a name="l06940"></a>06940 
    7005. 

  7005. <a name="l06941"></a>06941   <span class="keyword">if</span> ( resasc /= 0.0e+00.and.abserr /= 0.0e+00) &amp;
    7006. 

  7006. <a name="l06942"></a>06942     abserr = resasc*min ( 1.0e+00,(2.0e+02*abserr/resasc)**1.5e+00)
    7007. 

  7007. <a name="l06943"></a>06943 
    7008. 

  7008. <a name="l06944"></a>06944   <span class="keyword">if</span> ( resabs &gt; tiny ( resabs ) /(5.0e+01* epsilon ( resabs ) )) <span class="keyword">then</span>
    7009. 

  7009. <a name="l06945"></a>06945     abserr = max (( epsilon ( resabs ) *5.0e+01)*resabs,abserr)
    7010. 

  7010. <a name="l06946"></a>06946   <span class="keyword">end if</span>
    7011. 

  7011. <a name="l06947"></a>06947 
    7012. 

  7012. <a name="l06948"></a>06948   return
    7013. 

  7013. <a name="l06949"></a>06949 <span class="keyword">end</span>
    7014. 

  7014. <a name="l06950"></a><a class="code" href="quadpack_8f90.html#aface4edf24710a0b323f5aaeb6bdec34">06950</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#aface4edf24710a0b323f5aaeb6bdec34">qk41</a> ( f, a, b, result, abserr, resabs, resasc )
    7015. 

  7015. <a name="l06951"></a>06951 
    7016. 

  7016. <a name="l06952"></a>06952 <span class="comment">!*****************************************************************************80</span>
    7017. 

  7017. <a name="l06953"></a>06953 <span class="comment">!</span>
    7018. 

  7018. <a name="l06954"></a>06954 <span class="comment">!! QK41 carries out a 41 point Gauss-Kronrod quadrature rule.</span>
    7019. 

  7019. <a name="l06955"></a>06955 <span class="comment">!</span>
    7020. 

  7020. <a name="l06956"></a>06956 <span class="comment">!  Discussion:</span>
    7021. 

  7021. <a name="l06957"></a>06957 <span class="comment">!</span>
    7022. 

  7022. <a name="l06958"></a>06958 <span class="comment">!    This routine approximates</span>
    7023. 

  7023. <a name="l06959"></a>06959 <span class="comment">!      I = integral ( A &lt;= X &lt;= B ) F(X) dx</span>
    7024. 

  7024. <a name="l06960"></a>06960 <span class="comment">!    with an error estimate, and</span>
    7025. 

  7025. <a name="l06961"></a>06961 <span class="comment">!      J = integral ( A &lt;= X &lt;= B ) | F(X) | dx</span>
    7026. 

  7026. <a name="l06962"></a>06962 <span class="comment">!</span>
    7027. 

  7027. <a name="l06963"></a>06963 <span class="comment">!  Author:</span>
    7028. 

  7028. <a name="l06964"></a>06964 <span class="comment">!</span>
    7029. 

  7029. <a name="l06965"></a>06965 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    7030. 

  7030. <a name="l06966"></a>06966 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    7031. 

  7031. <a name="l06967"></a>06967 <span class="comment">!</span>
    7032. 

  7032. <a name="l06968"></a>06968 <span class="comment">!  Reference:</span>
    7033. 

  7033. <a name="l06969"></a>06969 <span class="comment">!</span>
    7034. 

  7034. <a name="l06970"></a>06970 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    7035. 

  7035. <a name="l06971"></a>06971 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    7036. 

  7036. <a name="l06972"></a>06972 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    7037. 

  7037. <a name="l06973"></a>06973 <span class="comment">!    Springer Verlag, 1983</span>
    7038. 

  7038. <a name="l06974"></a>06974 <span class="comment">!</span>
    7039. 

  7039. <a name="l06975"></a>06975 <span class="comment">!  Parameters:</span>
    7040. 

  7040. <a name="l06976"></a>06976 <span class="comment">!</span>
    7041. 

  7041. <a name="l06977"></a>06977 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    7042. 

  7042. <a name="l06978"></a>06978 <span class="comment">!      function f ( x )</span>
    7043. 

  7043. <a name="l06979"></a>06979 <span class="comment">!      real f</span>
    7044. 

  7044. <a name="l06980"></a>06980 <span class="comment">!      real x</span>
    7045. 

  7045. <a name="l06981"></a>06981 <span class="comment">!    which evaluates the integrand function.</span>
    7046. 

  7046. <a name="l06982"></a>06982 <span class="comment">!</span>
    7047. 

  7047. <a name="l06983"></a>06983 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    7048. 

  7048. <a name="l06984"></a>06984 <span class="comment">!</span>
    7049. 

  7049. <a name="l06985"></a>06985 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    7050. 

  7050. <a name="l06986"></a>06986 <span class="comment">!                       result is computed by applying the 41-point</span>
    7051. 

  7051. <a name="l06987"></a>06987 <span class="comment">!                       Gauss-Kronrod rule (resk) obtained by optimal</span>
    7052. 

  7052. <a name="l06988"></a>06988 <span class="comment">!                       addition of abscissae to the 20-point Gauss</span>
    7053. 

  7053. <a name="l06989"></a>06989 <span class="comment">!                       rule (resg).</span>
    7054. 

  7054. <a name="l06990"></a>06990 <span class="comment">!</span>
    7055. 

  7055. <a name="l06991"></a>06991 <span class="comment">!    Output, real ABSERR, an estimate of | I - RESULT |.</span>
    7056. 

  7056. <a name="l06992"></a>06992 <span class="comment">!</span>
    7057. 

  7057. <a name="l06993"></a>06993 <span class="comment">!    Output, real RESABS, approximation to the integral of the absolute</span>
    7058. 

  7058. <a name="l06994"></a>06994 <span class="comment">!    value of F.</span>
    7059. 

  7059. <a name="l06995"></a>06995 <span class="comment">!</span>
    7060. 

  7060. <a name="l06996"></a>06996 <span class="comment">!    Output, real RESASC, approximation to the integral | F-I/(B-A) | </span>
    7061. 

  7061. <a name="l06997"></a>06997 <span class="comment">!    over [A,B].</span>
    7062. 

  7062. <a name="l06998"></a>06998 <span class="comment">!</span>
    7063. 

  7063. <a name="l06999"></a>06999 <span class="comment">!  Local Parameters:</span>
    7064. 

  7064. <a name="l07000"></a>07000 <span class="comment">!</span>
    7065. 

  7065. <a name="l07001"></a>07001 <span class="comment">!           centr  - mid point of the interval</span>
    7066. 

  7066. <a name="l07002"></a>07002 <span class="comment">!           hlgth  - half-length of the interval</span>
    7067. 

  7067. <a name="l07003"></a>07003 <span class="comment">!           absc   - abscissa</span>
    7068. 

  7068. <a name="l07004"></a>07004 <span class="comment">!           fval*  - function value</span>
    7069. 

  7069. <a name="l07005"></a>07005 <span class="comment">!           resg   - result of the 20-point Gauss formula</span>
    7070. 

  7070. <a name="l07006"></a>07006 <span class="comment">!           resk   - result of the 41-point Kronrod formula</span>
    7071. 

  7071. <a name="l07007"></a>07007 <span class="comment">!           reskh  - approximation to mean value of f over (a,b), i.e.</span>
    7072. 

  7072. <a name="l07008"></a>07008 <span class="comment">!                    to i/(b-a)</span>
    7073. 

  7073. <a name="l07009"></a>07009 <span class="comment">!</span>
    7074. 

  7074. <a name="l07010"></a>07010   <span class="keyword">implicit none</span>
    7075. 

  7075. <a name="l07011"></a>07011 
    7076. 

  7076. <a name="l07012"></a>07012   <span class="keywordtype">real</span> a
    7077. 

  7077. <a name="l07013"></a>07013   <span class="keywordtype">real</span> absc
    7078. 

  7078. <a name="l07014"></a>07014   <span class="keywordtype">real</span> abserr
    7079. 

  7079. <a name="l07015"></a>07015   <span class="keywordtype">real</span> b
    7080. 

  7080. <a name="l07016"></a>07016   <span class="keywordtype">real</span> centr
    7081. 

  7081. <a name="l07017"></a>07017   <span class="keywordtype">real</span> dhlgth
    7082. 

  7082. <a name="l07018"></a>07018   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    7083. 

  7083. <a name="l07019"></a>07019   <span class="keywordtype">real</span> fc
    7084. 

  7084. <a name="l07020"></a>07020   <span class="keywordtype">real</span> fsum
    7085. 

  7085. <a name="l07021"></a>07021   <span class="keywordtype">real</span> fval1
    7086. 

  7086. <a name="l07022"></a>07022   <span class="keywordtype">real</span> fval2
    7087. 

  7087. <a name="l07023"></a>07023   <span class="keywordtype">real</span> fv1(20)
    7088. 

  7088. <a name="l07024"></a>07024   <span class="keywordtype">real</span> fv2(20)
    7089. 

  7089. <a name="l07025"></a>07025   <span class="keywordtype">real</span> hlgth
    7090. 

  7090. <a name="l07026"></a>07026   <span class="keywordtype">integer</span> j
    7091. 

  7091. <a name="l07027"></a>07027   <span class="keywordtype">integer</span> jtw
    7092. 

  7092. <a name="l07028"></a>07028   <span class="keywordtype">integer</span> jtwm1
    7093. 

  7093. <a name="l07029"></a>07029   <span class="keywordtype">real</span> resabs
    7094. 

  7094. <a name="l07030"></a>07030   <span class="keywordtype">real</span> resasc
    7095. 

  7095. <a name="l07031"></a>07031   <span class="keywordtype">real</span> resg
    7096. 

  7096. <a name="l07032"></a>07032   <span class="keywordtype">real</span> resk
    7097. 

  7097. <a name="l07033"></a>07033   <span class="keywordtype">real</span> reskh
    7098. 

  7098. <a name="l07034"></a>07034   <span class="keywordtype">real</span> result
    7099. 

  7099. <a name="l07035"></a>07035   <span class="keywordtype">real</span> wg(10)
    7100. 

  7100. <a name="l07036"></a>07036   <span class="keywordtype">real</span> wgk(21)
    7101. 

  7101. <a name="l07037"></a>07037   <span class="keywordtype">real</span> xgk(21)
    7102. 

  7102. <a name="l07038"></a>07038 <span class="comment">!</span>
    7103. 

  7103. <a name="l07039"></a>07039 <span class="comment">!           the abscissae and weights are given for the interval (-1,1).</span>
    7104. 

  7104. <a name="l07040"></a>07040 <span class="comment">!           because of symmetry only the positive abscissae and their</span>
    7105. 

  7105. <a name="l07041"></a>07041 <span class="comment">!           corresponding weights are given.</span>
    7106. 

  7106. <a name="l07042"></a>07042 <span class="comment">!</span>
    7107. 

  7107. <a name="l07043"></a>07043 <span class="comment">!           xgk    - abscissae of the 41-point Gauss-Kronrod rule</span>
    7108. 

  7108. <a name="l07044"></a>07044 <span class="comment">!                    xgk(2), xgk(4), ...  abscissae of the 20-point</span>
    7109. 

  7109. <a name="l07045"></a>07045 <span class="comment">!                    Gauss rule</span>
    7110. 

  7110. <a name="l07046"></a>07046 <span class="comment">!                    xgk(1), xgk(3), ...  abscissae which are optimally</span>
    7111. 

  7111. <a name="l07047"></a>07047 <span class="comment">!                    added to the 20-point Gauss rule</span>
    7112. 

  7112. <a name="l07048"></a>07048 <span class="comment">!</span>
    7113. 

  7113. <a name="l07049"></a>07049 <span class="comment">!           wgk    - weights of the 41-point Gauss-Kronrod rule</span>
    7114. 

  7114. <a name="l07050"></a>07050 <span class="comment">!</span>
    7115. 

  7115. <a name="l07051"></a>07051 <span class="comment">!           wg     - weights of the 20-point Gauss rule</span>
    7116. 

  7116. <a name="l07052"></a>07052 <span class="comment">!</span>
    7117. 

  7117. <a name="l07053"></a>07053   <span class="keyword">data</span> xgk(1),xgk(2),xgk(3),xgk(4),xgk(5),xgk(6),xgk(7),xgk(8), &amp;
    7118. 

  7118. <a name="l07054"></a>07054     xgk(9),xgk(10),xgk(11),xgk(12),xgk(13),xgk(14),xgk(15),xgk(16), &amp;
    7119. 

  7119. <a name="l07055"></a>07055     xgk(17),xgk(18),xgk(19),xgk(20),xgk(21)/ &amp;
    7120. 

  7120. <a name="l07056"></a>07056        9.988590315882777e-01,   9.931285991850949e-01, &amp;
    7121. 

  7121. <a name="l07057"></a>07057        9.815078774502503e-01,   9.639719272779138e-01, &amp;
    7122. 

  7122. <a name="l07058"></a>07058        9.408226338317548e-01,   9.122344282513259e-01, &amp;
    7123. 

  7123. <a name="l07059"></a>07059        8.782768112522820e-01,   8.391169718222188e-01, &amp;
    7124. 

  7124. <a name="l07060"></a>07060        7.950414288375512e-01,   7.463319064601508e-01, &amp;
    7125. 

  7125. <a name="l07061"></a>07061        6.932376563347514e-01,   6.360536807265150e-01, &amp;
    7126. 

  7126. <a name="l07062"></a>07062        5.751404468197103e-01,   5.108670019508271e-01, &amp;
    7127. 

  7127. <a name="l07063"></a>07063        4.435931752387251e-01,   3.737060887154196e-01, &amp;
    7128. 

  7128. <a name="l07064"></a>07064        3.016278681149130e-01,   2.277858511416451e-01, &amp;
    7129. 

  7129. <a name="l07065"></a>07065        1.526054652409227e-01,   7.652652113349733e-02, &amp;
    7130. 

  7130. <a name="l07066"></a>07066        0.0e+00               /
    7131. 

  7131. <a name="l07067"></a>07067   <span class="keyword">data</span> wgk(1),wgk(2),wgk(3),wgk(4),wgk(5),wgk(6),wgk(7),wgk(8), &amp;
    7132. 

  7132. <a name="l07068"></a>07068     wgk(9),wgk(10),wgk(11),wgk(12),wgk(13),wgk(14),wgk(15),wgk(16), &amp;
    7133. 

  7133. <a name="l07069"></a>07069     wgk(17),wgk(18),wgk(19),wgk(20),wgk(21)/ &amp;
    7134. 

  7134. <a name="l07070"></a>07070        3.073583718520532e-03,   8.600269855642942e-03, &amp;
    7135. 

  7135. <a name="l07071"></a>07071        1.462616925697125e-02,   2.038837346126652e-02, &amp;
    7136. 

  7136. <a name="l07072"></a>07072        2.588213360495116e-02,   3.128730677703280e-02, &amp;
    7137. 

  7137. <a name="l07073"></a>07073        3.660016975820080e-02,   4.166887332797369e-02, &amp;
    7138. 

  7138. <a name="l07074"></a>07074        4.643482186749767e-02,   5.094457392372869e-02, &amp;
    7139. 

  7139. <a name="l07075"></a>07075        5.519510534828599e-02,   5.911140088063957e-02, &amp;
    7140. 

  7140. <a name="l07076"></a>07076        6.265323755478117e-02,   6.583459713361842e-02, &amp;
    7141. 

  7141. <a name="l07077"></a>07077        6.864867292852162e-02,   7.105442355344407e-02, &amp;
    7142. 

  7142. <a name="l07078"></a>07078        7.303069033278667e-02,   7.458287540049919e-02, &amp;
    7143. 

  7143. <a name="l07079"></a>07079        7.570449768455667e-02,   7.637786767208074e-02, &amp;
    7144. 

  7144. <a name="l07080"></a>07080        7.660071191799966e-02/
    7145. 

  7145. <a name="l07081"></a>07081   <span class="keyword">data</span> wg(1),wg(2),wg(3),wg(4),wg(5),wg(6),wg(7),wg(8),wg(9),wg(10)/ &amp;
    7146. 

  7146. <a name="l07082"></a>07082        1.761400713915212e-02,   4.060142980038694e-02, &amp;
    7147. 

  7147. <a name="l07083"></a>07083        6.267204833410906e-02,   8.327674157670475e-02, &amp;
    7148. 

  7148. <a name="l07084"></a>07084        1.019301198172404e-01,   1.181945319615184e-01, &amp;
    7149. 

  7149. <a name="l07085"></a>07085        1.316886384491766e-01,   1.420961093183821e-01, &amp;
    7150. 

  7150. <a name="l07086"></a>07086        1.491729864726037e-01,   1.527533871307259e-01/
    7151. 

  7151. <a name="l07087"></a>07087 <span class="comment">!</span>
    7152. 

  7152. <a name="l07088"></a>07088   centr = 5.0e-01*(a+b)
    7153. 

  7153. <a name="l07089"></a>07089   hlgth = 5.0e-01*(b-a)
    7154. 

  7154. <a name="l07090"></a>07090   dhlgth = abs(hlgth)
    7155. 

  7155. <a name="l07091"></a>07091 <span class="comment">!</span>
    7156. 

  7156. <a name="l07092"></a>07092 <span class="comment">!  Compute 41-point Gauss-Kronrod approximation to the</span>
    7157. 

  7157. <a name="l07093"></a>07093 <span class="comment">!  the integral, and estimate the absolute error.</span>
    7158. 

  7158. <a name="l07094"></a>07094 <span class="comment">!</span>
    7159. 

  7159. <a name="l07095"></a>07095   resg = 0.0e+00
    7160. 

  7160. <a name="l07096"></a>07096   fc = f(centr)
    7161. 

  7161. <a name="l07097"></a>07097   resk = wgk(21)*fc
    7162. 

  7162. <a name="l07098"></a>07098   resabs = abs(resk)
    7163. 

  7163. <a name="l07099"></a>07099 
    7164. 

  7164. <a name="l07100"></a>07100   <span class="keyword">do</span> j = 1, 10
    7165. 

  7165. <a name="l07101"></a>07101     jtw = j*2
    7166. 

  7166. <a name="l07102"></a>07102     absc = hlgth*xgk(jtw)
    7167. 

  7167. <a name="l07103"></a>07103     fval1 = f(centr-absc)
    7168. 

  7168. <a name="l07104"></a>07104     fval2 = f(centr+absc)
    7169. 

  7169. <a name="l07105"></a>07105     fv1(jtw) = fval1
    7170. 

  7170. <a name="l07106"></a>07106     fv2(jtw) = fval2
    7171. 

  7171. <a name="l07107"></a>07107     fsum = fval1+fval2
    7172. 

  7172. <a name="l07108"></a>07108     resg = resg+wg(j)*fsum
    7173. 

  7173. <a name="l07109"></a>07109     resk = resk+wgk(jtw)*fsum
    7174. 

  7174. <a name="l07110"></a>07110     resabs = resabs+wgk(jtw)*(abs(fval1)+abs(fval2))
    7175. 

  7175. <a name="l07111"></a>07111   <span class="keyword">end do</span>
    7176. 

  7176. <a name="l07112"></a>07112 
    7177. 

  7177. <a name="l07113"></a>07113   <span class="keyword">do</span> j = 1, 10
    7178. 

  7178. <a name="l07114"></a>07114     jtwm1 = j*2-1
    7179. 

  7179. <a name="l07115"></a>07115     absc = hlgth*xgk(jtwm1)
    7180. 

  7180. <a name="l07116"></a>07116     fval1 = f(centr-absc)
    7181. 

  7181. <a name="l07117"></a>07117     fval2 = f(centr+absc)
    7182. 

  7182. <a name="l07118"></a>07118     fv1(jtwm1) = fval1
    7183. 

  7183. <a name="l07119"></a>07119     fv2(jtwm1) = fval2
    7184. 

  7184. <a name="l07120"></a>07120     fsum = fval1+fval2
    7185. 

  7185. <a name="l07121"></a>07121     resk = resk+wgk(jtwm1)*fsum
    7186. 

  7186. <a name="l07122"></a>07122     resabs = resabs+wgk(jtwm1)*(abs(fval1)+abs(fval2))
    7187. 

  7187. <a name="l07123"></a>07123   <span class="keyword">end do</span>
    7188. 

  7188. <a name="l07124"></a>07124 
    7189. 

  7189. <a name="l07125"></a>07125   reskh = resk*5.0e-01
    7190. 

  7190. <a name="l07126"></a>07126   resasc = wgk(21)*abs(fc-reskh)
    7191. 

  7191. <a name="l07127"></a>07127 
    7192. 

  7192. <a name="l07128"></a>07128   <span class="keyword">do</span> j = 1, 20
    7193. 

  7193. <a name="l07129"></a>07129     resasc = resasc+wgk(j)*(abs(fv1(j)-reskh)+abs(fv2(j)-reskh))
    7194. 

