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1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> 4<title>Sample Article (The Remez Method)</title> 5<link rel="stylesheet" href="../../../../doc/src/boostbook.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.75.2"> 7<link rel="home" href="../index.html" title="Document To Test Formatting"> 8<link rel="up" href="../index.html" title="Document To Test Formatting"> 9<link rel="prev" href="test.html" title="test HTML4 symbols"> 10<link rel="next" href="array.html" title="Array Example Boostbook XML Documentation"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../boost.png"></td> 15<td align="center"><a href="../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="test.html"><img src="../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../index.html"><img src="../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="array.html"><img src="../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 27<a name="document_to_test_formatting.remez"></a><a class="link" href="remez.html" title="Sample Article (The Remez Method)"> Sample Article (The 28 Remez Method)</a> 29</h2></div></div></div> 30<p> 31 The <a href="http://en.wikipedia.org/wiki/Remez_algorithm" target="_top">Remez algorithm</a> 32 is a methodology for locating the minimax rational approximation to a function. 33 This short article gives a brief overview of the method, but it should not 34 be regarded as a thorough theoretical treatment, for that you should consult 35 your favorite textbook. 36 </p> 37<p> 38 Imagine that you want to approximate some function f(x) by way of a rational 39 function R(x), where R(x) may be either a polynomial P(x) or a ratio of two 40 polynomials P(x)/Q(x) (a rational function). Initially we'll concentrate on 41 the polynomial case, as it's by far the easier to deal with, later we'll extend 42 to the full rational function case. 43 </p> 44<p> 45 We want to find the "best" rational approximation, where "best" 46 is defined to be the approximation that has the least deviation from f(x). 47 We can measure the deviation by way of an error function: 48 </p> 49<p> 50 E<sub>abs</sub>(x) = f(x) - R(x) 51 </p> 52<p> 53 which is expressed in terms of absolute error, but we can equally use relative 54 error: 55 </p> 56<p> 57 E<sub>rel</sub>(x) = (f(x) - R(x)) / |f(x)| 58 </p> 59<p> 60 And indeed in general we can scale the error function in any way we want, it 61 makes no difference to the maths, although the two forms above cover almost 62 every practical case that you're likely to encounter. 63 </p> 64<p> 65 The minimax rational function R(x) is then defined to be the function that 66 yields the smallest maximal value of the error function. Chebyshev showed that 67 there is a unique minimax solution for R(x) that has the following properties: 68 </p> 69<div class="itemizedlist"><ul class="itemizedlist" type="disc"> 70<li class="listitem"> 71 If R(x) is a polynomial of degree N, then there are N+2 unknowns: the N+1 72 coefficients of the polynomial, and maximal value of the error function. 73 </li> 74<li class="listitem"> 75 The error function has N+1 roots, and N+2 extrema (minima and maxima). 76 </li> 77<li class="listitem"> 78 The extrema alternate in sign, and all have the same magnitude. 79 </li> 80</ul></div> 81<p> 82 That means that if we know the location of the extrema of the error function 83 then we can write N+2 simultaneous equations: 84 </p> 85<p> 86 R(x<sub>i</sub>) + (-1)<sup>i</sup>E = f(x<sub>i</sub>) 87 </p> 88<p> 89 where E is the maximal error term, and x<sub>i</sub> are the abscissa values of the N+2 90 extrema of the error function. It is then trivial to solve the simultaneous 91 equations to obtain the polynomial coefficients and the error term. 92 </p> 93<p> 94 <span class="emphasis"><em>Unfortunately we don't know where the extrema of the error function 95 are located!</em></span> 96 </p> 97<a name="document_to_test_formatting.remez.the_remez_method"></a><h5> 98<a name="id771060"></a> 99 <a class="link" href="remez.html#document_to_test_formatting.remez.the_remez_method">The Remez 100 Method</a> 101 </h5> 102<p> 103 The Remez method is an iterative technique which, given a broad range of assumptions, 104 will converge on the extrema of the error function, and therefore the minimax 105 solution. 106 </p> 107<p> 108 In the following discussion we'll use a concrete example to illustrate the 109 Remez method: an approximation to the function e<sup>x</sup> over the range [-1, 1]. 110 </p> 111<p> 112 Before we can begin the Remez method, we must obtain an initial value for the 113 location of the extrema of the error function. We could "guess" these, 114 but a much closer first approximation can be obtained by first constructing 115 an interpolated polynomial approximation to f(x). 