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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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  64. <div class="contents">
    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">!  …
    608.

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