  7194. <a name="l07130"></a>07130   <span class="keyword">end do</span>
    7195. 

  7195. <a name="l07131"></a>07131 
    7196. 

  7196. <a name="l07132"></a>07132   result = resk*hlgth
    7197. 

  7197. <a name="l07133"></a>07133   resabs = resabs*dhlgth
    7198. 

  7198. <a name="l07134"></a>07134   resasc = resasc*dhlgth
    7199. 

  7199. <a name="l07135"></a>07135   abserr = abs((resk-resg)*hlgth)
    7200. 

  7200. <a name="l07136"></a>07136 
    7201. 

  7201. <a name="l07137"></a>07137   <span class="keyword">if</span> ( resasc /= 0.0e+00.and.abserr /= 0.0e+00) &amp;
    7202. 

  7202. <a name="l07138"></a>07138     abserr = resasc*min ( 1.0e+00,(2.0e+02*abserr/resasc)**1.5e+00)
    7203. 

  7203. <a name="l07139"></a>07139 
    7204. 

  7204. <a name="l07140"></a>07140   <span class="keyword">if</span> ( resabs &gt; tiny ( resabs ) /(5.0e+01* epsilon ( resabs ) )) <span class="keyword">then</span>
    7205. 

  7205. <a name="l07141"></a>07141     abserr = max (( epsilon ( resabs ) *5.0e+01)*resabs,abserr)
    7206. 

  7206. <a name="l07142"></a>07142   <span class="keyword">end if</span>
    7207. 

  7207. <a name="l07143"></a>07143 
    7208. 

  7208. <a name="l07144"></a>07144   return
    7209. 

  7209. <a name="l07145"></a>07145 <span class="keyword">end</span>
    7210. 

  7210. <a name="l07146"></a><a class="code" href="quadpack_8f90.html#a73edb4987a87a40ebf4731ab63d7f03e">07146</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a73edb4987a87a40ebf4731ab63d7f03e">qk51</a> ( f, a, b, result, abserr, resabs, resasc )
    7211. 

  7211. <a name="l07147"></a>07147 
    7212. 

  7212. <a name="l07148"></a>07148 <span class="comment">!*****************************************************************************80</span>
    7213. 

  7213. <a name="l07149"></a>07149 <span class="comment">!</span>
    7214. 

  7214. <a name="l07150"></a>07150 <span class="comment">!! QK51 carries out a 51 point Gauss-Kronrod quadrature rule.</span>
    7215. 

  7215. <a name="l07151"></a>07151 <span class="comment">!</span>
    7216. 

  7216. <a name="l07152"></a>07152 <span class="comment">!  Discussion:</span>
    7217. 

  7217. <a name="l07153"></a>07153 <span class="comment">!</span>
    7218. 

  7218. <a name="l07154"></a>07154 <span class="comment">!    This routine approximates</span>
    7219. 

  7219. <a name="l07155"></a>07155 <span class="comment">!      I = integral ( A &lt;= X &lt;= B ) F(X) dx</span>
    7220. 

  7220. <a name="l07156"></a>07156 <span class="comment">!    with an error estimate, and</span>
    7221. 

  7221. <a name="l07157"></a>07157 <span class="comment">!      J = integral ( A &lt;= X &lt;= B ) | F(X) | dx</span>
    7222. 

  7222. <a name="l07158"></a>07158 <span class="comment">!</span>
    7223. 

  7223. <a name="l07159"></a>07159 <span class="comment">!  Author:</span>
    7224. 

  7224. <a name="l07160"></a>07160 <span class="comment">!</span>
    7225. 

  7225. <a name="l07161"></a>07161 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    7226. 

  7226. <a name="l07162"></a>07162 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    7227. 

  7227. <a name="l07163"></a>07163 <span class="comment">!</span>
    7228. 

  7228. <a name="l07164"></a>07164 <span class="comment">!  Reference:</span>
    7229. 

  7229. <a name="l07165"></a>07165 <span class="comment">!</span>
    7230. 

  7230. <a name="l07166"></a>07166 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    7231. 

  7231. <a name="l07167"></a>07167 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    7232. 

  7232. <a name="l07168"></a>07168 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    7233. 

  7233. <a name="l07169"></a>07169 <span class="comment">!    Springer Verlag, 1983</span>
    7234. 

  7234. <a name="l07170"></a>07170 <span class="comment">!</span>
    7235. 

  7235. <a name="l07171"></a>07171 <span class="comment">!  Parameters:</span>
    7236. 

  7236. <a name="l07172"></a>07172 <span class="comment">!</span>
    7237. 

  7237. <a name="l07173"></a>07173 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    7238. 

  7238. <a name="l07174"></a>07174 <span class="comment">!      function f ( x )</span>
    7239. 

  7239. <a name="l07175"></a>07175 <span class="comment">!      real f</span>
    7240. 

  7240. <a name="l07176"></a>07176 <span class="comment">!      real x</span>
    7241. 

  7241. <a name="l07177"></a>07177 <span class="comment">!    which evaluates the integrand function.</span>
    7242. 

  7242. <a name="l07178"></a>07178 <span class="comment">!</span>
    7243. 

  7243. <a name="l07179"></a>07179 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    7244. 

  7244. <a name="l07180"></a>07180 <span class="comment">!</span>
    7245. 

  7245. <a name="l07181"></a>07181 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    7246. 

  7246. <a name="l07182"></a>07182 <span class="comment">!                       result is computed by applying the 51-point</span>
    7247. 

  7247. <a name="l07183"></a>07183 <span class="comment">!                       Kronrod rule (resk) obtained by optimal addition</span>
    7248. 

  7248. <a name="l07184"></a>07184 <span class="comment">!                       of abscissae to the 25-point Gauss rule (resg).</span>
    7249. 

  7249. <a name="l07185"></a>07185 <span class="comment">!</span>
    7250. 

  7250. <a name="l07186"></a>07186 <span class="comment">!    Output, real ABSERR, an estimate of | I - RESULT |.</span>
    7251. 

  7251. <a name="l07187"></a>07187 <span class="comment">!</span>
    7252. 

  7252. <a name="l07188"></a>07188 <span class="comment">!    Output, real RESABS, approximation to the integral of the absolute</span>
    7253. 

  7253. <a name="l07189"></a>07189 <span class="comment">!    value of F.</span>
    7254. 

  7254. <a name="l07190"></a>07190 <span class="comment">!</span>
    7255. 

  7255. <a name="l07191"></a>07191 <span class="comment">!    Output, real RESASC, approximation to the integral | F-I/(B-A) | </span>
    7256. 

  7256. <a name="l07192"></a>07192 <span class="comment">!    over [A,B].</span>
    7257. 

  7257. <a name="l07193"></a>07193 <span class="comment">!</span>
    7258. 

  7258. <a name="l07194"></a>07194 <span class="comment">!  Local Parameters:</span>
    7259. 

  7259. <a name="l07195"></a>07195 <span class="comment">!</span>
    7260. 

  7260. <a name="l07196"></a>07196 <span class="comment">!           centr  - mid point of the interval</span>
    7261. 

  7261. <a name="l07197"></a>07197 <span class="comment">!           hlgth  - half-length of the interval</span>
    7262. 

  7262. <a name="l07198"></a>07198 <span class="comment">!           absc   - abscissa</span>
    7263. 

  7263. <a name="l07199"></a>07199 <span class="comment">!           fval*  - function value</span>
    7264. 

  7264. <a name="l07200"></a>07200 <span class="comment">!           resg   - result of the 25-point Gauss formula</span>
    7265. 

  7265. <a name="l07201"></a>07201 <span class="comment">!           resk   - result of the 51-point Kronrod formula</span>
    7266. 

  7266. <a name="l07202"></a>07202 <span class="comment">!           reskh  - approximation to the mean value of f over (a,b),</span>
    7267. 

  7267. <a name="l07203"></a>07203 <span class="comment">!                    i.e. to i/(b-a)</span>
    7268. 

  7268. <a name="l07204"></a>07204 <span class="comment">!</span>
    7269. 

  7269. <a name="l07205"></a>07205   <span class="keyword">implicit none</span>
    7270. 

  7270. <a name="l07206"></a>07206 
    7271. 

  7271. <a name="l07207"></a>07207   <span class="keywordtype">real</span> a
    7272. 

  7272. <a name="l07208"></a>07208   <span class="keywordtype">real</span> absc
    7273. 

  7273. <a name="l07209"></a>07209   <span class="keywordtype">real</span> abserr
    7274. 

  7274. <a name="l07210"></a>07210   <span class="keywordtype">real</span> b
    7275. 

  7275. <a name="l07211"></a>07211   <span class="keywordtype">real</span> centr
    7276. 

  7276. <a name="l07212"></a>07212   <span class="keywordtype">real</span> dhlgth
    7277. 

  7277. <a name="l07213"></a>07213   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    7278. 

  7278. <a name="l07214"></a>07214   <span class="keywordtype">real</span> fc
    7279. 

  7279. <a name="l07215"></a>07215   <span class="keywordtype">real</span> fsum
    7280. 

  7280. <a name="l07216"></a>07216   <span class="keywordtype">real</span> fval1
    7281. 

  7281. <a name="l07217"></a>07217   <span class="keywordtype">real</span> fval2
    7282. 

  7282. <a name="l07218"></a>07218   <span class="keywordtype">real</span> fv1(25)
    7283. 

  7283. <a name="l07219"></a>07219   <span class="keywordtype">real</span> fv2(25)
    7284. 

  7284. <a name="l07220"></a>07220   <span class="keywordtype">real</span> hlgth
    7285. 

  7285. <a name="l07221"></a>07221   <span class="keywordtype">integer</span> j
    7286. 

  7286. <a name="l07222"></a>07222   <span class="keywordtype">integer</span> jtw
    7287. 

  7287. <a name="l07223"></a>07223   <span class="keywordtype">integer</span> jtwm1
    7288. 

  7288. <a name="l07224"></a>07224   <span class="keywordtype">real</span> resabs
    7289. 

  7289. <a name="l07225"></a>07225   <span class="keywordtype">real</span> resasc
    7290. 

  7290. <a name="l07226"></a>07226   <span class="keywordtype">real</span> resg
    7291. 

  7291. <a name="l07227"></a>07227   <span class="keywordtype">real</span> resk
    7292. 

  7292. <a name="l07228"></a>07228   <span class="keywordtype">real</span> reskh
    7293. 

  7293. <a name="l07229"></a>07229   <span class="keywordtype">real</span> result
    7294. 

  7294. <a name="l07230"></a>07230   <span class="keywordtype">real</span> wg(13)
    7295. 

  7295. <a name="l07231"></a>07231   <span class="keywordtype">real</span> wgk(26)
    7296. 

  7296. <a name="l07232"></a>07232   <span class="keywordtype">real</span> xgk(26)
    7297. 

  7297. <a name="l07233"></a>07233 <span class="comment">!</span>
    7298. 

  7298. <a name="l07234"></a>07234 <span class="comment">!           the abscissae and weights are given for the interval (-1,1).</span>
    7299. 

  7299. <a name="l07235"></a>07235 <span class="comment">!           because of symmetry only the positive abscissae and their</span>
    7300. 

  7300. <a name="l07236"></a>07236 <span class="comment">!           corresponding weights are given.</span>
    7301. 

  7301. <a name="l07237"></a>07237 <span class="comment">!</span>
    7302. 

  7302. <a name="l07238"></a>07238 <span class="comment">!           xgk    - abscissae of the 51-point Kronrod rule</span>
    7303. 

  7303. <a name="l07239"></a>07239 <span class="comment">!                    xgk(2), xgk(4), ...  abscissae of the 25-point</span>
    7304. 

  7304. <a name="l07240"></a>07240 <span class="comment">!                    Gauss rule</span>
    7305. 

  7305. <a name="l07241"></a>07241 <span class="comment">!                    xgk(1), xgk(3), ...  abscissae which are optimally</span>
    7306. 

  7306. <a name="l07242"></a>07242 <span class="comment">!                    added to the 25-point Gauss rule</span>
    7307. 

  7307. <a name="l07243"></a>07243 <span class="comment">!</span>
    7308. 

  7308. <a name="l07244"></a>07244 <span class="comment">!           wgk    - weights of the 51-point Kronrod rule</span>
    7309. 

  7309. <a name="l07245"></a>07245 <span class="comment">!</span>
    7310. 

  7310. <a name="l07246"></a>07246 <span class="comment">!           wg     - weights of the 25-point Gauss rule</span>
    7311. 

  7311. <a name="l07247"></a>07247 <span class="comment">!</span>
    7312. 

  7312. <a name="l07248"></a>07248   <span class="keyword">data</span> xgk(1),xgk(2),xgk(3),xgk(4),xgk(5),xgk(6),xgk(7),xgk(8), &amp;
    7313. 

  7313. <a name="l07249"></a>07249     xgk(9),xgk(10),xgk(11),xgk(12),xgk(13),xgk(14)/ &amp;
    7314. 

  7314. <a name="l07250"></a>07250        9.992621049926098e-01,   9.955569697904981e-01, &amp;
    7315. 

  7315. <a name="l07251"></a>07251        9.880357945340772e-01,   9.766639214595175e-01, &amp;
    7316. 

  7316. <a name="l07252"></a>07252        9.616149864258425e-01,   9.429745712289743e-01, &amp;
    7317. 

  7317. <a name="l07253"></a>07253        9.207471152817016e-01,   8.949919978782754e-01, &amp;
    7318. 

  7318. <a name="l07254"></a>07254        8.658470652932756e-01,   8.334426287608340e-01, &amp;
    7319. 

  7319. <a name="l07255"></a>07255        7.978737979985001e-01,   7.592592630373576e-01, &amp;
    7320. 

  7320. <a name="l07256"></a>07256        7.177664068130844e-01,   6.735663684734684e-01/
    7321. 

  7321. <a name="l07257"></a>07257    <span class="keyword">data</span> xgk(15),xgk(16),xgk(17),xgk(18),xgk(19),xgk(20),xgk(21), &amp;
    7322. 

  7322. <a name="l07258"></a>07258     xgk(22),xgk(23),xgk(24),xgk(25),xgk(26)/ &amp;
    7323. 

  7323. <a name="l07259"></a>07259        6.268100990103174e-01,   5.776629302412230e-01, &amp;
    7324. 

  7324. <a name="l07260"></a>07260        5.263252843347192e-01,   4.730027314457150e-01, &amp;
    7325. 

  7325. <a name="l07261"></a>07261        4.178853821930377e-01,   3.611723058093878e-01, &amp;
    7326. 

  7326. <a name="l07262"></a>07262        3.030895389311078e-01,   2.438668837209884e-01, &amp;
    7327. 

  7327. <a name="l07263"></a>07263        1.837189394210489e-01,   1.228646926107104e-01, &amp;
    7328. 

  7328. <a name="l07264"></a>07264        6.154448300568508e-02,   0.0e+00               /
    7329. 

  7329. <a name="l07265"></a>07265   <span class="keyword">data</span> wgk(1),wgk(2),wgk(3),wgk(4),wgk(5),wgk(6),wgk(7),wgk(8), &amp;
    7330. 

  7330. <a name="l07266"></a>07266     wgk(9),wgk(10),wgk(11),wgk(12),wgk(13),wgk(14)/ &amp;
    7331. 

  7331. <a name="l07267"></a>07267        1.987383892330316e-03,   5.561932135356714e-03, &amp;
    7332. 

  7332. <a name="l07268"></a>07268        9.473973386174152e-03,   1.323622919557167e-02, &amp;
    7333. 

  7333. <a name="l07269"></a>07269        1.684781770912830e-02,   2.043537114588284e-02, &amp;
    7334. 

  7334. <a name="l07270"></a>07270        2.400994560695322e-02,   2.747531758785174e-02, &amp;
    7335. 

  7335. <a name="l07271"></a>07271        3.079230016738749e-02,   3.400213027432934e-02, &amp;
    7336. 

  7336. <a name="l07272"></a>07272        3.711627148341554e-02,   4.008382550403238e-02, &amp;
    7337. 

  7337. <a name="l07273"></a>07273        4.287284502017005e-02,   4.550291304992179e-02/
    7338. 

  7338. <a name="l07274"></a>07274    <span class="keyword">data</span> wgk(15),wgk(16),wgk(17),wgk(18),wgk(19),wgk(20),wgk(21), &amp;
    7339. 

  7339. <a name="l07275"></a>07275     wgk(22),wgk(23),wgk(24),wgk(25),wgk(26)/ &amp;
    7340. 

  7340. <a name="l07276"></a>07276        4.798253713883671e-02,   5.027767908071567e-02, &amp;
    7341. 

  7341. <a name="l07277"></a>07277        5.236288580640748e-02,   5.425112988854549e-02, &amp;
    7342. 

  7342. <a name="l07278"></a>07278        5.595081122041232e-02,   5.743711636156783e-02, &amp;
    7343. 

  7343. <a name="l07279"></a>07279        5.868968002239421e-02,   5.972034032417406e-02, &amp;
    7344. 

  7344. <a name="l07280"></a>07280        6.053945537604586e-02,   6.112850971705305e-02, &amp;
    7345. 

  7345. <a name="l07281"></a>07281        6.147118987142532e-02,   6.158081806783294e-02/
    7346. 

  7346. <a name="l07282"></a>07282   <span class="keyword">data</span> wg(1),wg(2),wg(3),wg(4),wg(5),wg(6),wg(7),wg(8),wg(9),wg(10), &amp;
    7347. 

  7347. <a name="l07283"></a>07283     wg(11),wg(12),wg(13)/ &amp;
    7348. 

  7348. <a name="l07284"></a>07284        1.139379850102629e-02,   2.635498661503214e-02, &amp;
    7349. 

  7349. <a name="l07285"></a>07285        4.093915670130631e-02,   5.490469597583519e-02, &amp;
    7350. 

  7350. <a name="l07286"></a>07286        6.803833381235692e-02,   8.014070033500102e-02, &amp;
    7351. 

  7351. <a name="l07287"></a>07287        9.102826198296365e-02,   1.005359490670506e-01, &amp;
    7352. 

  7352. <a name="l07288"></a>07288        1.085196244742637e-01,   1.148582591457116e-01, &amp;
    7353. 

  7353. <a name="l07289"></a>07289        1.194557635357848e-01,   1.222424429903100e-01, &amp;
    7354. 

  7354. <a name="l07290"></a>07290        1.231760537267155e-01/
    7355. 

  7355. <a name="l07291"></a>07291 <span class="comment">!</span>
    7356. 

  7356. <a name="l07292"></a>07292   centr = 5.0e-01*(a+b)
    7357. 

  7357. <a name="l07293"></a>07293   hlgth = 5.0e-01*(b-a)
    7358. 

  7358. <a name="l07294"></a>07294   dhlgth = abs(hlgth)
    7359. 

  7359. <a name="l07295"></a>07295 <span class="comment">!</span>
    7360. 

  7360. <a name="l07296"></a>07296 <span class="comment">!  Compute the 51-point Kronrod approximation to the integral,</span>
    7361. 

  7361. <a name="l07297"></a>07297 <span class="comment">!  and estimate the absolute error.</span>
    7362. 

  7362. <a name="l07298"></a>07298 <span class="comment">!</span>
    7363. 

  7363. <a name="l07299"></a>07299   fc = f(centr)
    7364. 

  7364. <a name="l07300"></a>07300   resg = wg(13)*fc
    7365. 

  7365. <a name="l07301"></a>07301   resk = wgk(26)*fc
    7366. 

  7366. <a name="l07302"></a>07302   resabs = abs(resk)
    7367. 

  7367. <a name="l07303"></a>07303 
    7368. 

  7368. <a name="l07304"></a>07304   <span class="keyword">do</span> j = 1, 12
    7369. 

  7369. <a name="l07305"></a>07305     jtw = j*2
    7370. 

  7370. <a name="l07306"></a>07306     absc = hlgth*xgk(jtw)
    7371. 

  7371. <a name="l07307"></a>07307     fval1 = f(centr-absc)
    7372. 

  7372. <a name="l07308"></a>07308     fval2 = f(centr+absc)
    7373. 

  7373. <a name="l07309"></a>07309     fv1(jtw) = fval1
    7374. 

  7374. <a name="l07310"></a>07310     fv2(jtw) = fval2
    7375. 

  7375. <a name="l07311"></a>07311     fsum = fval1+fval2
    7376. 

  7376. <a name="l07312"></a>07312     resg = resg+wg(j)*fsum
    7377. 

  7377. <a name="l07313"></a>07313     resk = resk+wgk(jtw)*fsum
    7378. 

  7378. <a name="l07314"></a>07314     resabs = resabs+wgk(jtw)*(abs(fval1)+abs(fval2))
    7379. 

  7379. <a name="l07315"></a>07315   <span class="keyword">end do</span>
    7380. 

  7380. <a name="l07316"></a>07316 
    7381. 

  7381. <a name="l07317"></a>07317   <span class="keyword">do</span> j = 1, 13
    7382. 