116 </p> 117<p> 118 In order to obtain the N+1 coefficients of the interpolated polynomial we need 119 N+1 points (x<sub>0</sub>...x<sub>N</sub>): with our interpolated form passing through each of those 120 points that yields N+1 simultaneous equations: 121 </p> 122<p> 123 f(x<sub>i</sub>) = P(x<sub>i</sub>) = c<sub>0</sub> + c<sub>1</sub>x<sub>i</sub> ... + c<sub>N</sub>x<sub>i</sub><sup>N</sup> 124 </p> 125<p> 126 Which can be solved for the coefficients c<sub>0</sub>...c<sub>N</sub> in P(x). 127 </p> 128<p> 129 Obviously this is not a minimax solution, indeed our only guarantee is that 130 f(x) and P(x) touch at N+1 locations, away from those points the error may 131 be arbitrarily large. However, we would clearly like this initial approximation 132 to be as close to f(x) as possible, and it turns out that using the zeros of 133 an orthogonal polynomial as the initial interpolation points is a good choice. 134 In our example we'll use the zeros of a Chebyshev polynomial as these are particularly 135 easy to calculate, interpolating for a polynomial of degree 4, and measuring 136 <span class="emphasis"><em>relative error</em></span> we get the following error function: 137 </p> 138<p> 139 <span class="inlinemediaobject"><img src="../images/remez-2.png" alt="remez-2"></span> 140 </p> 141<p> 142 Which has a peak relative error of 1.2x10<sup>-3</sup>. 143 </p> 144<p> 145 While this is a pretty good approximation already, judging by the shape of 146 the error function we can clearly do better. Before starting on the Remez method 147 propper, we have one more step to perform: locate all the extrema of the error 148 function, and store these locations as our initial <span class="emphasis"><em>Chebyshev control 149 points</em></span>. 150 </p> 151<div class="note"><table border="0" summary="Note"> 152<tr> 153<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../doc/src/images/note.png"></td> 154<th align="left">Note</th> 155</tr> 156<tr><td align="left" valign="top"> 157<p> 158 In the simple case of a polynomial approximation, by interpolating through 159 the roots of a Chebyshev polynomial we have in fact created a <span class="emphasis"><em>Chebyshev 160 approximation</em></span> to the function: in terms of <span class="emphasis"><em>absolute 161 error</em></span> this is the best a priori choice for the interpolated form 162 we can achieve, and typically is very close to the minimax solution. 163 </p> 164<p> 165 However, if we want to optimise for <span class="emphasis"><em>relative error</em></span>, 166 or if the approximation is a rational function, then the initial Chebyshev 167 solution can be quite far from the ideal minimax solution. 168 </p> 169<p> 170 A more technical discussion of the theory involved can be found in this 171 <a href="http://math.fullerton.edu/mathews/n2003/ChebyshevPolyMod.html" target="_top">online 172 course</a>. 173 </p> 174</td></tr> 175</table></div> 176<a name="document_to_test_formatting.remez.remez_step_1"></a><h5> 177<a name="id771248"></a> 178 <a class="link" href="remez.html#document_to_test_formatting.remez.remez_step_1">Remez Step 1</a> 179 </h5> 180<p> 181 The first step in the Remez method, given our current set of N+2 Chebyshev 182 control points x<sub>i</sub>, is to solve the N+2 simultaneous equations: 183 </p> 184<p> 185 P(x<sub>i</sub>) + (-1)<sup>i</sup>E = f(x<sub>i</sub>) 186 </p> 187<p> 188 To obtain the error term E, and the coefficients of the polynomial P(x). 189 </p> 190<p> 191 This gives us a new approximation to f(x) that has the same error <span class="emphasis"><em>E</em></span> 192 at each of the control points, and whose error function <span class="emphasis"><em>alternates 193 in sign</em></span> at the control points. This is still not necessarily the 194 minimax solution though: since the control points may not be at the extrema 195 of the error function. After this first step here's what our approximation's 196 error function looks like: 197 </p> 198<p> 199 <span class="inlinemediaobject"><img src="../images/remez-3.png" alt="remez-3"></span> 200 </p> 201<p> 202 Clearly this is still not the minimax solution since the control points are 203 not located at the extrema, but the maximum relative error has now dropped 204 to 5.6x10<sup>-4</sup>. 205 </p> 206<a name="document_to_test_formatting.remez.remez_step_2"></a><h5> 207<a name="id771342"></a> 208 <a class="link" href="remez.html#document_to_test_formatting.remez.remez_step_2">Remez Step 2</a> 209 </h5> 210<p> 211 The second step is to locate the extrema of the new approximation, which we 212 do in two stages: first, since the error function changes sign at each control 213 point, we must have N+1 roots of the error function located between each pair 214 of N+2 control points. Once these roots are found by standard root finding 215 techniques, we know that N extrema are bracketed between each pair of roots, 216 plus two more between the endpoints of the range and the first and last roots. 217 The N+2 extrema can then be found using standard function minimisation techniques. 218 </p> 219<p> 220 We now have a choice: multi-point exchange, or single point exchange. 221 </p> 222<p> 223 In single point exchange, we move the control point nearest to the largest 224 extrema to the absissa value of the extrema. 