  7382. <a name="l07318"></a>07318     jtwm1 = j*2-1
    7383. 

  7383. <a name="l07319"></a>07319     absc = hlgth*xgk(jtwm1)
    7384. 

  7384. <a name="l07320"></a>07320     fval1 = f(centr-absc)
    7385. 

  7385. <a name="l07321"></a>07321     fval2 = f(centr+absc)
    7386. 

  7386. <a name="l07322"></a>07322     fv1(jtwm1) = fval1
    7387. 

  7387. <a name="l07323"></a>07323     fv2(jtwm1) = fval2
    7388. 

  7388. <a name="l07324"></a>07324     fsum = fval1+fval2
    7389. 

  7389. <a name="l07325"></a>07325     resk = resk+wgk(jtwm1)*fsum
    7390. 

  7390. <a name="l07326"></a>07326     resabs = resabs+wgk(jtwm1)*(abs(fval1)+abs(fval2))
    7391. 

  7391. <a name="l07327"></a>07327   <span class="keyword">end do</span>
    7392. 

  7392. <a name="l07328"></a>07328 
    7393. 

  7393. <a name="l07329"></a>07329   reskh = resk*5.0e-01
    7394. 

  7394. <a name="l07330"></a>07330   resasc = wgk(26)*abs(fc-reskh)
    7395. 

  7395. <a name="l07331"></a>07331 
    7396. 

  7396. <a name="l07332"></a>07332   <span class="keyword">do</span> j = 1, 25
    7397. 

  7397. <a name="l07333"></a>07333     resasc = resasc+wgk(j)*(abs(fv1(j)-reskh)+abs(fv2(j)-reskh))
    7398. 

  7398. <a name="l07334"></a>07334   <span class="keyword">end do</span>
    7399. 

  7399. <a name="l07335"></a>07335 
    7400. 

  7400. <a name="l07336"></a>07336   result = resk*hlgth
    7401. 

  7401. <a name="l07337"></a>07337   resabs = resabs*dhlgth
    7402. 

  7402. <a name="l07338"></a>07338   resasc = resasc*dhlgth
    7403. 

  7403. <a name="l07339"></a>07339   abserr = abs((resk-resg)*hlgth)
    7404. 

  7404. <a name="l07340"></a>07340 
    7405. 

  7405. <a name="l07341"></a>07341   <span class="keyword">if</span> ( resasc /= 0.0e+00.and.abserr /= 0.0e+00) <span class="keyword">then</span>
    7406. 

  7406. <a name="l07342"></a>07342     abserr = resasc*min ( 1.0e+00,(2.0e+02*abserr/resasc)**1.5e+00)
    7407. 

  7407. <a name="l07343"></a>07343   <span class="keyword">end if</span>
    7408. 

  7408. <a name="l07344"></a>07344 
    7409. 

  7409. <a name="l07345"></a>07345   <span class="keyword">if</span> ( resabs &gt; tiny ( resabs ) / (5.0e+01* epsilon ( resabs ) ) ) <span class="keyword">then</span>
    7410. 

  7410. <a name="l07346"></a>07346     abserr = max (( epsilon ( resabs ) *5.0e+01)*resabs,abserr)
    7411. 

  7411. <a name="l07347"></a>07347   <span class="keyword">end if</span>
    7412. 

  7412. <a name="l07348"></a>07348 
    7413. 

  7413. <a name="l07349"></a>07349   return
    7414. 

  7414. <a name="l07350"></a>07350 <span class="keyword">end</span>
    7415. 

  7415. <a name="l07351"></a><a class="code" href="quadpack_8f90.html#acb4a48f5e54a2c5f951d0828e8f8146d">07351</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#acb4a48f5e54a2c5f951d0828e8f8146d">qk61</a> ( f, a, b, result, abserr, resabs, resasc ) 
    7416. 

  7416. <a name="l07352"></a>07352 
    7417. 

  7417. <a name="l07353"></a>07353 <span class="comment">!*****************************************************************************80</span>
    7418. 

  7418. <a name="l07354"></a>07354 <span class="comment">!</span>
    7419. 

  7419. <a name="l07355"></a>07355 <span class="comment">!! QK61 carries out a 61 point Gauss-Kronrod quadrature rule.</span>
    7420. 

  7420. <a name="l07356"></a>07356 <span class="comment">!</span>
    7421. 

  7421. <a name="l07357"></a>07357 <span class="comment">!  Discussion:</span>
    7422. 

  7422. <a name="l07358"></a>07358 <span class="comment">!</span>
    7423. 

  7423. <a name="l07359"></a>07359 <span class="comment">!    This routine approximates</span>
    7424. 

  7424. <a name="l07360"></a>07360 <span class="comment">!      I = integral ( A &lt;= X &lt;= B ) F(X) dx</span>
    7425. 

  7425. <a name="l07361"></a>07361 <span class="comment">!    with an error estimate, and</span>
    7426. 

  7426. <a name="l07362"></a>07362 <span class="comment">!      J = integral ( A &lt;= X &lt;= B ) | F(X) | dx</span>
    7427. 

  7427. <a name="l07363"></a>07363 <span class="comment">!</span>
    7428. 

  7428. <a name="l07364"></a>07364 <span class="comment">!  Author:</span>
    7429. 

  7429. <a name="l07365"></a>07365 <span class="comment">!</span>
    7430. 

  7430. <a name="l07366"></a>07366 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    7431. 

  7431. <a name="l07367"></a>07367 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    7432. 

  7432. <a name="l07368"></a>07368 <span class="comment">!</span>
    7433. 

  7433. <a name="l07369"></a>07369 <span class="comment">!  Reference:</span>
    7434. 

  7434. <a name="l07370"></a>07370 <span class="comment">!</span>
    7435. 

  7435. <a name="l07371"></a>07371 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    7436. 

  7436. <a name="l07372"></a>07372 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    7437. 

  7437. <a name="l07373"></a>07373 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    7438. 

  7438. <a name="l07374"></a>07374 <span class="comment">!    Springer Verlag, 1983</span>
    7439. 

  7439. <a name="l07375"></a>07375 <span class="comment">!</span>
    7440. 

  7440. <a name="l07376"></a>07376 <span class="comment">!  Parameters:</span>
    7441. 

  7441. <a name="l07377"></a>07377 <span class="comment">!</span>
    7442. 

  7442. <a name="l07378"></a>07378 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    7443. 

  7443. <a name="l07379"></a>07379 <span class="comment">!      function f ( x )</span>
    7444. 

  7444. <a name="l07380"></a>07380 <span class="comment">!      real f</span>
    7445. 

  7445. <a name="l07381"></a>07381 <span class="comment">!      real x</span>
    7446. 

  7446. <a name="l07382"></a>07382 <span class="comment">!    which evaluates the integrand function.</span>
    7447. 

  7447. <a name="l07383"></a>07383 <span class="comment">!</span>
    7448. 

  7448. <a name="l07384"></a>07384 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    7449. 

  7449. <a name="l07385"></a>07385 <span class="comment">!</span>
    7450. 

  7450. <a name="l07386"></a>07386 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    7451. 

  7451. <a name="l07387"></a>07387 <span class="comment">!                    result is computed by applying the 61-point</span>
    7452. 

  7452. <a name="l07388"></a>07388 <span class="comment">!                    Kronrod rule (resk) obtained by optimal addition of</span>
    7453. 

  7453. <a name="l07389"></a>07389 <span class="comment">!                    abscissae to the 30-point Gauss rule (resg).</span>
    7454. 

  7454. <a name="l07390"></a>07390 <span class="comment">!</span>
    7455. 

  7455. <a name="l07391"></a>07391 <span class="comment">!    Output, real ABSERR, an estimate of | I - RESULT |.</span>
    7456. 

  7456. <a name="l07392"></a>07392 <span class="comment">!</span>
    7457. 

  7457. <a name="l07393"></a>07393 <span class="comment">!    Output, real RESABS, approximation to the integral of the absolute</span>
    7458. 

  7458. <a name="l07394"></a>07394 <span class="comment">!    value of F.</span>
    7459. 

  7459. <a name="l07395"></a>07395 <span class="comment">!</span>
    7460. 

  7460. <a name="l07396"></a>07396 <span class="comment">!    Output, real RESASC, approximation to the integral | F-I/(B-A) | </span>
    7461. 

  7461. <a name="l07397"></a>07397 <span class="comment">!    over [A,B].</span>
    7462. 

  7462. <a name="l07398"></a>07398 <span class="comment">!</span>
    7463. 

  7463. <a name="l07399"></a>07399 <span class="comment">!  Local Parameters:</span>
    7464. 

  7464. <a name="l07400"></a>07400 <span class="comment">!</span>
    7465. 

  7465. <a name="l07401"></a>07401 <span class="comment">!           centr  - mid point of the interval</span>
    7466. 

  7466. <a name="l07402"></a>07402 <span class="comment">!           hlgth  - half-length of the interval</span>
    7467. 

  7467. <a name="l07403"></a>07403 <span class="comment">!           absc   - abscissa</span>
    7468. 

  7468. <a name="l07404"></a>07404 <span class="comment">!           fval*  - function value</span>
    7469. 

  7469. <a name="l07405"></a>07405 <span class="comment">!           resg   - result of the 30-point Gauss rule</span>
    7470. 

  7470. <a name="l07406"></a>07406 <span class="comment">!           resk   - result of the 61-point Kronrod rule</span>
    7471. 

  7471. <a name="l07407"></a>07407 <span class="comment">!           reskh  - approximation to the mean value of f</span>
    7472. 

  7472. <a name="l07408"></a>07408 <span class="comment">!                    over (a,b), i.e. to i/(b-a)</span>
    7473. 

  7473. <a name="l07409"></a>07409 <span class="comment">!</span>
    7474. 

  7474. <a name="l07410"></a>07410   <span class="keyword">implicit none</span>
    7475. 

  7475. <a name="l07411"></a>07411 
    7476. 

  7476. <a name="l07412"></a>07412   <span class="keywordtype">real</span> a
    7477. 

  7477. <a name="l07413"></a>07413   <span class="keywordtype">real</span> absc
    7478. 

  7478. <a name="l07414"></a>07414   <span class="keywordtype">real</span> abserr
    7479. 

  7479. <a name="l07415"></a>07415   <span class="keywordtype">real</span> b
    7480. 

  7480. <a name="l07416"></a>07416   <span class="keywordtype">real</span> centr
    7481. 

  7481. <a name="l07417"></a>07417   <span class="keywordtype">real</span> dhlgth
    7482. 

  7482. <a name="l07418"></a>07418   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    7483. 

  7483. <a name="l07419"></a>07419   <span class="keywordtype">real</span> fc
    7484. 

  7484. <a name="l07420"></a>07420   <span class="keywordtype">real</span> fsum
    7485. 

  7485. <a name="l07421"></a>07421   <span class="keywordtype">real</span> fval1
    7486. 

  7486. <a name="l07422"></a>07422   <span class="keywordtype">real</span> fval2
    7487. 

  7487. <a name="l07423"></a>07423   <span class="keywordtype">real</span> fv1(30)
    7488. 

  7488. <a name="l07424"></a>07424   <span class="keywordtype">real</span> fv2(30)
    7489. 

  7489. <a name="l07425"></a>07425   <span class="keywordtype">real</span> hlgth
    7490. 

  7490. <a name="l07426"></a>07426   <span class="keywordtype">integer</span> j
    7491. 

  7491. <a name="l07427"></a>07427   <span class="keywordtype">integer</span> jtw
    7492. 

  7492. <a name="l07428"></a>07428   <span class="keywordtype">integer</span> jtwm1
    7493. 

  7493. <a name="l07429"></a>07429   <span class="keywordtype">real</span> resabs
    7494. 

  7494. <a name="l07430"></a>07430   <span class="keywordtype">real</span> resasc
    7495. 

  7495. <a name="l07431"></a>07431   <span class="keywordtype">real</span> resg
    7496. 

  7496. <a name="l07432"></a>07432   <span class="keywordtype">real</span> resk
    7497. 

  7497. <a name="l07433"></a>07433   <span class="keywordtype">real</span> reskh
    7498. 

  7498. <a name="l07434"></a>07434   <span class="keywordtype">real</span> result
    7499. 

  7499. <a name="l07435"></a>07435   <span class="keywordtype">real</span> wg(15)
    7500. 

  7500. <a name="l07436"></a>07436   <span class="keywordtype">real</span> wgk(31)
    7501. 

  7501. <a name="l07437"></a>07437   <span class="keywordtype">real</span> xgk(31)
    7502. 

  7502. <a name="l07438"></a>07438 <span class="comment">!</span>
    7503. 

  7503. <a name="l07439"></a>07439 <span class="comment">!           the abscissae and weights are given for the</span>
    7504. 

  7504. <a name="l07440"></a>07440 <span class="comment">!           interval (-1,1). because of symmetry only the positive</span>
    7505. 

  7505. <a name="l07441"></a>07441 <span class="comment">!           abscissae and their corresponding weights are given.</span>
    7506. 

  7506. <a name="l07442"></a>07442 <span class="comment">!</span>
    7507. 

  7507. <a name="l07443"></a>07443 <span class="comment">!           xgk   - abscissae of the 61-point Kronrod rule</span>
    7508. 

  7508. <a name="l07444"></a>07444 <span class="comment">!                   xgk(2), xgk(4)  ... abscissae of the 30-point</span>
    7509. 

  7509. <a name="l07445"></a>07445 <span class="comment">!                   Gauss rule</span>
    7510. 

  7510. <a name="l07446"></a>07446 <span class="comment">!                   xgk(1), xgk(3)  ... optimally added abscissae</span>
    7511. 

  7511. <a name="l07447"></a>07447 <span class="comment">!                   to the 30-point Gauss rule</span>
    7512. 

  7512. <a name="l07448"></a>07448 <span class="comment">!</span>
    7513. 

  7513. <a name="l07449"></a>07449 <span class="comment">!           wgk   - weights of the 61-point Kronrod rule</span>
    7514. 

  7514. <a name="l07450"></a>07450 <span class="comment">!</span>
    7515. 

  7515. <a name="l07451"></a>07451 <span class="comment">!           wg    - weigths of the 30-point Gauss rule</span>
    7516. 

  7516. <a name="l07452"></a>07452 <span class="comment">!</span>
    7517. 

  7517. <a name="l07453"></a>07453   <span class="keyword">data</span> xgk(1),xgk(2),xgk(3),xgk(4),xgk(5),xgk(6),xgk(7),xgk(8), &amp;
    7518. 

  7518. <a name="l07454"></a>07454      xgk(9),xgk(10)/ &amp;
    7519. 

  7519. <a name="l07455"></a>07455        9.994844100504906e-01,     9.968934840746495e-01, &amp;
    7520. 

  7520. <a name="l07456"></a>07456        9.916309968704046e-01,     9.836681232797472e-01, &amp;
    7521. 

  7521. <a name="l07457"></a>07457        9.731163225011263e-01,     9.600218649683075e-01, &amp;
    7522. 

  7522. <a name="l07458"></a>07458        9.443744447485600e-01,     9.262000474292743e-01, &amp;
    7523. 

  7523. <a name="l07459"></a>07459        9.055733076999078e-01,     8.825605357920527e-01/
    7524. 

  7524. <a name="l07460"></a>07460   <span class="keyword">data</span> xgk(11),xgk(12),xgk(13),xgk(14),xgk(15),xgk(16),xgk(17), &amp;
    7525. 

  7525. <a name="l07461"></a>07461     xgk(18),xgk(19),xgk(20)/ &amp;
    7526. 

  7526. <a name="l07462"></a>07462        8.572052335460611e-01,     8.295657623827684e-01, &amp;
    7527. 

  7527. <a name="l07463"></a>07463        7.997278358218391e-01,     7.677774321048262e-01, &amp;
    7528. 

  7528. <a name="l07464"></a>07464        7.337900624532268e-01,     6.978504947933158e-01, &amp;
    7529. 

  7529. <a name="l07465"></a>07465        6.600610641266270e-01,     6.205261829892429e-01, &amp;
    7530. 

  7530. <a name="l07466"></a>07466        5.793452358263617e-01,     5.366241481420199e-01/
    7531. 

  7531. <a name="l07467"></a>07467   <span class="keyword">data</span> xgk(21),xgk(22),xgk(23),xgk(24),xgk(25),xgk(26),xgk(27), &amp;
    7532. 

  7532. <a name="l07468"></a>07468     xgk(28),xgk(29),xgk(30),xgk(31)/ &amp;
    7533. 

  7533. <a name="l07469"></a>07469        4.924804678617786e-01,     4.470337695380892e-01, &amp;
    7534. 

  7534. <a name="l07470"></a>07470        4.004012548303944e-01,     3.527047255308781e-01, &amp;
    7535. 

  7535. <a name="l07471"></a>07471        3.040732022736251e-01,     2.546369261678898e-01, &amp;
    7536. 

  7536. <a name="l07472"></a>07472        2.045251166823099e-01,     1.538699136085835e-01, &amp;
    7537. 

  7537. <a name="l07473"></a>07473        1.028069379667370e-01,     5.147184255531770e-02, &amp;
    7538. 

  7538. <a name="l07474"></a>07474        0.0e+00                   /
    7539. 

  7539. <a name="l07475"></a>07475   <span class="keyword">data</span> wgk(1),wgk(2),wgk(3),wgk(4),wgk(5),wgk(6),wgk(7),wgk(8), &amp;
    7540. 

  7540. <a name="l07476"></a>07476     wgk(9),wgk(10)/ &amp;
    7541. 

  7541. <a name="l07477"></a>07477        1.389013698677008e-03,     3.890461127099884e-03, &amp;
    7542. 

  7542. <a name="l07478"></a>07478        6.630703915931292e-03,     9.273279659517763e-03, &amp;
    7543. 

  7543. <a name="l07479"></a>07479        1.182301525349634e-02,     1.436972950704580e-02, &amp;
    7544. 

  7544. <a name="l07480"></a>07480        1.692088918905327e-02,     1.941414119394238e-02, &amp;
    7545. 

  7545. <a name="l07481"></a>07481        2.182803582160919e-02,     2.419116207808060e-02/
    7546. 

  7546. <a name="l07482"></a>07482   <span class="keyword">data</span> wgk(11),wgk(12),wgk(13),wgk(14),wgk(15),wgk(16),wgk(17), &amp;
    7547. 

  7547. <a name="l07483"></a>07483     wgk(18),wgk(19),wgk(20)/ &amp;
    7548. 

  7548. <a name="l07484"></a>07484        2.650995488233310e-02,     2.875404876504129e-02, &amp;
    7549. 

  7549. <a name="l07485"></a>07485        3.090725756238776e-02,     3.298144705748373e-02, &amp;
    7550. 

  7550. <a name="l07486"></a>07486        3.497933802806002e-02,     3.688236465182123e-02, &amp;
    7551. 

  7551. <a name="l07487"></a>07487        3.867894562472759e-02,     4.037453895153596e-02, &amp;
    7552. 

  7552. <a name="l07488"></a>07488        4.196981021516425e-02,     4.345253970135607e-02/
    7553. 

  7553. <a name="l07489"></a>07489   <span class="keyword">data</span> wgk(21),wgk(22),wgk(23),wgk(24),wgk(25),wgk(26),wgk(27), &amp;
    7554. 

  7554. <a name="l07490"></a>07490     wgk(28),wgk(29),wgk(30),wgk(31)/ &amp;
    7555. 

  7555. <a name="l07491"></a>07491        4.481480013316266e-02,     4.605923827100699e-02, &amp;
    7556. 

  7556. <a name="l07492"></a>07492        4.718554656929915e-02,     4.818586175708713e-02, &amp;
    7557. 

  7557. <a name="l07493"></a>07493        4.905543455502978e-02,     4.979568342707421e-02, &amp;
    7558. 

  7558. <a name="l07494"></a>07494        5.040592140278235e-02,     5.088179589874961e-02, &amp;
    7559. 

  7559. <a name="l07495"></a>07495        5.122154784925877e-02,     5.142612853745903e-02, &amp;
    7560. 

  7560. <a name="l07496"></a>07496        5.149472942945157e-02/
    7561. 

  7561. <a name="l07497"></a>07497   <span class="keyword">data</span> wg(1),wg(2),wg(3),wg(4),wg(5),wg(6),wg(7),wg(8)/ &amp;
    7562. 

  7562. <a name="l07498"></a>07498        7.968192496166606e-03,     1.846646831109096e-02, &amp;
    7563. 

  7563. <a name="l07499"></a>07499        2.878470788332337e-02,     3.879919256962705e-02, &amp;
    7564. 

  7564. <a name="l07500"></a>07500        4.840267283059405e-02,     5.749315621761907e-02, &amp;
    7565. 

  7565. <a name="l07501"></a>07501        6.597422988218050e-02,     7.375597473770521e-02/
    7566. 

  7566. <a name="l07502"></a>07502   <span class="keyword">data</span> wg(9),wg(10),wg(11),wg(12),wg(13),wg(14),wg(15)/ &amp;
    7567. 

  7567. <a name="l07503"></a>07503        8.075589522942022e-02,     8.689978720108298e-02, &amp;
    7568. 

  7568. <a name="l07504"></a>07504        9.212252223778613e-02,     9.636873717464426e-02, &amp;
    7569. 

  7569. <a name="l07505"></a>07505        9.959342058679527e-02,     1.017623897484055e-01, &amp;
    7570. 

  7570. <a name="l07506"></a>07506        1.028526528935588e-01/
    7571. 

  7571. <a name="l07507"></a>07507 
    7572. 

  7572. <a name="l07508"></a>07508   centr = 5.0e-01*(b+a)
    7573. 

  7573. <a name="l07509"></a>07509   hlgth = 5.0e-01*(b-a)
    7574. 

  7574. <a name="l07510"></a>07510   dhlgth = abs(hlgth)
    7575. 

  7575. <a name="l07511"></a>07511 <span class="comment">!</span>
    7576. 

  7576. <a name="l07512"></a>07512 <span class="comment">!  Compute the 61-point Kronrod approximation to the integral,</span>
    7577. 

  7577. <a name="l07513"></a>07513 <span class="comment">!  and estimate the absolute error.</span>
    7578. 

  7578. <a name="l07514"></a>07514 <span class="comment">!</span>
    7579. 

  7579. <a name="l07515"></a>07515   resg = 0.0e+00
    7580. 

  7580. <a name="l07516"></a>07516   fc = f(centr)
    7581. 

  7581. <a name="l07517"></a>07517   resk = wgk(31)*fc
    7582. 

  7582. <a name="l07518"></a>07518   resabs = abs(resk)
    7583. 

  7583. <a name="l07519"></a>07519 
    7584. 

  7584. <a name="l07520"></a>07520   <span class="keyword">do</span> j = 1, 15
    7585. 

  7585. <a name="l07521"></a>07521     jtw = j*2
    7586. 

  7586. <a name="l07522"></a>07522     absc = hlgth*xgk(jtw)
    7587. 

  7587. <a name="l07523"></a>07523     fval1 = f(centr-absc)
    7588. 

  7588. <a name="l07524"></a>07524     fval2 = f(centr+absc)
    7589. 

  7589. <a name="l07525"></a>07525     fv1(jtw) = fval1
    7590. 

  7590. <a name="l07526"></a>07526     fv2(jtw) = fval2
    7591. 

  7591. <a name="l07527"></a>07527     fsum = fval1+fval2
    7592. 

  7592. <a name="l07528"></a>07528     resg = resg+wg(j)*fsum
    7593. 

  7593. <a name="l07529"></a>07529     resk = resk+wgk(jtw)*fsum
    7594. 

  7594. <a name="l07530"></a>07530     resabs = resabs+wgk(jtw)*(abs(fval1)+abs(fval2))
    7595. 

  7595. <a name="l07531"></a>07531   <span class="keyword">end do</span>
    7596. 

  7596. <a name="l07532"></a>07532 
    7597. 

  7597. <a name="l07533"></a>07533   <span class="keyword">do</span> j = 1, 15
    7598. 

  7598. <a name="l07534"></a>07534     jtwm1 = j*2-1
    7599. 

  7599. <a name="l07535"></a>07535     absc = hlgth*xgk(jtwm1)
    7600. 

  7600. <a name="l07536"></a>07536     fval1 = f(centr-absc)
    7601. 

  7601. <a name="l07537"></a>07537     fval2 = f(centr+absc)
    7602. 

  7602. <a name="l07538"></a>07538     fv1(jtwm1) = fval1
    7603. 

  7603. <a name="l07539"></a>07539     fv2(jtwm1) = fval2
    7604. 

  7604. <a name="l07540"></a>07540     fsum = fval1+fval2
    7605. 

  7605. <a name="l07541"></a>07541     resk = resk+wgk(jtwm1)*fsum
    7606. 

  7606. <a name="l07542"></a>07542     resabs = resabs+wgk(jtwm1)*(abs(fval1)+abs(fval2))
    7607. 

  7607. <a name="l07543"></a>07543   <span class="keyword">end do</span>
    7608. 

  7608. <a name="l07544"></a>07544 
    7609. 

  7609. <a name="l07545"></a>07545   reskh = resk * 5.0e-01
    7610. 

  7610. <a name="l07546"></a>07546   resasc = wgk(31)*abs(fc-reskh)
    7611. 

  7611. <a name="l07547"></a>07547 
    7612. 

  7612. <a name="l07548"></a>07548   <span class="keyword">do</span> j = 1, 30
    7613. 

  7613. <a name="l07549"></a>07549     resasc = resasc+wgk(j)*(abs(fv1(j)-reskh)+abs(fv2(j)-reskh))
    7614. 

  7614. <a name="l07550"></a>07550   <span class="keyword">end do</span>
    7615. 

  7615. <a name="l07551"></a>07551 
    7616. 

  7616. <a name="l07552"></a>07552   result = resk*hlgth
    7617. 

  7617. <a name="l07553"></a>07553   resabs = resabs*dhlgth
    7618. 

  7618. <a name="l07554"></a>07554   resasc = resasc*dhlgth
    7619. 

  7619. <a name="l07555"></a>07555   abserr = abs((resk-resg)*hlgth)
    7620. 

  7620. <a name="l07556"></a>07556 
    7621. 

  7621. <a name="l07557"></a>07557   <span class="keyword">if</span> ( resasc /= 0.0e+00 .and. abserr /= 0.0e+00) <span class="keyword">then</span>
    7622. 

  7622. <a name="l07558"></a>07558     abserr = resasc*min ( 1.0e+00,(2.0e+02*abserr/resasc)**1.5e+00)
    7623. 

  7623. <a name="l07559"></a>07559   <span class="keyword">end if</span>
    7624. 

  7624. <a name="l07560"></a>07560 
    7625. 

  7625. <a name="l07561"></a>07561   <span class="keyword">if</span> ( resabs &gt; tiny ( resabs ) / (5.0e+01* epsilon ( resabs ) )) <span class="keyword">then</span>
    7626. 

  7626. <a name="l07562"></a>07562     abserr = max ( ( epsilon ( resabs ) *5.0e+01)*resabs, abserr )
    7627. 

  7627. <a name="l07563"></a>07563   <span class="keyword">end if</span>
    7628. 

  7628. <a name="l07564"></a>07564 
    7629. 

  7629. <a name="l07565"></a>07565   return
    7630. 

  7630. <a name="l07566"></a>07566 <span class="keyword">end</span>
    7631. 

  7631. <a name="l07567"></a><a class="code" href="quadpack_8f90.html#aa732651ae77f9486d6e3d17999c699ab">07567</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#aa732651ae77f9486d6e3d17999c699ab">qmomo</a> ( alfa, beta, ri, rj, rg, rh, integr )
    7632. 

  7632. <a name="l07568"></a>07568 
    7633. 

  7633. <a name="l07569"></a>07569 <span class="comment">!*****************************************************************************80</span>
    7634. 

  7634. <a name="l07570"></a>07570 <span class="comment">!</span>
    7635. 

  7635. <a name="l07571"></a>07571 <span class="comment">!! QMOMO computes modified Chebyshev moments.</span>
    7636. 

  7636. <a name="l07572"></a>07572 <span class="comment">!</span>
    7637. 

  7637. <a name="l07573"></a>07573 <span class="comment">!  Discussion:</span>
    7638. 

  7638. <a name="l07574"></a>07574 <span class="comment">!</span>
    7639. 

  7639. <a name="l07575"></a>07575 <span class="comment">!    This routine computes modified Chebyshev moments.</span>
    7640. 

  7640. <a name="l07576"></a>07576 <span class="comment">!    The K-th modified Chebyshev moment is defined as the</span>
    7641. 

  7641. <a name="l07577"></a>07577 <span class="comment">!    integral over (-1,1) of W(X)*T(K,X), where T(K,X) is the</span>
    7642. 

  7642. <a name="l07578"></a>07578 <span class="comment">!    Chebyshev polynomial of degree K.</span>
    7643. 

  7643. <a name="l07579"></a>07579 <span class="comment">!</span>
    7644. 

  7644. <a name="l07580"></a>07580 <span class="comment">!  Author:</span>
    7645. 

  7645. <a name="l07581"></a>07581 <span class="comment">!</span>
    7646. 

  7646. <a name="l07582"></a>07582 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    7647. 

  7647. <a name="l07583"></a>07583 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    7648. 

  7648. <a name="l07584"></a>07584 <span class="comment">!</span>
    7649. 

  7649. <a name="l07585"></a>07585 <span class="comment">!  Reference:</span>
    7650. 

  7650. <a name="l07586"></a>07586 <span class="comment">!</span>
    7651. 

  7651. <a name="l07587"></a>07587 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    7652. 

  7652. <a name="l07588"></a>07588 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    7653. 

  7653. <a name="l07589"></a>07589 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    7654. 

  7654. <a name="l07590"></a>07590 <span class="comment">!    Springer Verlag, 1983</span>
    7655. 

  7655. <a name="l07591"></a>07591 <span class="comment">!</span>
    7656. 

  7656. <a name="l07592"></a>07592 <span class="comment">!  Parameters:</span>
    7657. 

  7657. <a name="l07593"></a>07593 <span class="comment">!</span>
    7658. 

  7658. <a name="l07594"></a>07594 <span class="comment">!    Input, real ALFA, a parameter in the weight function w(x), ALFA &gt; -1.</span>
    7659. 

  7659. <a name="l07595"></a>07595 <span class="comment">!</span>
    7660. 

  7660. <a name="l07596"></a>07596 <span class="comment">!    Input, real BETA, a parameter in the weight function w(x), BETA &gt; -1.</span>
    7661. 

  7661. <a name="l07597"></a>07597 <span class="comment">!</span>
    7662. 

  7662. <a name="l07598"></a>07598 <span class="comment">!           ri     - real</span>
    7663. 

  7663. <a name="l07599"></a>07599 <span class="comment">!                    vector of dimension 25</span>
    7664. 

  7664. <a name="l07600"></a>07600 <span class="comment">!                    ri(k) is the integral over (-1,1) of</span>
    7665. 

  7665. <a name="l07601"></a>07601 <span class="comment">!                    (1+x)**alfa*t(k-1,x), k = 1, ..., 25.</span>
    7666. 

  7666. <a name="l07602"></a>07602 <span class="comment">!</span>
    7667. 

  7667. <a name="l07603"></a>07603 <span class="comment">!           rj     - real</span>
    7668. 

  7668. <a name="l07604"></a>07604 <span class="comment">!                    vector of dimension 25</span>
    7669. 

  7669. <a name="l07605"></a>07605 <span class="comment">!                    rj(k) is the integral over (-1,1) of</span>
    7670. 

  7670. <a name="l07606"></a>07606 <span class="comment">!                    (1-x)**beta*t(k-1,x), k = 1, ..., 25.</span>
    7671. 

  7671. <a name="l07607"></a>07607 <span class="comment">!</span>
    7672. 

  7672. <a name="l07608"></a>07608 <span class="comment">!           rg     - real</span>
    7673. 

  7673. <a name="l07609"></a>07609 <span class="comment">!                    vector of dimension 25</span>
    7674. 

  7674. <a name="l07610"></a>07610 <span class="comment">!                    rg(k) is the integral over (-1,1) of</span>
    7675. 

  7675. <a name="l07611"></a>07611 <span class="comment">!                    (1+x)**alfa*log((1+x)/2)*t(k-1,x), k = 1, ...,25.</span>
    7676. 

  7676. <a name="l07612"></a>07612 <span class="comment">!</span>
    7677. 

  7677. <a name="l07613"></a>07613 <span class="comment">!           rh     - real</span>
    7678. 

  7678. <a name="l07614"></a>07614 <span class="comment">!                    vector of dimension 25</span>
    7679. 

  7679. <a name="l07615"></a>07615 <span class="comment">!                    rh(k) is the integral over (-1,1) of</span>
    7680. 

  7680. <a name="l07616"></a>07616 <span class="comment">!                    (1-x)**beta*log((1-x)/2)*t(k-1,x), k = 1, ..., 25.</span>
    7681. 

  7681. <a name="l07617"></a>07617 <span class="comment">!</span>
    7682. 

  7682. <a name="l07618"></a>07618 <span class="comment">!           integr - integer</span>
    7683. 

  7683. <a name="l07619"></a>07619 <span class="comment">!                    input parameter indicating the modified moments</span>
    7684. 

  7684. <a name="l07620"></a>07620 <span class="comment">!                    to be computed</span>
    7685. 

  7685. <a name="l07621"></a>07621 <span class="comment">!                    integr = 1 compute ri, rj</span>
    7686. 

  7686. <a name="l07622"></a>07622 <span class="comment">!                           = 2 compute ri, rj, rg</span>
    7687. 

  7687. <a name="l07623"></a>07623 <span class="comment">!                           = 3 compute ri, rj, rh</span>
    7688. 

  7688. <a name="l07624"></a>07624 <span class="comment">!                           = 4 compute ri, rj, rg, rh</span>
    7689. 

  7689. <a name="l07625"></a>07625 <span class="comment">!</span>
    7690. 

  7690. <a name="l07626"></a>07626   <span class="keyword">implicit none</span>
    7691. 

  7691. <a name="l07627"></a>07627 
    7692. 

  7692. <a name="l07628"></a>07628   <span class="keywordtype">real</span> alfa
    7693. 

  7693. <a name="l07629"></a>07629   <span class="keywordtype">real</span> alfp1
    7694. 

  7694. <a name="l07630"></a>07630   <span class="keywordtype">real</span> alfp2
    7695. 

  7695. <a name="l07631"></a>07631   <span class="keywordtype">real</span> an
    7696. 

  7696. <a name="l07632"></a>07632   <span class="keywordtype">real</span> anm1
    7697. 

  7697. <a name="l07633"></a>07633   <span class="keywordtype">real</span> beta
    7698. 

  7698. <a name="l07634"></a>07634   <span class="keywordtype">real</span> betp1
    7699. 

  7699. <a name="l07635"></a>07635   <span class="keywordtype">real</span> betp2
    7700. 

  7700. <a name="l07636"></a>07636   <span class="keywordtype">integer</span> i
    7701. 

  7701. <a name="l07637"></a>07637   <span class="keywordtype">integer</span> im1
    7702. 

  7702. <a name="l07638"></a>07638   <span class="keywordtype">integer</span> integr
    7703. 

  7703. <a name="l07639"></a>07639   <span class="keywordtype">real</span> ralf
    7704. 

  7704. <a name="l07640"></a>07640   <span class="keywordtype">real</span> rbet
    7705. 

  7705. <a name="l07641"></a>07641   <span class="keywordtype">real</span> rg(25)
    7706. 

  7706. <a name="l07642"></a>07642   <span class="keywordtype">real</span> rh(25)
    7707. 

  7707. <a name="l07643"></a>07643   <span class="keywordtype">real</span> ri(25)
    7708. 

  7708. <a name="l07644"></a>07644   <span class="keywordtype">real</span> rj(25)
    7709. 

  7709. <a name="l07645"></a>07645 <span class="comment">!</span>
    7710. 

  7710. <a name="l07646"></a>07646   alfp1 = alfa+1.0e+00
    7711. 

  7711. <a name="l07647"></a>07647   betp1 = beta+1.0e+00
    7712. 

  7712. <a name="l07648"></a>07648   alfp2 = alfa+2.0e+00
    7713. 

  7713. <a name="l07649"></a>07649   betp2 = beta+2.0e+00
    7714. 

  7714. <a name="l07650"></a>07650   ralf = 2.0e+00**alfp1
    7715. 

  7715. <a name="l07651"></a>07651   rbet = 2.0e+00**betp1
    7716. 

  7716. <a name="l07652"></a>07652 <span class="comment">!</span>
    7717. 

  7717. <a name="l07653"></a>07653 <span class="comment">!  Compute RI, RJ using a forward recurrence relation.</span>
    7718. 

  7718. <a name="l07654"></a>07654 <span class="comment">!</span>
    7719. 

  7719. <a name="l07655"></a>07655   ri(1) = ralf/alfp1
    7720. 

  7720. <a name="l07656"></a>07656   rj(1) = rbet/betp1
    7721. 

  7721. <a name="l07657"></a>07657   ri(2) = ri(1)*alfa/alfp2
    7722. 

  7722. <a name="l07658"></a>07658   rj(2) = rj(1)*beta/betp2
    7723. 

  7723. <a name="l07659"></a>07659   an = 2.0e+00
    7724. 

  7724. <a name="l07660"></a>07660   anm1 = 1.0e+00
    7725. 

  7725. <a name="l07661"></a>07661 
    7726. 

  7726. <a name="l07662"></a>07662   <span class="keyword">do</span> i = 3, 25
    7727. 

  7727. <a name="l07663"></a>07663     ri(i) = -(ralf+an*(an-alfp2)*ri(i-1))/(anm1*(an+alfp1))
    7728. 

  7728. <a name="l07664"></a>07664     rj(i) = -(rbet+an*(an-betp2)*rj(i-1))/(anm1*(an+betp1))
    7729. 

  7729. <a name="l07665"></a>07665     anm1 = an
    7730. 

  7730. <a name="l07666"></a>07666     an = an+1.0e+00
    7731. 

  7731. <a name="l07667"></a>07667   <span class="keyword">end do</span>
    7732. 

  7732. <a name="l07668"></a>07668 
    7733. 

  7733. <a name="l07669"></a>07669   <span class="keyword">if</span> ( integr == 1 ) go to 70
    7734. 

  7734. <a name="l07670"></a>07670   <span class="keyword">if</span> ( integr == 3 ) go to 40
    7735. 

  7735. <a name="l07671"></a>07671 <span class="comment">!</span>
    7736. 

  7736. <a name="l07672"></a>07672 <span class="comment">!  Compute RG using a forward recurrence relation.</span>
    7737. 

  7737. <a name="l07673"></a>07673 <span class="comment">!</span>
    7738. 

  7738. <a name="l07674"></a>07674   rg(1) = -ri(1)/alfp1
    7739. 

  7739. <a name="l07675"></a>07675   rg(2) = -(ralf+ralf)/(alfp2*alfp2)-rg(1)
    7740. 

  7740. <a name="l07676"></a>07676   an = 2.0e+00
    7741. 

  7741. <a name="l07677"></a>07677   anm1 = 1.0e+00
    7742. 

  7742. <a name="l07678"></a>07678   im1 = 2
    7743. 

  7743. <a name="l07679"></a>07679 
    7744. 

  7744. <a name="l07680"></a>07680   <span class="keyword">do</span> i = 3, 25
    7745. 

  7745. <a name="l07681"></a>07681     rg(i) = -(an*(an-alfp2)*rg(im1)-an*ri(im1)+anm1*ri(i))/ &amp;
    7746. 

  7746. <a name="l07682"></a>07682     (anm1*(an+alfp1))
    7747. 

  7747. <a name="l07683"></a>07683     anm1 = an
    7748. 

  7748. <a name="l07684"></a>07684     an = an+1.0e+00
    7749. 

  7749. <a name="l07685"></a>07685     im1 = i
    7750. 

  7750. <a name="l07686"></a>07686   <span class="keyword">end do</span>
    7751. 

  7751. <a name="l07687"></a>07687 
    7752. 

  7752. <a name="l07688"></a>07688   <span class="keyword">if</span> ( integr == 2 ) go to 70
    7753. 

  7753. <a name="l07689"></a>07689 <span class="comment">!</span>
    7754. 

  7754. <a name="l07690"></a>07690 <span class="comment">!  Compute RH using a forward recurrence relation.</span>
    7755. 

  7755. <a name="l07691"></a>07691 <span class="comment">!</span>
    7756. 

  7756. <a name="l07692"></a>07692 40 continue
    7757. 

  7757. <a name="l07693"></a>07693 
    7758. 

  7758. <a name="l07694"></a>07694   rh(1) = -rj(1) / betp1
    7759. 

  7759. <a name="l07695"></a>07695   rh(2) = -(rbet+rbet)/(betp2*betp2)-rh(1)
    7760. 

  7760. <a name="l07696"></a>07696   an = 2.0e+00
    7761. 

  7761. <a name="l07697"></a>07697   anm1 = 1.0e+00
    7762. 

  7762. <a name="l07698"></a>07698   im1 = 2
    7763. 

  7763. <a name="l07699"></a>07699 
    7764. 

  7764. <a name="l07700"></a>07700   <span class="keyword">do</span> i = 3, 25
    7765. 

  7765. <a name="l07701"></a>07701     rh(i) = -(an*(an-betp2)*rh(im1)-an*rj(im1)+ &amp;
    7766. 

  7766. <a name="l07702"></a>07702     anm1*rj(i))/(anm1*(an+betp1))
    7767. 

  7767. <a name="l07703"></a>07703     anm1 = an
    7768. 

  7768. <a name="l07704"></a>07704     an = an+1.0e+00
    7769. 