225 </p> 226<p> 227 In multi-point exchange we swap all the current control points, for the locations 228 of the extrema. 229 </p> 230<p> 231 In our example we perform multi-point exchange. 232 </p> 233<a name="document_to_test_formatting.remez.iteration"></a><h5> 234<a name="id771387"></a> 235 <a class="link" href="remez.html#document_to_test_formatting.remez.iteration">Iteration</a> 236 </h5> 237<p> 238 The Remez method then performs steps 1 and 2 above iteratively until the control 239 points are located at the extrema of the error function: this is then the minimax 240 solution. 241 </p> 242<p> 243 For our current example, two more iterations converges on a minimax solution 244 with a peak relative error of 5x10<sup>-4</sup> and an error function that looks like: 245 </p> 246<p> 247 <span class="inlinemediaobject"><img src="../images/remez-4.png" alt="remez-4"></span> 248 </p> 249<a name="document_to_test_formatting.remez.rational_approximations"></a><h5> 250<a name="id771441"></a> 251 <a class="link" href="remez.html#document_to_test_formatting.remez.rational_approximations">Rational 252 Approximations</a> 253 </h5> 254<p> 255 If we wish to extend the Remez method to a rational approximation of the form 256 </p> 257<p> 258 f(x) = R(x) = P(x) / Q(x) 259 </p> 260<p> 261 where P(x) and Q(x) are polynomials, then we proceed as before, except that 262 now we have N+M+2 unknowns if P(x) is of order N and Q(x) is of order M. This 263 assumes that Q(x) is normalised so that it's leading coefficient is 1, giving 264 N+M+1 polynomial coefficients in total, plus the error term E. 265 </p> 266<p> 267 The simultaneous equations to be solved are now: 268 </p> 269<p> 270 P(x<sub>i</sub>) / Q(x<sub>i</sub>) + (-1)<sup>i</sup>E = f(x<sub>i</sub>) 271 </p> 272<p> 273 Evaluated at the N+M+2 control points x<sub>i</sub>. 274 </p> 275<p> 276 Unfortunately these equations are non-linear in the error term E: we can only 277 solve them if we know E, and yet E is one of the unknowns! 278 </p> 279<p> 280 The method usually adopted to solve these equations is an iterative one: we 281 guess the value of E, solve the equations to obtain a new value for E (as well 282 as the polynomial coefficients), then use the new value of E as the next guess. 283 The method is repeated until E converges on a stable value. 284 </p> 285<p> 286 These complications extend the running time required for the development of 287 rational approximations quite considerably. It is often desirable to obtain 288 a rational rather than polynomial approximation none the less: rational approximations 289 will often match more difficult to approximate functions, to greater accuracy, 290 and with greater efficiency, than their polynomial alternatives. For example, 291 if we takes our previous example of an approximation to e<sup>x</sup>, we obtained 5x10<sup>-4</sup> accuracy 292 with an order 4 polynomial. If we move two of the unknowns into the denominator 293 to give a pair of order 2 polynomials, and re-minimise, then the peak relative 294 error drops to 8.7x10<sup>-5</sup>. That's a 5 fold increase in accuracy, for the same 295 number of terms overall. 296 </p> 297<a name="document_to_test_formatting.remez.practical_considerations"></a><h5> 298<a name="id771550"></a> 299 <a class="link" href="remez.html#document_to_test_formatting.remez.practical_considerations">Practical 300 Considerations</a> 301 </h5> 302<p> 303 Most treatises on approximation theory stop at this point. However, from a 304 practical point of view, most of the work involves finding the right approximating 305 form, and then persuading the Remez method to converge on a solution. 306 </p> 307<p> 308 So far we have used a direct approximation: 309 </p> 310<p> 311 f(x) = R(x) 312 </p> 313<p> 314 But this will converge to a useful approximation only if f(x) is smooth. In 315 addition round-off errors when evaluating the rational form mean that this 316 will never get closer than within a few epsilon of machine precision. Therefore 317 this form of direct approximation is often reserved for situations where we 318 want efficiency, rather than accuracy. 319 </p> 320<p> 321 The first step in improving the situation is generally to split f(x) into a 322 dominant part that we can compute accurately by another method, and a slowly 323 changing remainder which can be approximated by a rational approximation. We 324 might be tempted to write: 325 </p> 326<p> 327 f(x) = g(x) + R(x) 328 </p> 329<p> 330 where g(x) is the dominant part of f(x), but if f(x)/g(x) is approximately 331 constant over the interval of interest then: 332 </p> 333<p> 334 f(x) = g(x)(c + R(x)) 335 </p> 336<p> 337 Will yield a much better solution: here <span class="emphasis"><em>c</em></span> is a constant 338 that is the approximate value of f(x)/g(x) and R(x) is typically tiny compared 339 to <span class="emphasis"><em>c</em></span>. In this situation if R(x) is optimised for absolute 340 error, then as long as its error is small compared to the constant <span class="emphasis"><em>c</em></span>, 341 that error will effectively get wiped out when R(x) is added to <span class="emphasis"><em>c</em></span>. 