  7769. <a name="l07705"></a>07705     im1 = i
    7770. 

  7770. <a name="l07706"></a>07706   <span class="keyword">end do</span>
    7771. 

  7771. <a name="l07707"></a>07707 
    7772. 

  7772. <a name="l07708"></a>07708   <span class="keyword">do</span> i = 2, 25, 2
    7773. 

  7773. <a name="l07709"></a>07709     rh(i) = -rh(i)
    7774. 

  7774. <a name="l07710"></a>07710   <span class="keyword">end do</span>
    7775. 

  7775. <a name="l07711"></a>07711 
    7776. 

  7776. <a name="l07712"></a>07712    70 continue
    7777. 

  7777. <a name="l07713"></a>07713 
    7778. 

  7778. <a name="l07714"></a>07714   <span class="keyword">do</span> i = 2, 25, 2
    7779. 

  7779. <a name="l07715"></a>07715     rj(i) = -rj(i)
    7780. 

  7780. <a name="l07716"></a>07716   <span class="keyword">end do</span>
    7781. 

  7781. <a name="l07717"></a>07717 
    7782. 

  7782. <a name="l07718"></a>07718 <span class="comment">!  90 continue</span>
    7783. 

  7783. <a name="l07719"></a>07719 
    7784. 

  7784. <a name="l07720"></a>07720   return
    7785. 

  7785. <a name="l07721"></a>07721 <span class="keyword">end</span>
    7786. 

  7786. <a name="l07722"></a><a class="code" href="quadpack_8f90.html#a611318e7c2d2bafa649b25519dc1b55d">07722</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a611318e7c2d2bafa649b25519dc1b55d">qng</a> ( f, a, b, epsabs, epsrel, result, abserr, neval, ier )
    7787. 

  7787. <a name="l07723"></a>07723 
    7788. 

  7788. <a name="l07724"></a>07724 <span class="comment">!*****************************************************************************80</span>
    7789. 

  7789. <a name="l07725"></a>07725 <span class="comment">!</span>
    7790. 

  7790. <a name="l07726"></a>07726 <span class="comment">!! QNG estimates an integral, using non-adaptive integration.</span>
    7791. 

  7791. <a name="l07727"></a>07727 <span class="comment">!</span>
    7792. 

  7792. <a name="l07728"></a>07728 <span class="comment">!  Discussion:</span>
    7793. 

  7793. <a name="l07729"></a>07729 <span class="comment">!</span>
    7794. 

  7794. <a name="l07730"></a>07730 <span class="comment">!    The routine calculates an approximation RESULT to a definite integral   </span>
    7795. 

  7795. <a name="l07731"></a>07731 <span class="comment">!      I = integral of F over (A,B),</span>
    7796. 

  7796. <a name="l07732"></a>07732 <span class="comment">!    hopefully satisfying</span>
    7797. 

  7797. <a name="l07733"></a>07733 <span class="comment">!      || I - RESULT || &lt;= max ( EPSABS, EPSREL * ||I|| ).</span>
    7798. 

  7798. <a name="l07734"></a>07734 <span class="comment">!</span>
    7799. 

  7799. <a name="l07735"></a>07735 <span class="comment">!    The routine is a simple non-adaptive automatic integrator, based on</span>
    7800. 

  7800. <a name="l07736"></a>07736 <span class="comment">!    a sequence of rules with increasing degree of algebraic</span>
    7801. 

  7801. <a name="l07737"></a>07737 <span class="comment">!    precision (Patterson, 1968).</span>
    7802. 

  7802. <a name="l07738"></a>07738 <span class="comment">!</span>
    7803. 

  7803. <a name="l07739"></a>07739 <span class="comment">!  Author:</span>
    7804. 

  7804. <a name="l07740"></a>07740 <span class="comment">!</span>
    7805. 

  7805. <a name="l07741"></a>07741 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    7806. 

  7806. <a name="l07742"></a>07742 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    7807. 

  7807. <a name="l07743"></a>07743 <span class="comment">!</span>
    7808. 

  7808. <a name="l07744"></a>07744 <span class="comment">!  Reference:</span>
    7809. 

  7809. <a name="l07745"></a>07745 <span class="comment">!</span>
    7810. 

  7810. <a name="l07746"></a>07746 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    7811. 

  7811. <a name="l07747"></a>07747 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    7812. 

  7812. <a name="l07748"></a>07748 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    7813. 

  7813. <a name="l07749"></a>07749 <span class="comment">!    Springer Verlag, 1983</span>
    7814. 

  7814. <a name="l07750"></a>07750 <span class="comment">!</span>
    7815. 

  7815. <a name="l07751"></a>07751 <span class="comment">!  Parameters:</span>
    7816. 

  7816. <a name="l07752"></a>07752 <span class="comment">!</span>
    7817. 

  7817. <a name="l07753"></a>07753 <span class="comment">!    Input, external real F, the name of the function routine, of the form</span>
    7818. 

  7818. <a name="l07754"></a>07754 <span class="comment">!      function f ( x )</span>
    7819. 

  7819. <a name="l07755"></a>07755 <span class="comment">!      real f</span>
    7820. 

  7820. <a name="l07756"></a>07756 <span class="comment">!      real x</span>
    7821. 

  7821. <a name="l07757"></a>07757 <span class="comment">!    which evaluates the integrand function.</span>
    7822. 

  7822. <a name="l07758"></a>07758 <span class="comment">!</span>
    7823. 

  7823. <a name="l07759"></a>07759 <span class="comment">!    Input, real A, B, the limits of integration.</span>
    7824. 

  7824. <a name="l07760"></a>07760 <span class="comment">!</span>
    7825. 

  7825. <a name="l07761"></a>07761 <span class="comment">!    Input, real EPSABS, EPSREL, the absolute and relative accuracy requested.</span>
    7826. 

  7826. <a name="l07762"></a>07762 <span class="comment">!</span>
    7827. 

  7827. <a name="l07763"></a>07763 <span class="comment">!    Output, real RESULT, the estimated value of the integral.</span>
    7828. 

  7828. <a name="l07764"></a>07764 <span class="comment">!    RESULT is obtained by applying the 21-point Gauss-Kronrod rule (RES21)</span>
    7829. 

  7829. <a name="l07765"></a>07765 <span class="comment">!    obtained  by optimal addition of abscissae to the 10-point Gauss rule</span>
    7830. 

  7830. <a name="l07766"></a>07766 <span class="comment">!    (RES10), or by applying the 43-point rule (RES43) obtained by optimal</span>
    7831. 

  7831. <a name="l07767"></a>07767 <span class="comment">!    addition of abscissae to the 21-point Gauss-Kronrod rule, or by </span>
    7832. 

  7832. <a name="l07768"></a>07768 <span class="comment">!    applying the 87-point rule (RES87) obtained by optimal addition of</span>
    7833. 

  7833. <a name="l07769"></a>07769 <span class="comment">!    abscissae to the 43-point rule.</span>
    7834. 

  7834. <a name="l07770"></a>07770 <span class="comment">!</span>
    7835. 

  7835. <a name="l07771"></a>07771 <span class="comment">!    Output, real ABSERR, an estimate of || I - RESULT ||.</span>
    7836. 

  7836. <a name="l07772"></a>07772 <span class="comment">!</span>
    7837. 

  7837. <a name="l07773"></a>07773 <span class="comment">!    Output, integer NEVAL, the number of times the integral was evaluated.</span>
    7838. 

  7838. <a name="l07774"></a>07774 <span class="comment">!</span>
    7839. 

  7839. <a name="l07775"></a>07775 <span class="comment">!           ier    - ier = 0 normal and reliable termination of the</span>
    7840. 

  7840. <a name="l07776"></a>07776 <span class="comment">!                            routine. it is assumed that the requested</span>
    7841. 

  7841. <a name="l07777"></a>07777 <span class="comment">!                            accuracy has been achieved.</span>
    7842. 

  7842. <a name="l07778"></a>07778 <span class="comment">!                    ier &gt; 0 abnormal termination of the routine. it is</span>
    7843. 

  7843. <a name="l07779"></a>07779 <span class="comment">!                            assumed that the requested accuracy has</span>
    7844. 

  7844. <a name="l07780"></a>07780 <span class="comment">!                            not been achieved.</span>
    7845. 

  7845. <a name="l07781"></a>07781 <span class="comment">!                    ier = 1 the maximum number of steps has been</span>
    7846. 

  7846. <a name="l07782"></a>07782 <span class="comment">!                            executed. the integral is probably too</span>
    7847. 

  7847. <a name="l07783"></a>07783 <span class="comment">!                            difficult to be calculated by qng.</span>
    7848. 

  7848. <a name="l07784"></a>07784 <span class="comment">!                        = 6 the input is invalid, because</span>
    7849. 

  7849. <a name="l07785"></a>07785 <span class="comment">!                            epsabs &lt; 0 and epsrel &lt; 0,</span>
    7850. 

  7850. <a name="l07786"></a>07786 <span class="comment">!                            result, abserr and neval are set to zero.</span>
    7851. 

  7851. <a name="l07787"></a>07787 <span class="comment">!</span>
    7852. 

  7852. <a name="l07788"></a>07788 <span class="comment">!  Local Parameters:</span>
    7853. 

  7853. <a name="l07789"></a>07789 <span class="comment">!</span>
    7854. 

  7854. <a name="l07790"></a>07790 <span class="comment">!           centr  - mid point of the integration interval</span>
    7855. 

  7855. <a name="l07791"></a>07791 <span class="comment">!           hlgth  - half-length of the integration interval</span>
    7856. 

  7856. <a name="l07792"></a>07792 <span class="comment">!           fcentr - function value at mid point</span>
    7857. 

  7857. <a name="l07793"></a>07793 <span class="comment">!           absc   - abscissa</span>
    7858. 

  7858. <a name="l07794"></a>07794 <span class="comment">!           fval   - function value</span>
    7859. 

  7859. <a name="l07795"></a>07795 <span class="comment">!           savfun - array of function values which have already</span>
    7860. 

  7860. <a name="l07796"></a>07796 <span class="comment">!                    been computed</span>
    7861. 

  7861. <a name="l07797"></a>07797 <span class="comment">!           res10  - 10-point Gauss result</span>
    7862. 

  7862. <a name="l07798"></a>07798 <span class="comment">!           res21  - 21-point Kronrod result</span>
    7863. 

  7863. <a name="l07799"></a>07799 <span class="comment">!           res43  - 43-point result</span>
    7864. 

  7864. <a name="l07800"></a>07800 <span class="comment">!           res87  - 87-point result</span>
    7865. 

  7865. <a name="l07801"></a>07801 <span class="comment">!           resabs - approximation to the integral of abs(f)</span>
    7866. 

  7866. <a name="l07802"></a>07802 <span class="comment">!           resasc - approximation to the integral of abs(f-i/(b-a))</span>
    7867. 

  7867. <a name="l07803"></a>07803 <span class="comment">!</span>
    7868. 

  7868. <a name="l07804"></a>07804   <span class="keyword">implicit none</span>
    7869. 

  7869. <a name="l07805"></a>07805 
    7870. 

  7870. <a name="l07806"></a>07806   <span class="keywordtype">real</span> a
    7871. 

  7871. <a name="l07807"></a>07807   <span class="keywordtype">real</span> absc
    7872. 

  7872. <a name="l07808"></a>07808   <span class="keywordtype">real</span> abserr
    7873. 

  7873. <a name="l07809"></a>07809   <span class="keywordtype">real</span> b
    7874. 

  7874. <a name="l07810"></a>07810   <span class="keywordtype">real</span> centr
    7875. 

  7875. <a name="l07811"></a>07811   <span class="keywordtype">real</span> dhlgth
    7876. 

  7876. <a name="l07812"></a>07812   <span class="keywordtype">real</span> epsabs
    7877. 

  7877. <a name="l07813"></a>07813   <span class="keywordtype">real</span> epsrel
    7878. 

  7878. <a name="l07814"></a>07814   <span class="keywordtype">real</span>, <span class="keywordtype">external</span> :: f
    7879. 

  7879. <a name="l07815"></a>07815   <span class="keywordtype">real</span> fcentr
    7880. 

  7880. <a name="l07816"></a>07816   <span class="keywordtype">real</span> fval
    7881. 

  7881. <a name="l07817"></a>07817   <span class="keywordtype">real</span> fval1
    7882. 

  7882. <a name="l07818"></a>07818   <span class="keywordtype">real</span> fval2
    7883. 

  7883. <a name="l07819"></a>07819   <span class="keywordtype">real</span> fv1(5)
    7884. 

  7884. <a name="l07820"></a>07820   <span class="keywordtype">real</span> fv2(5)
    7885. 

  7885. <a name="l07821"></a>07821   <span class="keywordtype">real</span> fv3(5)
    7886. 

  7886. <a name="l07822"></a>07822   <span class="keywordtype">real</span> fv4(5)
    7887. 

  7887. <a name="l07823"></a>07823   <span class="keywordtype">real</span> hlgth
    7888. 

  7888. <a name="l07824"></a>07824   <span class="keywordtype">integer</span> ier
    7889. 

  7889. <a name="l07825"></a>07825   <span class="keywordtype">integer</span> ipx
    7890. 

  7890. <a name="l07826"></a>07826   <span class="keywordtype">integer</span> k
    7891. 

  7891. <a name="l07827"></a>07827   <span class="keywordtype">integer</span> l
    7892. 

  7892. <a name="l07828"></a>07828   <span class="keywordtype">integer</span> neval
    7893. 

  7893. <a name="l07829"></a>07829   <span class="keywordtype">real</span> result
    7894. 

  7894. <a name="l07830"></a>07830   <span class="keywordtype">real</span> res10
    7895. 

  7895. <a name="l07831"></a>07831   <span class="keywordtype">real</span> res21
    7896. 

  7896. <a name="l07832"></a>07832   <span class="keywordtype">real</span> res43
    7897. 

  7897. <a name="l07833"></a>07833   <span class="keywordtype">real</span> res87
    7898. 

  7898. <a name="l07834"></a>07834   <span class="keywordtype">real</span> resabs
    7899. 

  7899. <a name="l07835"></a>07835   <span class="keywordtype">real</span> resasc
    7900. 

  7900. <a name="l07836"></a>07836   <span class="keywordtype">real</span> reskh
    7901. 

  7901. <a name="l07837"></a>07837   <span class="keywordtype">real</span> savfun(21)
    7902. 

  7902. <a name="l07838"></a>07838   <span class="keywordtype">real</span> w10(5)
    7903. 

  7903. <a name="l07839"></a>07839   <span class="keywordtype">real</span> w21a(5)
    7904. 

  7904. <a name="l07840"></a>07840   <span class="keywordtype">real</span> w21b(6)
    7905. 

  7905. <a name="l07841"></a>07841   <span class="keywordtype">real</span> w43a(10)
    7906. 

  7906. <a name="l07842"></a>07842   <span class="keywordtype">real</span> w43b(12)
    7907. 

  7907. <a name="l07843"></a>07843   <span class="keywordtype">real</span> w87a(21)
    7908. 

  7908. <a name="l07844"></a>07844   <span class="keywordtype">real</span> w87b(23)
    7909. 

  7909. <a name="l07845"></a>07845   <span class="keywordtype">real</span> x1(5)
    7910. 

  7910. <a name="l07846"></a>07846   <span class="keywordtype">real</span> x2(5)
    7911. 

  7911. <a name="l07847"></a>07847   <span class="keywordtype">real</span> x3(11)
    7912. 

  7912. <a name="l07848"></a>07848   <span class="keywordtype">real</span> x4(22)
    7913. 

  7913. <a name="l07849"></a>07849 <span class="comment">!</span>
    7914. 

  7914. <a name="l07850"></a>07850 <span class="comment">!           the following data statements contain the abscissae</span>
    7915. 

  7915. <a name="l07851"></a>07851 <span class="comment">!           and weights of the integration rules used.</span>
    7916. 

  7916. <a name="l07852"></a>07852 <span class="comment">!</span>
    7917. 

  7917. <a name="l07853"></a>07853 <span class="comment">!           x1      abscissae common to the 10-, 21-, 43- and 87-point</span>
    7918. 

  7918. <a name="l07854"></a>07854 <span class="comment">!                   rule</span>
    7919. 

  7919. <a name="l07855"></a>07855 <span class="comment">!           x2      abscissae common to the 21-, 43- and 87-point rule</span>
    7920. 

  7920. <a name="l07856"></a>07856 <span class="comment">!           x3      abscissae common to the 43- and 87-point rule</span>
    7921. 

  7921. <a name="l07857"></a>07857 <span class="comment">!           x4      abscissae of the 87-point rule</span>
    7922. 

  7922. <a name="l07858"></a>07858 <span class="comment">!           w10     weights of the 10-point formula</span>
    7923. 

  7923. <a name="l07859"></a>07859 <span class="comment">!           w21a    weights of the 21-point formula for abscissae x1</span>
    7924. 

  7924. <a name="l07860"></a>07860 <span class="comment">!           w21b    weights of the 21-point formula for abscissae x2</span>
    7925. 

  7925. <a name="l07861"></a>07861 <span class="comment">!           w43a    weights of the 43-point formula for absissae x1, x3</span>
    7926. 

  7926. <a name="l07862"></a>07862 <span class="comment">!           w43b    weights of the 43-point formula for abscissae x3</span>
    7927. 

  7927. <a name="l07863"></a>07863 <span class="comment">!           w87a    weights of the 87-point formula for abscissae x1,</span>
    7928. 

  7928. <a name="l07864"></a>07864 <span class="comment">!                   x2 and x3</span>
    7929. 

  7929. <a name="l07865"></a>07865 <span class="comment">!           w87b    weights of the 87-point formula for abscissae x4</span>
    7930. 

  7930. <a name="l07866"></a>07866 <span class="comment">!</span>
    7931. 

  7931. <a name="l07867"></a>07867   <span class="keyword">data</span> x1(1),x1(2),x1(3),x1(4),x1(5)/ &amp;
    7932. 

  7932. <a name="l07868"></a>07868        9.739065285171717e-01,     8.650633666889845e-01, &amp;
    7933. 

  7933. <a name="l07869"></a>07869        6.794095682990244e-01,     4.333953941292472e-01, &amp;
    7934. 

  7934. <a name="l07870"></a>07870        1.488743389816312e-01/
    7935. 

  7935. <a name="l07871"></a>07871   <span class="keyword">data</span> x2(1),x2(2),x2(3),x2(4),x2(5)/ &amp;
    7936. 

  7936. <a name="l07872"></a>07872        9.956571630258081e-01,     9.301574913557082e-01, &amp;
    7937. 

  7937. <a name="l07873"></a>07873        7.808177265864169e-01,     5.627571346686047e-01, &amp;
    7938. 

  7938. <a name="l07874"></a>07874        2.943928627014602e-01/
    7939. 

  7939. <a name="l07875"></a>07875   <span class="keyword">data</span> x3(1),x3(2),x3(3),x3(4),x3(5),x3(6),x3(7),x3(8),x3(9),x3(10), &amp;
    7940. 

  7940. <a name="l07876"></a>07876     x3(11)/ &amp;
    7941. 

  7941. <a name="l07877"></a>07877        9.993333609019321e-01,     9.874334029080889e-01, &amp;
    7942. 

  7942. <a name="l07878"></a>07878        9.548079348142663e-01,     9.001486957483283e-01, &amp;
    7943. 

  7943. <a name="l07879"></a>07879        8.251983149831142e-01,     7.321483889893050e-01, &amp;
    7944. 

  7944. <a name="l07880"></a>07880        6.228479705377252e-01,     4.994795740710565e-01, &amp;
    7945. 

  7945. <a name="l07881"></a>07881        3.649016613465808e-01,     2.222549197766013e-01, &amp;
    7946. 

  7946. <a name="l07882"></a>07882        7.465061746138332e-02/
    7947. 

  7947. <a name="l07883"></a>07883   <span class="keyword">data</span> x4(1),x4(2),x4(3),x4(4),x4(5),x4(6),x4(7),x4(8),x4(9),x4(10), &amp;
    7948. 

  7948. <a name="l07884"></a>07884     x4(11),x4(12),x4(13),x4(14),x4(15),x4(16),x4(17),x4(18),x4(19), &amp;
    7949. 

  7949. <a name="l07885"></a>07885     x4(20),x4(21),x4(22)/         9.999029772627292e-01, &amp;
    7950. 

  7950. <a name="l07886"></a>07886        9.979898959866787e-01,     9.921754978606872e-01, &amp;
    7951. 

  7951. <a name="l07887"></a>07887        9.813581635727128e-01,     9.650576238583846e-01, &amp;
    7952. 

  7952. <a name="l07888"></a>07888        9.431676131336706e-01,     9.158064146855072e-01, &amp;
    7953. 

  7953. <a name="l07889"></a>07889        8.832216577713165e-01,     8.457107484624157e-01, &amp;
    7954. 

  7954. <a name="l07890"></a>07890        8.035576580352310e-01,     7.570057306854956e-01, &amp;
    7955. 

  7955. <a name="l07891"></a>07891        7.062732097873218e-01,     6.515894665011779e-01, &amp;
    7956. 

  7956. <a name="l07892"></a>07892        5.932233740579611e-01,     5.314936059708319e-01, &amp;
    7957. 

  7957. <a name="l07893"></a>07893        4.667636230420228e-01,     3.994248478592188e-01, &amp;
    7958. 

  7958. <a name="l07894"></a>07894        3.298748771061883e-01,     2.585035592021616e-01, &amp;
    7959. 

  7959. <a name="l07895"></a>07895        1.856953965683467e-01,     1.118422131799075e-01, &amp;
    7960. 

  7960. <a name="l07896"></a>07896        3.735212339461987e-02/
    7961. 

  7961. <a name="l07897"></a>07897   <span class="keyword">data</span> w10(1),w10(2),w10(3),w10(4),w10(5)/ &amp;
    7962. 

  7962. <a name="l07898"></a>07898        6.667134430868814e-02,     1.494513491505806e-01, &amp;
    7963. 

  7963. <a name="l07899"></a>07899        2.190863625159820e-01,     2.692667193099964e-01, &amp;
    7964. 

  7964. <a name="l07900"></a>07900        2.955242247147529e-01/
    7965. 

  7965. <a name="l07901"></a>07901   <span class="keyword">data</span> w21a(1),w21a(2),w21a(3),w21a(4),w21a(5)/ &amp;
    7966. 

  7966. <a name="l07902"></a>07902        3.255816230796473e-02,     7.503967481091995e-02, &amp;
    7967. 

  7967. <a name="l07903"></a>07903        1.093871588022976e-01,     1.347092173114733e-01, &amp;
    7968. 

  7968. <a name="l07904"></a>07904        1.477391049013385e-01/
    7969. 

  7969. <a name="l07905"></a>07905   <span class="keyword">data</span> w21b(1),w21b(2),w21b(3),w21b(4),w21b(5),w21b(6)/ &amp;
    7970. 

  7970. <a name="l07906"></a>07906        1.169463886737187e-02,     5.475589657435200e-02, &amp;
    7971. 

  7971. <a name="l07907"></a>07907        9.312545458369761e-02,     1.234919762620659e-01, &amp;
    7972. 

  7972. <a name="l07908"></a>07908        1.427759385770601e-01,     1.494455540029169e-01/
    7973. 

  7973. <a name="l07909"></a>07909   <span class="keyword">data</span> w43a(1),w43a(2),w43a(3),w43a(4),w43a(5),w43a(6),w43a(7), &amp;
    7974. 

  7974. <a name="l07910"></a>07910     w43a(8),w43a(9),w43a(10)/     1.629673428966656e-02, &amp;
    7975. 

  7975. <a name="l07911"></a>07911        3.752287612086950e-02,     5.469490205825544e-02, &amp;
    7976. 

  7976. <a name="l07912"></a>07912        6.735541460947809e-02,     7.387019963239395e-02, &amp;
    7977. 

  7977. <a name="l07913"></a>07913        5.768556059769796e-03,     2.737189059324884e-02, &amp;
    7978. 

  7978. <a name="l07914"></a>07914        4.656082691042883e-02,     6.174499520144256e-02, &amp;
    7979. 

  7979. <a name="l07915"></a>07915        7.138726726869340e-02/
    7980. 

  7980. <a name="l07916"></a>07916   <span class="keyword">data</span> w43b(1),w43b(2),w43b(3),w43b(4),w43b(5),w43b(6),w43b(7), &amp;
    7981. 

  7981. <a name="l07917"></a>07917     w43b(8),w43b(9),w43b(10),w43b(11),w43b(12)/ &amp;
    7982. 

  7982. <a name="l07918"></a>07918        1.844477640212414e-03,     1.079868958589165e-02, &amp;
    7983. 

  7983. <a name="l07919"></a>07919        2.189536386779543e-02,     3.259746397534569e-02, &amp;
    7984. 

  7984. <a name="l07920"></a>07920        4.216313793519181e-02,     5.074193960018458e-02, &amp;
    7985. 

  7985. <a name="l07921"></a>07921        5.837939554261925e-02,     6.474640495144589e-02, &amp;
    7986. 

  7986. <a name="l07922"></a>07922        6.956619791235648e-02,     7.282444147183321e-02, &amp;
    7987. 

  7987. <a name="l07923"></a>07923        7.450775101417512e-02,     7.472214751740301e-02/
    7988. 

  7988. <a name="l07924"></a>07924   <span class="keyword">data</span> w87a(1),w87a(2),w87a(3),w87a(4),w87a(5),w87a(6),w87a(7), &amp;
    7989. 

  7989. <a name="l07925"></a>07925     w87a(8),w87a(9),w87a(10),w87a(11),w87a(12),w87a(13),w87a(14), &amp;
    7990. 

  7990. <a name="l07926"></a>07926     w87a(15),w87a(16),w87a(17),w87a(18),w87a(19),w87a(20),w87a(21)/ &amp;
    7991. 

  7991. <a name="l07927"></a>07927        8.148377384149173e-03,     1.876143820156282e-02, &amp;
    7992. 

  7992. <a name="l07928"></a>07928        2.734745105005229e-02,     3.367770731163793e-02, &amp;
    7993. 

  7993. <a name="l07929"></a>07929        3.693509982042791e-02,     2.884872430211531e-03, &amp;
    7994. 

  7994. <a name="l07930"></a>07930        1.368594602271270e-02,     2.328041350288831e-02, &amp;
    7995. 

  7995. <a name="l07931"></a>07931        3.087249761171336e-02,     3.569363363941877e-02, &amp;
    7996. 

  7996. <a name="l07932"></a>07932        9.152833452022414e-04,     5.399280219300471e-03, &amp;
    7997. 

  7997. <a name="l07933"></a>07933        1.094767960111893e-02,     1.629873169678734e-02, &amp;
    7998. 

  7998. <a name="l07934"></a>07934        2.108156888920384e-02,     2.537096976925383e-02, &amp;
    7999. 

  7999. <a name="l07935"></a>07935        2.918969775647575e-02,     3.237320246720279e-02, &amp;
    8000. 

  8000. <a name="l07936"></a>07936        3.478309895036514e-02,     3.641222073135179e-02, &amp;
    8001. 

  8001. <a name="l07937"></a>07937        3.725387550304771e-02/
    8002. 

  8002. <a name="l07938"></a>07938   <span class="keyword">data</span> w87b(1),w87b(2),w87b(3),w87b(4),w87b(5),w87b(6),w87b(7), &amp;
    8003. 

  8003. <a name="l07939"></a>07939     w87b(8),w87b(9),w87b(10),w87b(11),w87b(12),w87b(13),w87b(14), &amp;
    8004. 

  8004. <a name="l07940"></a>07940     w87b(15),w87b(16),w87b(17),w87b(18),w87b(19),w87b(20),w87b(21), &amp;
    8005. 

  8005. <a name="l07941"></a>07941     w87b(22),w87b(23)/            2.741455637620724e-04, &amp;
    8006. 

  8006. <a name="l07942"></a>07942        1.807124155057943e-03,     4.096869282759165e-03, &amp;
    8007. 

  8007. <a name="l07943"></a>07943        6.758290051847379e-03,     9.549957672201647e-03, &amp;
    8008. 

  8008. <a name="l07944"></a>07944        1.232944765224485e-02,     1.501044734638895e-02, &amp;
    8009. 

  8009. <a name="l07945"></a>07945        1.754896798624319e-02,     1.993803778644089e-02, &amp;
    8010. 

  8010. <a name="l07946"></a>07946        2.219493596101229e-02,     2.433914712600081e-02, &amp;
    8011. 

  8011. <a name="l07947"></a>07947        2.637450541483921e-02,     2.828691078877120e-02, &amp;
    8012. 

  8012. <a name="l07948"></a>07948        3.005258112809270e-02,     3.164675137143993e-02, &amp;
    8013. 

  8013. <a name="l07949"></a>07949        3.305041341997850e-02,     3.425509970422606e-02, &amp;
    8014. 

  8014. <a name="l07950"></a>07950        3.526241266015668e-02,     3.607698962288870e-02, &amp;
    8015. 

  8015. <a name="l07951"></a>07951        3.669860449845609e-02,     3.712054926983258e-02, &amp;
    8016. 

  8016. <a name="l07952"></a>07952        3.733422875193504e-02,     3.736107376267902e-02/
    8017. 

  8017. <a name="l07953"></a>07953 <span class="comment">!</span>
    8018. 

  8018. <a name="l07954"></a>07954 <span class="comment">!  Test on validity of parameters.</span>
    8019. 

  8019. <a name="l07955"></a>07955 <span class="comment">!</span>
    8020. 

  8020. <a name="l07956"></a>07956   result = 0.0e+00
    8021. 

  8021. <a name="l07957"></a>07957   abserr = 0.0e+00
    8022. 

  8022. <a name="l07958"></a>07958   neval = 0
    8023. 

  8023. <a name="l07959"></a>07959 
    8024. 

  8024. <a name="l07960"></a>07960   <span class="keyword">if</span> ( epsabs &lt; 0.0e+00 .and. epsrel &lt; 0.0e+00 ) <span class="keyword">then</span>
    8025. 

  8025. <a name="l07961"></a>07961     ier = 6
    8026. 

  8026. <a name="l07962"></a>07962     return
    8027. 

  8027. <a name="l07963"></a>07963   <span class="keyword">end if</span>
    8028. 

  8028. <a name="l07964"></a>07964 
    8029. 

  8029. <a name="l07965"></a>07965   hlgth = 5.0e-01 * ( b - a )
    8030. 

  8030. <a name="l07966"></a>07966   dhlgth = abs ( hlgth )
    8031. 

  8031. <a name="l07967"></a>07967   centr = 5.0e-01 * ( b + a )
    8032. 

  8032. <a name="l07968"></a>07968   fcentr = f(centr)
    8033. 

  8033. <a name="l07969"></a>07969   neval = 21
    8034. 

  8034. <a name="l07970"></a>07970   ier = 1
    8035. 

  8035. <a name="l07971"></a>07971 <span class="comment">!</span>
    8036. 

  8036. <a name="l07972"></a>07972 <span class="comment">!  Compute the integral using the 10- and 21-point formula.</span>
    8037. 

  8037. <a name="l07973"></a>07973 <span class="comment">!</span>
    8038. 

  8038. <a name="l07974"></a>07974   <span class="keyword">do</span> l = 1, 3
    8039. 

  8039. <a name="l07975"></a>07975 
    8040. 

  8040. <a name="l07976"></a>07976     <span class="keyword">if</span> ( l == 1 ) <span class="keyword">then</span>
    8041. 

  8041. <a name="l07977"></a>07977 
    8042. 

  8042. <a name="l07978"></a>07978       res10 = 0.0e+00
    8043. 

  8043. <a name="l07979"></a>07979       res21 = w21b(6) * fcentr
    8044. 

  8044. <a name="l07980"></a>07980       resabs = w21b(6) * abs(fcentr)
    8045. 

  8045. <a name="l07981"></a>07981 
    8046. 

  8046. <a name="l07982"></a>07982       <span class="keyword">do</span> k = 1, 5
    8047. 

  8047. <a name="l07983"></a>07983         absc = hlgth * x1(k)
    8048. 

  8048. <a name="l07984"></a>07984         fval1 = f(centr+absc)
    8049. 

  8049. <a name="l07985"></a>07985         fval2 = f(centr-absc)
    8050. 

  8050. <a name="l07986"></a>07986         fval = fval1 + fval2
    8051. 

  8051. <a name="l07987"></a>07987         res10 = res10 + w10(k)*fval
    8052. 

  8052. <a name="l07988"></a>07988         res21 = res21 + w21a(k)*fval
    8053. 

  8053. <a name="l07989"></a>07989         resabs = resabs + w21a(k)*(abs(fval1)+abs(fval2))
    8054. 

  8054. <a name="l07990"></a>07990         savfun(k) = fval
    8055. 

  8055. <a name="l07991"></a>07991         fv1(k) = fval1
    8056. 

  8056. <a name="l07992"></a>07992         fv2(k) = fval2
    8057. 

  8057. <a name="l07993"></a>07993       <span class="keyword">end do</span>
    8058. 

  8058. <a name="l07994"></a>07994 
    8059. 

  8059. <a name="l07995"></a>07995       ipx = 5
    8060. 

  8060. <a name="l07996"></a>07996 
    8061. 

  8061. <a name="l07997"></a>07997       <span class="keyword">do</span> k = 1, 5
    8062. 

  8062. <a name="l07998"></a>07998         ipx = ipx + 1
    8063. 

  8063. <a name="l07999"></a>07999         absc = hlgth * x2(k)
    8064. 

  8064. <a name="l08000"></a>08000         fval1 = f(centr+absc)
    8065. 

  8065. <a name="l08001"></a>08001         fval2 = f(centr-absc)
    8066. 

  8066. <a name="l08002"></a>08002         fval = fval1 + fval2
    8067. 

  8067. <a name="l08003"></a>08003         res21 = res21 + w21b(k) * fval
    8068. 

  8068. <a name="l08004"></a>08004         resabs = resabs + w21b(k) * ( abs ( fval1 ) + abs ( fval2 ) )
    8069. 

  8069. <a name="l08005"></a>08005         savfun(ipx) = fval
    8070. 

  8070. <a name="l08006"></a>08006         fv3(k) = fval1
    8071. 

  8071. <a name="l08007"></a>08007         fv4(k) = fval2
    8072. 

  8072. <a name="l08008"></a>08008       <span class="keyword">end do</span>
    8073. 

  8073. <a name="l08009"></a>08009 <span class="comment">!</span>
    8074. 

  8074. <a name="l08010"></a>08010 <span class="comment">!  Test for convergence.</span>
    8075. 

  8075. <a name="l08011"></a>08011 <span class="comment">!</span>
    8076. 

  8076. <a name="l08012"></a>08012       result = res21 * hlgth
    8077. 

  8077. <a name="l08013"></a>08013       resabs = resabs * dhlgth
    8078. 

  8078. <a name="l08014"></a>08014       reskh = 5.0e-01 * res21
    8079. 

  8079. <a name="l08015"></a>08015       resasc = w21b(6) * abs ( fcentr - reskh )
    8080. 

  8080. <a name="l08016"></a>08016 
    8081. 

  8081. <a name="l08017"></a>08017       <span class="keyword">do</span> k = 1, 5
    8082. 

  8082. <a name="l08018"></a>08018         resasc = resasc+w21a(k)*(abs(fv1(k)-reskh)+abs(fv2(k)-reskh)) &amp;
    8083. 

  8083. <a name="l08019"></a>08019                      +w21b(k)*(abs(fv3(k)-reskh)+abs(fv4(k)-reskh))
    8084. 

  8084. <a name="l08020"></a>08020       <span class="keyword">end do</span>
    8085. 

  8085. <a name="l08021"></a>08021 
    8086. 

  8086. <a name="l08022"></a>08022       abserr = abs ( ( res21 - res10 ) * hlgth )
    8087. 

  8087. <a name="l08023"></a>08023       resasc = resasc * dhlgth
    8088. 

  8088. <a name="l08024"></a>08024 <span class="comment">!</span>
    8089. 

  8089. <a name="l08025"></a>08025 <span class="comment">!  Compute the integral using the 43-point formula.</span>
    8090. 

  8090. <a name="l08026"></a>08026 <span class="comment">!</span>
    8091. 

  8091. <a name="l08027"></a>08027     <span class="keyword">else</span> <span class="keyword">if</span> ( l == 2 ) <span class="keyword">then</span>
    8092. 

  8092. <a name="l08028"></a>08028 
    8093. 

  8093. <a name="l08029"></a>08029       res43 = w43b(12)*fcentr
    8094. 

  8094. <a name="l08030"></a>08030       neval = 43
    8095. 

  8095. <a name="l08031"></a>08031 
    8096. 

  8096. <a name="l08032"></a>08032       <span class="keyword">do</span> k = 1, 10
    8097. 

  8097. <a name="l08033"></a>08033         res43 = res43 + savfun(k) * w43a(k)
    8098. 

  8098. <a name="l08034"></a>08034       <span class="keyword">end do</span>
    8099. 

  8099. <a name="l08035"></a>08035 
    8100. 

  8100. <a name="l08036"></a>08036       <span class="keyword">do</span> k = 1, 11
    8101. 

  8101. <a name="l08037"></a>08037         ipx = ipx + 1
    8102. 

  8102. <a name="l08038"></a>08038         absc = hlgth * x3(k)
    8103. 

  8103. <a name="l08039"></a>08039         fval = f(absc+centr) + f(centr-absc)
    8104. 

  8104. <a name="l08040"></a>08040         res43 = res43 + fval * w43b(k)
    8105. 

  8105. <a name="l08041"></a>08041         savfun(ipx) = fval
    8106. 

  8106. <a name="l08042"></a>08042       <span class="keyword">end do</span>
    8107. 

  8107. <a name="l08043"></a>08043 <span class="comment">!</span>
    8108. 

  8108. <a name="l08044"></a>08044 <span class="comment">!  Test for convergence.</span>
    8109. 

  8109. <a name="l08045"></a>08045 <span class="comment">!</span>
    8110. 

  8110. <a name="l08046"></a>08046       result = res43 * hlgth
    8111. 

  8111. <a name="l08047"></a>08047       abserr = abs((res43-res21)*hlgth)
    8112. 

  8112. <a name="l08048"></a>08048 <span class="comment">!</span>
    8113. 

  8113. <a name="l08049"></a>08049 <span class="comment">!  Compute the integral using the 87-point formula.</span>
    8114. 

  8114. <a name="l08050"></a>08050 <span class="comment">!</span>
    8115. 

  8115. <a name="l08051"></a>08051     <span class="keyword">else</span> <span class="keyword">if</span> ( l == 3 ) <span class="keyword">then</span>
    8116. 

  8116. <a name="l08052"></a>08052 
    8117. 

  8117. <a name="l08053"></a>08053       res87 = w87b(23) * fcentr
    8118. 

  8118. <a name="l08054"></a>08054       neval = 87
    8119. 

  8119. <a name="l08055"></a>08055 
    8120. 

  8120. <a name="l08056"></a>08056       <span class="keyword">do</span> k = 1, 21
    8121. 

  8121. <a name="l08057"></a>08057         res87 = res87 + savfun(k) * w87a(k)
    8122. 

  8122. <a name="l08058"></a>08058       <span class="keyword">end do</span>
    8123. 

  8123. <a name="l08059"></a>08059 
    8124. 

  8124. <a name="l08060"></a>08060       <span class="keyword">do</span> k = 1, 22
    8125. 

  8125. <a name="l08061"></a>08061         absc = hlgth * x4(k)
    8126. 

  8126. <a name="l08062"></a>08062         res87 = res87 + w87b(k) * ( f(absc+centr) + f(centr-absc) )
    8127. 

  8127. <a name="l08063"></a>08063       <span class="keyword">end do</span>
    8128. 

  8128. <a name="l08064"></a>08064 
    8129. 

  8129. <a name="l08065"></a>08065       result = res87 * hlgth
    8130. 

  8130. <a name="l08066"></a>08066       abserr = abs ( ( res87 - res43) * hlgth )
    8131. 

  8131. <a name="l08067"></a>08067 
    8132. 

  8132. <a name="l08068"></a>08068     <span class="keyword">end if</span>
    8133. 

  8133. <a name="l08069"></a>08069 
    8134. 

  8134. <a name="l08070"></a>08070     <span class="keyword">if</span> ( resasc /= 0.0e+00.and.abserr /= 0.0e+00 ) <span class="keyword">then</span>
    8135. 

  8135. <a name="l08071"></a>08071       abserr = resasc * min ( 1.0e+00,(2.0e+02*abserr/resasc)**1.5e+00)
    8136. 

  8136. <a name="l08072"></a>08072     <span class="keyword">end if</span>
    8137. 

  8137. <a name="l08073"></a>08073 
    8138. 

  8138. <a name="l08074"></a>08074     <span class="keyword">if</span> ( resabs &gt; tiny ( resabs ) / ( 5.0e+01 * epsilon ( resabs ) ) ) <span class="keyword">then</span>
    8139. 

  8139. <a name="l08075"></a>08075       abserr = max (( epsilon ( resabs ) *5.0e+01) * resabs, abserr )
    8140. 

  8140. <a name="l08076"></a>08076     <span class="keyword">end if</span>
    8141. 

  8141. <a name="l08077"></a>08077 
    8142. 

  8142. <a name="l08078"></a>08078     <span class="keyword">if</span> ( abserr &lt;= max ( epsabs, epsrel*abs(result))) <span class="keyword">then</span>
    8143. 

  8143. <a name="l08079"></a>08079       ier = 0
    8144. 

  8144. <a name="l08080"></a>08080     <span class="keyword">end if</span>
    8145. 

  8145. <a name="l08081"></a>08081 
    8146. 

  8146. <a name="l08082"></a>08082     <span class="keyword">if</span> ( ier == 0 ) <span class="keyword">then</span>
    8147. 

  8147. <a name="l08083"></a>08083       exit
    8148. 

  8148. <a name="l08084"></a>08084     <span class="keyword">end if</span>
    8149. 

  8149. <a name="l08085"></a>08085 
    8150. 

  8150. <a name="l08086"></a>08086   <span class="keyword">end do</span>
    8151. 

  8151. <a name="l08087"></a>08087 
    8152. 

  8152. <a name="l08088"></a>08088   return
    8153. 

  8153. <a name="l08089"></a>08089 <span class="keyword">end</span>
    8154. 

  8154. <a name="l08090"></a><a class="code" href="quadpack_8f90.html#a55e08a684c5a6315fb37dd0fdc66d8e6">08090</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a55e08a684c5a6315fb37dd0fdc66d8e6">qsort</a> ( limit, last, maxerr, ermax, elist, iord, nrmax )
    8155. 

  8155. <a name="l08091"></a>08091 
    8156. 

  8156. <a name="l08092"></a>08092 <span class="comment">!*****************************************************************************80</span>
    8157. 

  8157. <a name="l08093"></a>08093 <span class="comment">!</span>
    8158. 

  8158. <a name="l08094"></a>08094 <span class="comment">!! QSORT maintains the order of a list of local error estimates.</span>
    8159. 

  8159. <a name="l08095"></a>08095 <span class="comment">!</span>
    8160. 

  8160. <a name="l08096"></a>08096 <span class="comment">!  Discussion:</span>
    8161. 

  8161. <a name="l08097"></a>08097 <span class="comment">!</span>
    8162. 

  8162. <a name="l08098"></a>08098 <span class="comment">!    This routine maintains the descending ordering in the list of the </span>
    8163. 

  8163. <a name="l08099"></a>08099 <span class="comment">!    local error estimates resulting from the interval subdivision process. </span>
    8164. 

  8164. <a name="l08100"></a>08100 <span class="comment">!    At each call two error estimates are inserted using the sequential </span>
    8165. 

  8165. <a name="l08101"></a>08101 <span class="comment">!    search top-down for the largest error estimate and bottom-up for the</span>
    8166. 

  8166. <a name="l08102"></a>08102 <span class="comment">!    smallest error estimate.</span>
    8167. 

  8167. <a name="l08103"></a>08103 <span class="comment">!</span>
    8168. 

  8168. <a name="l08104"></a>08104 <span class="comment">!  Author:</span>
    8169. 

  8169. <a name="l08105"></a>08105 <span class="comment">!</span>
    8170. 

  8170. <a name="l08106"></a>08106 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    8171. 

  8171. <a name="l08107"></a>08107 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    8172. 

  8172. <a name="l08108"></a>08108 <span class="comment">!</span>
    8173. 

  8173. <a name="l08109"></a>08109 <span class="comment">!  Reference:</span>
    8174. 

  8174. <a name="l08110"></a>08110 <span class="comment">!</span>
    8175. 

  8175. <a name="l08111"></a>08111 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    8176. 

  8176. <a name="l08112"></a>08112 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    8177. 

  8177. <a name="l08113"></a>08113 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    8178. 

  8178. <a name="l08114"></a>08114 <span class="comment">!    Springer Verlag, 1983</span>
    8179. 

  8179. <a name="l08115"></a>08115 <span class="comment">!</span>
    8180. 

  8180. <a name="l08116"></a>08116 <span class="comment">!  Parameters:</span>
    8181. 

  8181. <a name="l08117"></a>08117 <span class="comment">!</span>
    8182. 

  8182. <a name="l08118"></a>08118 <span class="comment">!    Input, integer LIMIT, the maximum number of error estimates the list can</span>
    8183. 

  8183. <a name="l08119"></a>08119 <span class="comment">!    contain.</span>
    8184. 

  8184. <a name="l08120"></a>08120 <span class="comment">!</span>
    8185. 

  8185. <a name="l08121"></a>08121 <span class="comment">!    Input, integer LAST, the current number of error estimates.</span>
    8186. 

  8186. <a name="l08122"></a>08122 <span class="comment">!</span>
    8187. 

  8187. <a name="l08123"></a>08123 <span class="comment">!    Input/output, integer MAXERR, the index in the list of the NRMAX-th </span>
    8188. 

  8188. <a name="l08124"></a>08124 <span class="comment">!    largest error.</span>
    8189. 

  8189. <a name="l08125"></a>08125 <span class="comment">!</span>
    8190. 

  8190. <a name="l08126"></a>08126 <span class="comment">!    Output, real ERMAX, the NRMAX-th largest error = ELIST(MAXERR).</span>
    8191. 

  8191. <a name="l08127"></a>08127 <span class="comment">!</span>
    8192. 

  8192. <a name="l08128"></a>08128 <span class="comment">!    Input, real ELIST(LIMIT), contains the error estimates.</span>
    8193. 

  8193. <a name="l08129"></a>08129 <span class="comment">!</span>
    8194. 

  8194. <a name="l08130"></a>08130 <span class="comment">!    Input/output, integer IORD(LAST).  The first K elements contain </span>
    8195. 

  8195. <a name="l08131"></a>08131 <span class="comment">!    pointers to the error estimates such that ELIST(IORD(1)) through</span>
    8196. 

  8196. <a name="l08132"></a>08132 <span class="comment">!    ELIST(IORD(K)) form a decreasing sequence, with</span>
    8197. 

  8197. <a name="l08133"></a>08133 <span class="comment">!      K = LAST </span>
    8198. 

  8198. <a name="l08134"></a>08134 <span class="comment">!    if </span>
    8199. 

  8199. <a name="l08135"></a>08135 <span class="comment">!      LAST &lt;= (LIMIT/2+2), </span>
    8200. 

  8200. <a name="l08136"></a>08136 <span class="comment">!    and otherwise</span>
    8201. 

  8201. <a name="l08137"></a>08137 <span class="comment">!      K = LIMIT+1-LAST.</span>
    8202. 

  8202. <a name="l08138"></a>08138 <span class="comment">!</span>
    8203. 

  8203. <a name="l08139"></a>08139 <span class="comment">!    Input/output, integer NRMAX.</span>
    8204. 

  8204. <a name="l08140"></a>08140 <span class="comment">!</span>
    8205. 

  8205. <a name="l08141"></a>08141   <span class="keyword">implicit none</span>
    8206. 

  8206. <a name="l08142"></a>08142 
    8207. 

  8207. <a name="l08143"></a>08143   <span class="keywordtype">integer</span> last
    8208. 

  8208. <a name="l08144"></a>08144 
    8209. 

  8209. <a name="l08145"></a>08145   <span class="keywordtype">real</span> elist(last)
    8210. 

  8210. <a name="l08146"></a>08146   <span class="keywordtype">real</span> ermax
    8211. 

  8211. <a name="l08147"></a>08147   <span class="keywordtype">real</span> errmax
    8212. 

  8212. <a name="l08148"></a>08148   <span class="keywordtype">real</span> errmin
    8213. 

  8213. <a name="l08149"></a>08149   <span class="keywordtype">integer</span> i
    8214. 

  8214. <a name="l08150"></a>08150   <span class="keywordtype">integer</span> ibeg
    8215. 

  8215. <a name="l08151"></a>08151   <span class="keywordtype">integer</span> iord(last)
    8216. 

  8216. <a name="l08152"></a>08152   <span class="keywordtype">integer</span> isucc
    8217. 

  8217. <a name="l08153"></a>08153   <span class="keywordtype">integer</span> j
    8218. 

  8218. <a name="l08154"></a>08154   <span class="keywordtype">integer</span> jbnd
    8219. 

  8219. <a name="l08155"></a>08155   <span class="keywordtype">integer</span> jupbn
    8220. 

  8220. <a name="l08156"></a>08156   <span class="keywordtype">integer</span> k
    8221. 

  8221. <a name="l08157"></a>08157   <span class="keywordtype">integer</span> limit
    8222. 

  8222. <a name="l08158"></a>08158   <span class="keywordtype">integer</span> maxerr
    8223. 

  8223. <a name="l08159"></a>08159   <span class="keywordtype">integer</span> nrmax
    8224. 

  8224. <a name="l08160"></a>08160 <span class="comment">!</span>
    8225. 

  8225. <a name="l08161"></a>08161 <span class="comment">!  Check whether the list contains more than two error estimates.</span>
    8226. 

  8226. <a name="l08162"></a>08162 <span class="comment">!</span>
    8227. 

  8227. <a name="l08163"></a>08163   <span class="keyword">if</span> ( last &lt;= 2 ) <span class="keyword">then</span>
    8228. 

  8228. <a name="l08164"></a>08164     iord(1) = 1
    8229. 

  8229. <a name="l08165"></a>08165     iord(2) = 2
    8230. 

  8230. <a name="l08166"></a>08166     go to 90
    8231. 

  8231. <a name="l08167"></a>08167   <span class="keyword">end if</span>
    8232. 

  8232. <a name="l08168"></a>08168 <span class="comment">!</span>
    8233. 

  8233. <a name="l08169"></a>08169 <span class="comment">!  This part of the routine is only executed if, due to a</span>
    8234. 

  8234. <a name="l08170"></a>08170 <span class="comment">!  difficult integrand, subdivision increased the error</span>
    8235. 

  8235. <a name="l08171"></a>08171 <span class="comment">!  estimate. in the normal case the insert procedure should</span>
    8236. 

  8236. <a name="l08172"></a>08172 <span class="comment">!  start after the nrmax-th largest error estimate.</span>
    8237. 

  8237. <a name="l08173"></a>08173 <span class="comment">!</span>
    8238. 

  8238. <a name="l08174"></a>08174   errmax = elist(maxerr)
    8239. 

  8239. <a name="l08175"></a>08175 
    8240. 

  8240. <a name="l08176"></a>08176   <span class="keyword">do</span> i = 1, nrmax-1
    8241. 

  8241. <a name="l08177"></a>08177 
    8242. 

  8242. <a name="l08178"></a>08178     isucc = iord(nrmax-1)
    8243. 

  8243. <a name="l08179"></a>08179 
    8244. 

  8244. <a name="l08180"></a>08180     <span class="keyword">if</span> ( errmax &lt;= elist(isucc) ) <span class="keyword">then</span>
    8245. 

  8245. <a name="l08181"></a>08181       exit
    8246. 

  8246. <a name="l08182"></a>08182     <span class="keyword">end if</span>
    8247. 

  8247. <a name="l08183"></a>08183 
    8248. 

  8248. <a name="l08184"></a>08184     iord(nrmax) = isucc
    8249. 

  8249. <a name="l08185"></a>08185     nrmax = nrmax-1
    8250. 

  8250. <a name="l08186"></a>08186 
    8251. 

  8251. <a name="l08187"></a>08187   <span class="keyword">end do</span>
    8252. 

  8252. <a name="l08188"></a>08188 <span class="comment">!</span>
    8253. 

  8253. <a name="l08189"></a>08189 <span class="comment">!  Compute the number of elements in the list to be maintained</span>
    8254. 

  8254. <a name="l08190"></a>08190 <span class="comment">!  in descending order.  This number depends on the number of</span>
    8255. 

  8255. <a name="l08191"></a>08191 <span class="comment">!  subdivisions still allowed.</span>
    8256. 

  8256. <a name="l08192"></a>08192 <span class="comment">!</span>
    8257. 

  8257. <a name="l08193"></a>08193   jupbn = last
    8258. 

  8258. <a name="l08194"></a>08194 
    8259. 

  8259. <a name="l08195"></a>08195   <span class="keyword">if</span> ( (limit/2+2) &lt; last ) <span class="keyword">then</span>
    8260. 

  8260. <a name="l08196"></a>08196     jupbn = limit+3-last
    8261. 

  8261. <a name="l08197"></a>08197   <span class="keyword">end if</span>
    8262. 

  8262. <a name="l08198"></a>08198 
    8263. 

  8263. <a name="l08199"></a>08199   errmin = elist(last)
    8264. 

  8264. <a name="l08200"></a>08200 <span class="comment">!</span>
    8265. 

  8265. <a name="l08201"></a>08201 <span class="comment">!  Insert errmax by traversing the list top-down, starting</span>
    8266. 

  8266. <a name="l08202"></a>08202 <span class="comment">!  comparison from the element elist(iord(nrmax+1)).</span>
    8267. 

  8267. <a name="l08203"></a>08203 <span class="comment">!</span>
    8268. 

  8268. <a name="l08204"></a>08204   jbnd = jupbn-1
    8269. 

  8269. <a name="l08205"></a>08205   ibeg = nrmax+1
    8270. 

  8270. <a name="l08206"></a>08206 
    8271. 

  8271. <a name="l08207"></a>08207   <span class="keyword">do</span> i = ibeg, jbnd
    8272. 

  8272. <a name="l08208"></a>08208     isucc = iord(i)
    8273. 

  8273. <a name="l08209"></a>08209     <span class="keyword">if</span> ( elist(isucc) &lt;= errmax ) <span class="keyword">then</span>
    8274. 

  8274. <a name="l08210"></a>08210       go to 60
    8275. 

  8275. <a name="l08211"></a>08211     <span class="keyword">end if</span>
    8276. 

  8276. <a name="l08212"></a>08212     iord(i-1) = isucc
    8277. 

  8277. <a name="l08213"></a>08213   <span class="keyword">end do</span>
    8278. 

  8278. <a name="l08214"></a>08214 
    8279. 

  8279. <a name="l08215"></a>08215   iord(jbnd) = maxerr
    8280. 

  8280. <a name="l08216"></a>08216   iord(jupbn) = last
    8281. 

  8281. <a name="l08217"></a>08217   go to 90
    8282. 

  8282. <a name="l08218"></a>08218 <span class="comment">!</span>
    8283. 

  8283. <a name="l08219"></a>08219 <span class="comment">!  Insert errmin by traversing the list bottom-up.</span>
    8284. 

  8284. <a name="l08220"></a>08220 <span class="comment">!</span>
    8285. 

  8285. <a name="l08221"></a>08221 60 continue
    8286. 

  8286. <a name="l08222"></a>08222 
    8287. 

  8287. <a name="l08223"></a>08223   iord(i-1) = maxerr
    8288. 

  8288. <a name="l08224"></a>08224   k = jbnd
    8289. 

  8289. <a name="l08225"></a>08225 
    8290. 

  8290. <a name="l08226"></a>08226   <span class="keyword">do</span> j = i, jbnd
    8291. 

  8291. <a name="l08227"></a>08227     isucc = iord(k)
    8292. 

  8292. <a name="l08228"></a>08228     <span class="keyword">if</span> ( errmin &lt; elist(isucc) ) <span class="keyword">then</span>
    8293. 

  8293. <a name="l08229"></a>08229       go to 80
    8294. 

  8294. <a name="l08230"></a>08230     <span class="keyword">end if</span>
    8295. 

  8295. <a name="l08231"></a>08231     iord(k+1) = isucc
    8296. 

  8296. <a name="l08232"></a>08232     k = k-1
    8297. 

  8297. <a name="l08233"></a>08233   <span class="keyword">end do</span>
    8298. 

  8298. <a name="l08234"></a>08234 
    8299. 

  8299. <a name="l08235"></a>08235   iord(i) = last
    8300. 

  8300. <a name="l08236"></a>08236   go to 90
    8301. 

  8301. <a name="l08237"></a>08237 
    8302. 

  8302. <a name="l08238"></a>08238 80 continue
    8303. 

  8303. <a name="l08239"></a>08239 
    8304. 

  8304. <a name="l08240"></a>08240   iord(k+1) = last
    8305. 

  8305. <a name="l08241"></a>08241 <span class="comment">!</span>
    8306. 

  8306. <a name="l08242"></a>08242 <span class="comment">!  Set maxerr and ermax.</span>
    8307. 

  8307. <a name="l08243"></a>08243 <span class="comment">!</span>
    8308. 

  8308. <a name="l08244"></a>08244 90 continue
    8309. 

  8309. <a name="l08245"></a>08245 
    8310. 

  8310. <a name="l08246"></a>08246   maxerr = iord(nrmax)
    8311. 

  8311. <a name="l08247"></a>08247   ermax = elist(maxerr)
    8312. 

  8312. <a name="l08248"></a>08248 
    8313. 

  8313. <a name="l08249"></a>08249   return
    8314. 

  8314. <a name="l08250"></a>08250 <span class="keyword">end</span>
    8315. 

  8315. <a name="l08251"></a><a class="code" href="quadpack_8f90.html#aa5883d04ddfdfe9356462d43567a2a08">08251</a> <span class="keyword">function </span>qwgtc ( x, c, p2, p3, p4, kp )
    8316. 

  8316. <a name="l08252"></a>08252 
    8317. 

  8317. <a name="l08253"></a>08253 <span class="comment">!*****************************************************************************80</span>
    8318. 

  8318. <a name="l08254"></a>08254 <span class="comment">!</span>
    8319. 

  8319. <a name="l08255"></a>08255 <span class="comment">!! QWGTC defines the weight function used by QC25C.</span>
    8320. 

  8320. <a name="l08256"></a>08256 <span class="comment">!</span>
    8321. 

  8321. <a name="l08257"></a>08257 <span class="comment">!  Discussion:</span>
    8322. 

  8322. <a name="l08258"></a>08258 <span class="comment">!</span>
    8323. 

  8323. <a name="l08259"></a>08259 <span class="comment">!    The weight function has the form 1 / ( X - C ).</span>
    8324. 

  8324. <a name="l08260"></a>08260 <span class="comment">!</span>
    8325. 

  8325. <a name="l08261"></a>08261 <span class="comment">!  Author:</span>
    8326. 

  8326. <a name="l08262"></a>08262 <span class="comment">!</span>
    8327. 

  8327. <a name="l08263"></a>08263 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    8328. 

  8328. <a name="l08264"></a>08264 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    8329. 

  8329. <a name="l08265"></a>08265 <span class="comment">!</span>
    8330. 

  8330. <a name="l08266"></a>08266 <span class="comment">!  Reference:</span>
    8331. 

  8331. <a name="l08267"></a>08267 <span class="comment">!</span>
    8332. 

  8332. <a name="l08268"></a>08268 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    8333. 

  8333. <a name="l08269"></a>08269 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    8334. 

  8334. <a name="l08270"></a>08270 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    8335. 

  8335. <a name="l08271"></a>08271 <span class="comment">!    Springer Verlag, 1983</span>
    8336. 

  8336. <a name="l08272"></a>08272 <span class="comment">!</span>
    8337. 

  8337. <a name="l08273"></a>08273 <span class="comment">!  Parameters:</span>
    8338. 

  8338. <a name="l08274"></a>08274 <span class="comment">!</span>
    8339. 

  8339. <a name="l08275"></a>08275 <span class="comment">!    Input, real X, the point at which the weight function is evaluated.</span>
    8340. 

  8340. <a name="l08276"></a>08276 <span class="comment">!</span>
    8341. 

  8341. <a name="l08277"></a>08277 <span class="comment">!    Input, real C, the location of the singularity.</span>
    8342. 

  8342. <a name="l08278"></a>08278 <span class="comment">!</span>
    8343. 

  8343. <a name="l08279"></a>08279 <span class="comment">!    Input, real P2, P3, P4, parameters that are not used.</span>
    8344. 

  8344. <a name="l08280"></a>08280 <span class="comment">!</span>
    8345. 

  8345. <a name="l08281"></a>08281 <span class="comment">!    Input, integer KP, a parameter that is not used.</span>
    8346. 

  8346. <a name="l08282"></a>08282 <span class="comment">!</span>
    8347. 

  8347. <a name="l08283"></a>08283 <span class="comment">!    Output, real QWGTC, the value of the weight function at X.</span>
    8348. 

  8348. <a name="l08284"></a>08284 <span class="comment">!</span>
    8349. 

  8349. <a name="l08285"></a>08285   <span class="keyword">implicit none</span>
    8350. 

  8350. <a name="l08286"></a>08286 
    8351. 

  8351. <a name="l08287"></a>08287   <span class="keywordtype">real</span> c
    8352. 

  8352. <a name="l08288"></a>08288   <span class="keywordtype">integer</span> kp
    8353. 

  8353. <a name="l08289"></a>08289   <span class="keywordtype">real</span> p2
    8354. 

  8354. <a name="l08290"></a>08290   <span class="keywordtype">real</span> p3
    8355. 

  8355. <a name="l08291"></a>08291   <span class="keywordtype">real</span> p4
    8356. 

  8356. <a name="l08292"></a>08292   <span class="keywordtype">real</span> qwgtc
    8357. 

  8357. <a name="l08293"></a>08293   <span class="keywordtype">real</span> x
    8358. 

  8358. <a name="l08294"></a>08294 
    8359. 

  8359. <a name="l08295"></a>08295   qwgtc = 1.0E+00 / ( x - c )
    8360. 

  8360. <a name="l08296"></a>08296 
    8361. 

  8361. <a name="l08297"></a>08297   return
    8362. 

  8362. <a name="l08298"></a>08298 <span class="keyword">end</span>
    8363. 

  8363. <a name="l08299"></a><a class="code" href="quadpack_8f90.html#a24e54bf3e5c0bd729e50fd85f02b9cda">08299</a> <span class="keyword">function </span>qwgto ( x, omega, p2, p3, p4, integr )
    8364. 

  8364. <a name="l08300"></a>08300 
    8365. 

  8365. <a name="l08301"></a>08301 <span class="comment">!*****************************************************************************80</span>
    8366. 

  8366. <a name="l08302"></a>08302 <span class="comment">!</span>
    8367. 

  8367. <a name="l08303"></a>08303 <span class="comment">!! QWGTO defines the weight functions used by QC25O.</span>
    8368. 

  8368. <a name="l08304"></a>08304 <span class="comment">!</span>
    8369. 

  8369. <a name="l08305"></a>08305 <span class="comment">!  Author:</span>
    8370. 

  8370. <a name="l08306"></a>08306 <span class="comment">!</span>
    8371. 

  8371. <a name="l08307"></a>08307 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    8372. 

  8372. <a name="l08308"></a>08308 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    8373. 

  8373. <a name="l08309"></a>08309 <span class="comment">!</span>
    8374. 

  8374. <a name="l08310"></a>08310 <span class="comment">!  Reference:</span>
    8375. 

  8375. <a name="l08311"></a>08311 <span class="comment">!</span>
    8376. 

  8376. <a name="l08312"></a>08312 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    8377. 

  8377. <a name="l08313"></a>08313 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    8378. 

  8378. <a name="l08314"></a>08314 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    8379. 

  8379. <a name="l08315"></a>08315 <span class="comment">!    Springer Verlag, 1983</span>
    8380. 

  8380. <a name="l08316"></a>08316 <span class="comment">!</span>
    8381. 

  8381. <a name="l08317"></a>08317 <span class="comment">!  Parameters:</span>
    8382. 

  8382. <a name="l08318"></a>08318 <span class="comment">!</span>
    8383. 

  8383. <a name="l08319"></a>08319 <span class="comment">!    Input, real X, the point at which the weight function is evaluated.</span>
    8384. 

  8384. <a name="l08320"></a>08320 <span class="comment">!</span>
    8385. 

  8385. <a name="l08321"></a>08321 <span class="comment">!    Input, real OMEGA, the factor multiplying X.</span>
    8386. 

  8386. <a name="l08322"></a>08322 <span class="comment">!</span>
    8387. 

  8387. <a name="l08323"></a>08323 <span class="comment">!    Input, real P2, P3, P4, parameters that are not used.</span>
    8388. 

  8388. <a name="l08324"></a>08324 <span class="comment">!</span>
    8389. 

  8389. <a name="l08325"></a>08325 <span class="comment">!    Input, integer INTEGR, specifies which weight function is used:</span>
    8390. 

  8390. <a name="l08326"></a>08326 <span class="comment">!    1. W(X) = cos ( OMEGA * X )</span>
    8391. 

  8391. <a name="l08327"></a>08327 <span class="comment">!    2, W(X) = sin ( OMEGA * X )</span>
    8392. 

  8392. <a name="l08328"></a>08328 <span class="comment">!</span>
    8393. 

  8393. <a name="l08329"></a>08329 <span class="comment">!    Output, real QWGTO, the value of the weight function at X.</span>
    8394. 

  8394. <a name="l08330"></a>08330 <span class="comment">!</span>
    8395. 

  8395. <a name="l08331"></a>08331   <span class="keyword">implicit none</span>
    8396. 

  8396. <a name="l08332"></a>08332 
    8397. 

  8397. <a name="l08333"></a>08333   <span class="keywordtype">integer</span> integr
    8398. 

  8398. <a name="l08334"></a>08334   <span class="keywordtype">real</span> omega
    8399. 

  8399. <a name="l08335"></a>08335   <span class="keywordtype">real</span> p2
    8400. 

  8400. <a name="l08336"></a>08336   <span class="keywordtype">real</span> p3
    8401. 

  8401. <a name="l08337"></a>08337   <span class="keywordtype">real</span> p4
    8402. 

  8402. <a name="l08338"></a>08338   <span class="keywordtype">real</span> qwgto
    8403. 

  8403. <a name="l08339"></a>08339   <span class="keywordtype">real</span> x
    8404. 

  8404. <a name="l08340"></a>08340 
    8405. 

  8405. <a name="l08341"></a>08341   <span class="keyword">if</span> ( integr == 1 ) <span class="keyword">then</span>
    8406. 

  8406. <a name="l08342"></a>08342     qwgto = cos ( omega * x )
    8407. 

  8407. <a name="l08343"></a>08343   <span class="keyword">else</span> <span class="keyword">if</span> ( integr == 2 ) <span class="keyword">then</span>
    8408. 

  8408. <a name="l08344"></a>08344     qwgto = sin ( omega * x )
    8409. 

  8409. <a name="l08345"></a>08345   <span class="keyword">end if</span>
    8410. 

  8410. <a name="l08346"></a>08346 
    8411. 

  8411. <a name="l08347"></a>08347   return
    8412. 

  8412. <a name="l08348"></a>08348 <span class="keyword">end</span>
    8413. 

  8413. <a name="l08349"></a><a class="code" href="quadpack_8f90.html#a018e1602a4bb65f8d28bbff344bf8150">08349</a> <span class="keyword">function </span>qwgts ( x, a, b, alfa, beta, integr )
    8414. 

  8414. <a name="l08350"></a>08350 
    8415. 

  8415. <a name="l08351"></a>08351 <span class="comment">!*****************************************************************************80</span>
    8416. 

  8416. <a name="l08352"></a>08352 <span class="comment">!</span>
    8417. 

  8417. <a name="l08353"></a>08353 <span class="comment">!! QWGTS defines the weight functions used by QC25S.</span>
    8418. 

  8418. <a name="l08354"></a>08354 <span class="comment">!</span>
    8419. 

  8419. <a name="l08355"></a>08355 <span class="comment">!  Author:</span>
    8420. 

  8420. <a name="l08356"></a>08356 <span class="comment">!</span>
    8421. 

  8421. <a name="l08357"></a>08357 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    8422. 

  8422. <a name="l08358"></a>08358 <span class="comment">!    Christian Ueberhuber, David Kahaner</span>
    8423. 

  8423. <a name="l08359"></a>08359 <span class="comment">!</span>
    8424. 

  8424. <a name="l08360"></a>08360 <span class="comment">!  Reference:</span>
    8425. 

  8425. <a name="l08361"></a>08361 <span class="comment">!</span>
    8426. 

  8426. <a name="l08362"></a>08362 <span class="comment">!    Robert Piessens, Elise de Doncker-Kapenger, </span>
    8427. 

  8427. <a name="l08363"></a>08363 <span class="comment">!    Christian Ueberhuber, David Kahaner,</span>
    8428. 

  8428. <a name="l08364"></a>08364 <span class="comment">!    QUADPACK, a Subroutine Package for Automatic Integration,</span>
    8429. 

  8429. <a name="l08365"></a>08365 <span class="comment">!    Springer Verlag, 1983</span>
    8430. 

  8430. <a name="l08366"></a>08366 <span class="comment">!</span>
    8431. 

  8431. <a name="l08367"></a>08367 <span class="comment">!  Parameters:</span>
    8432. 

  8432. <a name="l08368"></a>08368 <span class="comment">!</span>
    8433. 

  8433. <a name="l08369"></a>08369 <span class="comment">!    Input, real X, the point at which the weight function is evaluated.</span>
    8434. 

  8434. <a name="l08370"></a>08370 <span class="comment">!</span>
    8435. 

  8435. <a name="l08371"></a>08371 <span class="comment">!    Input, real A, B, the endpoints of the integration interval.</span>
    8436. 

  8436. <a name="l08372"></a>08372 <span class="comment">!</span>
    8437. 

  8437. <a name="l08373"></a>08373 <span class="comment">!    Input, real ALFA, BETA, exponents that occur in the weight function.</span>
    8438. 

  8438. <a name="l08374"></a>08374 <span class="comment">!</span>
    8439. 

  8439. <a name="l08375"></a>08375 <span class="comment">!    Input, integer INTEGR, specifies which weight function is used:</span>
    8440. 

  8440. <a name="l08376"></a>08376 <span class="comment">!    1. W(X) = (X-A)**ALFA * (B-X)**BETA</span>
    8441. 

  8441. <a name="l08377"></a>08377 <span class="comment">!    2, W(X) = (X-A)**ALFA * (B-X)**BETA * log (X-A)</span>
    8442. 

  8442. <a name="l08378"></a>08378 <span class="comment">!    3, W(X) = (X-A)**ALFA * (B-X)**BETA * log (B-X)</span>
    8443. 

  8443. <a name="l08379"></a>08379 <span class="comment">!    4, W(X) = (X-A)**ALFA * (B-X)**BETA * log (X-A) * log(B-X)</span>
    8444. 

  8444. <a name="l08380"></a>08380 <span class="comment">!</span>
    8445. 

  8445. <a name="l08381"></a>08381 <span class="comment">!    Output, real QWGTS, the value of the weight function at X.</span>
    8446. 

  8446. <a name="l08382"></a>08382 <span class="comment">!</span>
    8447. 

  8447. <a name="l08383"></a>08383   <span class="keyword">implicit none</span>
    8448. 

  8448. <a name="l08384"></a>08384 
    8449. 

  8449. <a name="l08385"></a>08385   <span class="keywordtype">real</span> a
    8450. 

  8450. <a name="l08386"></a>08386   <span class="keywordtype">real</span> alfa
    8451. 

  8451. <a name="l08387"></a>08387   <span class="keywordtype">real</span> b
    8452. 

  8452. <a name="l08388"></a>08388   <span class="keywordtype">real</span> beta
    8453. 

  8453. <a name="l08389"></a>08389   <span class="keywordtype">integer</span> integr
    8454. 

  8454. <a name="l08390"></a>08390   <span class="keywordtype">real</span> qwgts
    8455. 

  8455. <a name="l08391"></a>08391   <span class="keywordtype">real</span> x
    8456. 

  8456. <a name="l08392"></a>08392 
    8457. 

  8457. <a name="l08393"></a>08393   <span class="keyword">if</span> ( integr == 1 ) <span class="keyword">then</span>
    8458. 

  8458. <a name="l08394"></a>08394     qwgts = ( x - a )**alfa * ( b - x )**beta
    8459. 

  8459. <a name="l08395"></a>08395   <span class="keyword">else</span> <span class="keyword">if</span> ( integr == 2 ) <span class="keyword">then</span>
    8460. 

  8460. <a name="l08396"></a>08396     qwgts = ( x - a )**alfa * ( b - x )**beta * log ( x - a )
    8461. 

  8461. <a name="l08397"></a>08397   <span class="keyword">else</span> <span class="keyword">if</span> ( integr == 3 ) <span class="keyword">then</span>
    8462. 

  8462. <a name="l08398"></a>08398     qwgts = ( x - a )**alfa * ( b - x )**beta * log ( b - x )
    8463. 

  8463. <a name="l08399"></a>08399   <span class="keyword">else</span> <span class="keyword">if</span> ( integr == 4 ) <span class="keyword">then</span>
    8464. 

  8464. <a name="l08400"></a>08400     qwgts = ( x - a )**alfa * ( b - x )**beta * log ( x - a ) * log ( b - x )
    8465. 

  8465. <a name="l08401"></a>08401   <span class="keyword">end if</span>
    8466. 

  8466. <a name="l08402"></a>08402 
    8467. 

  8467. <a name="l08403"></a>08403   return
    8468. 

  8468. <a name="l08404"></a>08404 <span class="keyword">end</span>
    8469. 

  8469. <a name="l08405"></a><a class="code" href="quadpack_8f90.html#a44fdaa08281ed3009542bc9257fa965d">08405</a> <span class="keyword">subroutine </span><a class="code" href="quadpack_8f90.html#a44fdaa08281ed3009542bc9257fa965d">timestamp</a> ( )
    8470. 

  8470. <a name="l08406"></a>08406 
    8471. 

  8471. <a name="l08407"></a>08407 <span class="comment">!*****************************************************************************80*</span>
    8472. 

  8472. <a name="l08408"></a>08408 <span class="comment">!</span>
    8473. 

  8473. <a name="l08409"></a>08409 <span class="comment">!! TIMESTAMP prints the current YMDHMS date as a time stamp.</span>
    8474. 

  8474. <a name="l08410"></a>08410 <span class="comment">!</span>
    8475. 

  8475. <a name="l08411"></a>08411 <span class="comment">!  Example:</span>
    8476. 

  8476. <a name="l08412"></a>08412 <span class="comment">!</span>
    8477. 

  8477. <a name="l08413"></a>08413 <span class="comment">!    May 31 2001   9:45:54.872 AM</span>
    8478. 

  8478. <a name="l08414"></a>08414 <span class="comment">!</span>
    8479. 

  8479. <a name="l08415"></a>08415 <span class="comment">!  Modified:</span>
    8480. 

  8480. <a name="l08416"></a>08416 <span class="comment">!</span>
    8481. 

  8481. <a name="l08417"></a>08417 <span class="comment">!    31 May 2001</span>
    8482. 

  8482. <a name="l08418"></a>08418 <span class="comment">!</span>
    8483. 

  8483. <a name="l08419"></a>08419 <span class="comment">!  Author:</span>
    8484. 

  8484. <a name="l08420"></a>08420 <span class="comment">!</span>
    8485. 

  8485. <a name="l08421"></a>08421 <span class="comment">!    John Burkardt</span>
    8486. 

  8486. <a name="l08422"></a>08422 <span class="comment">!</span>
    8487. 

  8487. <a name="l08423"></a>08423 <span class="comment">!  Parameters:</span>
    8488. 

  8488. <a name="l08424"></a>08424 <span class="comment">!</span>
    8489. 

  8489. <a name="l08425"></a>08425 <span class="comment">!    None</span>
    8490. 

  8490. <a name="l08426"></a>08426 <span class="comment">!</span>
    8491. 

  8491. <a name="l08427"></a>08427   <span class="keyword">implicit none</span>
    8492. 

  8492. <a name="l08428"></a>08428 
    8493. 

  8493. <a name="l08429"></a>08429   <span class="keywordtype">character ( len = 8 )</span> ampm
    8494. 

  8494. <a name="l08430"></a>08430   <span class="keywordtype">integer</span> d
    8495. 

  8495. <a name="l08431"></a>08431   <span class="keywordtype">character ( len = 8 )</span> date
    8496. 

  8496. <a name="l08432"></a>08432   <span class="keywordtype">integer</span> h
    8497. 

  8497. <a name="l08433"></a>08433   <span class="keywordtype">integer</span> m
    8498. 

  8498. <a name="l08434"></a>08434   <span class="keywordtype">integer</span> mm
    8499. 

  8499. <a name="l08435"></a>08435   <span class="keywordtype">character ( len = 9 )</span>, <span class="keywordtype">parameter</span>, <span class="keywordtype">dimension(12)</span> :: month = (/ 
    8500. 

  8500. <a name="l08436"></a>08436     <span class="stringliteral">&#39;January  &#39;</span>, <span class="stringliteral">&#39;February &#39;</span>, <span class="stringliteral">&#39;March    &#39;</span>, <span class="stringliteral">&#39;April    &#39;</span>, 
    8501. 

  8501. <a name="l08437"></a>08437     <span class="stringliteral">&#39;May      &#39;</span>, <span class="stringliteral">&#39;June     &#39;</span>, <span class="stringliteral">&#39;July     &#39;</span>, <span class="stringliteral">&#39;August   &#39;</span>, 
    8502. 

  8502. <a name="l08438"></a>08438     <span class="stringliteral">&#39;September&#39;</span>, <span class="stringliteral">&#39;October  &#39;</span>, <span class="stringliteral">&#39;November &#39;</span>, <span class="stringliteral">&#39;December &#39;</span> /)
    8503. 

  8503. <a name="l08439"></a>08439   <span class="keywordtype">integer</span> n
    8504. 

  8504. <a name="l08440"></a>08440   <span class="keywordtype">integer</span> s
    8505. 

  8505. <a name="l08441"></a>08441   <span class="keywordtype">character ( len = 10 )</span>  time
    8506. 

  8506. <a name="l08442"></a>08442   <span class="keywordtype">integer</span> values(8)
    8507. 

  8507. <a name="l08443"></a>08443   <span class="keywordtype">integer</span> y
    8508. 

  8508. <a name="l08444"></a>08444   <span class="keywordtype">character ( len = 5 )</span> zone
    8509. 

  8509. <a name="l08445"></a>08445 
    8510. 

  8510. <a name="l08446"></a>08446   call date_and_time ( date, time, zone, values )
    8511. 

  8511. <a name="l08447"></a>08447 
    8512. 

  8512. <a name="l08448"></a>08448   y = values(1)
    8513. 

  8513. <a name="l08449"></a>08449   m = values(2)
    8514. 

  8514. <a name="l08450"></a>08450   d = values(3)
    8515. 

  8515. <a name="l08451"></a>08451   h = values(5)
    8516. 

  8516. <a name="l08452"></a>08452   n = values(6)
    8517. 

  8517. <a name="l08453"></a>08453   s = values(7)
    8518. 

  8518. <a name="l08454"></a>08454   mm = values(8)
    8519. 

  8519. <a name="l08455"></a>08455 
    8520. 

  8520. <a name="l08456"></a>08456   <span class="keyword">if</span> ( h &lt; 12 ) <span class="keyword">then</span>
    8521. 

  8521. <a name="l08457"></a>08457     ampm = <span class="stringliteral">&#39;AM&#39;</span>
    8522. 

  8522. <a name="l08458"></a>08458   <span class="keyword">else</span> <span class="keyword">if</span> ( h == 12 ) <span class="keyword">then</span>
    8523. 

  8523. <a name="l08459"></a>08459     <span class="keyword">if</span> ( n == 0 .and. s == 0 ) <span class="keyword">then</span>
    8524. 

  8524. <a name="l08460"></a>08460       ampm = <span class="stringliteral">&#39;Noon&#39;</span>
    8525. 

  8525. <a name="l08461"></a>08461     <span class="keyword">else</span>
    8526. 

  8526. <a name="l08462"></a>08462       ampm = <span class="stringliteral">&#39;PM&#39;</span>
    8527. 

  8527. <a name="l08463"></a>08463     <span class="keyword">end if</span>
    8528. 

  8528. <a name="l08464"></a>08464   <span class="keyword">else</span>
    8529. 

  8529. <a name="l08465"></a>08465     h = h - 12
    8530. 

  8530. <a name="l08466"></a>08466     <span class="keyword">if</span> ( h &lt; 12 ) <span class="keyword">then</span>
    8531. 

  8531. <a name="l08467"></a>08467       ampm = <span class="stringliteral">&#39;PM&#39;</span>
    8532. 

  8532. <a name="l08468"></a>08468     <span class="keyword">else</span> <span class="keyword">if</span> ( h == 12 ) <span class="keyword">then</span>
    8533. 

  8533. <a name="l08469"></a>08469       <span class="keyword">if</span> ( n == 0 .and. s == 0 ) <span class="keyword">then</span>
    8534. 

  8534. <a name="l08470"></a>08470         ampm = <span class="stringliteral">&#39;Midnight&#39;</span>
    8535. 

  8535. <a name="l08471"></a>08471       <span class="keyword">else</span>
    8536. 

  8536. <a name="l08472"></a>08472         ampm = <span class="stringliteral">&#39;AM&#39;</span>
    8537. 

  8537. <a name="l08473"></a>08473       <span class="keyword">end if</span>
    8538. 

  8538. <a name="l08474"></a>08474     <span class="keyword">end if</span>
    8539. 

  8539. <a name="l08475"></a>08475   <span class="keyword">end if</span>
    8540. 

  8540. <a name="l08476"></a>08476 
    8541. 

  8541. <a name="l08477"></a>08477   <span class="keyword">write</span> ( *, <span class="stringliteral">&#39;(a,1x,i2,1x,i4,2x,i2,a1,i2.2,a1,i2.2,a1,i3.3,1x,a)&#39;</span> ) &amp;
    8542. 

  8542. <a name="l08478"></a>08478     trim ( month(m) ), d, y, h, <span class="stringliteral">&#39;:&#39;</span>, n, <span class="stringliteral">&#39;:&#39;</span>, s, <span class="stringliteral">&#39;.&#39;</span>, mm, trim ( ampm )
    8543. 

  8543. <a name="l08479"></a>08479 
    8544. 

  8544. <a name="l08480"></a>08480   return
    8545. 

  8545. <a name="l08481"></a>08481 <span class="keyword">end</span>
    8546. 

  8546. </pre></div></div>
    8547. 

  8547. </div>
    8548. 

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    8550. 

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    8551. 

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    8554. 

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    8564. 

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