342 </p> 343<p> 344 The difficult part is obviously finding the right g(x) to extract from your 345 function: often the asymptotic behaviour of the function will give a clue, 346 so for example the function __erfc becomes proportional to e<sup>-x<sup>2</sup></sup>/x as x becomes 347 large. Therefore using: 348 </p> 349<p> 350 erfc(z) = (C + R(x)) e<sup>-x<sup>2</sup></sup>/x 351 </p> 352<p> 353 as the approximating form seems like an obvious thing to try, and does indeed 354 yield a useful approximation. 355 </p> 356<p> 357 However, the difficulty then becomes one of converging the minimax solution. 358 Unfortunately, it is known that for some functions the Remez method can lead 359 to divergent behaviour, even when the initial starting approximation is quite 360 good. Furthermore, it is not uncommon for the solution obtained in the first 361 Remez step above to be a bad one: the equations to be solved are generally 362 "stiff", often very close to being singular, and assuming a solution 363 is found at all, round-off errors and a rapidly changing error function, can 364 lead to a situation where the error function does not in fact change sign at 365 each control point as required. If this occurs, it is fatal to the Remez method. 366 It is also possible to obtain solutions that are perfectly valid mathematically, 367 but which are quite useless computationally: either because there is an unavoidable 368 amount of roundoff error in the computation of the rational function, or because 369 the denominator has one or more roots over the interval of the approximation. 370 In the latter case while the approximation may have the correct limiting value 371 at the roots, the approximation is nonetheless useless. 372 </p> 373<p> 374 Assuming that the approximation does not have any fatal errors, and that the 375 only issue is converging adequately on the minimax solution, the aim is to 376 get as close as possible to the minimax solution before beginning the Remez 377 method. Using the zeros of a Chebyshev polynomial for the initial interpolation 378 is a good start, but may not be ideal when dealing with relative errors and/or 379 rational (rather than polynomial) approximations. One approach is to skew the 380 initial interpolation points to one end: for example if we raise the roots 381 of the Chebyshev polynomial to a positive power greater than 1 then the roots 382 will be skewed towards the middle of the [-1,1] interval, while a positive 383 power less than one will skew them towards either end. More usefully, if we 384 initially rescale the points over [0,1] and then raise to a positive power, 385 we can skew them to the left or right. Returning to our example of e<sup>x</sup> over [-1,1], 386 the initial interpolated form was some way from the minimax solution: 387 </p> 388<p> 389 <span class="inlinemediaobject"><img src="../images/remez-2.png" alt="remez-2"></span> 390 </p> 391<p> 392 However, if we first skew the interpolation points to the left (rescale them 393 to [0, 1], raise to the power 1.3, and then rescale back to [-1,1]) we reduce 394 the error from 1.3x10<sup>-3</sup>to 6x10<sup>-4</sup>: 395 </p> 396<p> 397 <span class="inlinemediaobject"><img src="../images/remez-5.png" alt="remez-5"></span> 398 </p> 399<p> 400 It's clearly still not ideal, but it is only a few percent away from our desired 401 minimax solution (5x10<sup>-4</sup>). 402 </p> 403<a name="document_to_test_formatting.remez.remez_method_checklist"></a><h5> 404<a name="id771737"></a> 405 <a class="link" href="remez.html#document_to_test_formatting.remez.remez_method_checklist">Remez 406 Method Checklist</a> 407 </h5> 408<p> 409 The following lists some of the things to check if the Remez method goes wrong, 410 it is by no means an exhaustive list, but is provided in the hopes that it 411 will prove useful. 412 </p> 413<div class="itemizedlist"><ul class="itemizedlist" type="disc"> 414<li class="listitem"> 415 Is the function smooth enough? Can it be better separated into a rapidly 416 changing part, and an asymptotic part? 417 </li> 418<li class="listitem"> 419 Does the function being approximated have any "blips" in it? 420 Check for problems as the function changes computation method, or if a 421 root, or an infinity has been divided out. The telltale sign is if there 422 is a narrow region where the Remez method will not converge. 423 </li> 424<li class="listitem"> 425 Check you have enough accuracy in your calculations: remember that the 426 Remez method works on the difference between the approximation and the 427 function being approximated: so you must have more digits of precision 428 available than the precision of the approximation being constructed. So 429 for example at double precision, you shouldn't expect to be able to get 430 better than a float precision approximation. 431 </li> 432<li class="listitem"> 433 Try skewing the initial interpolated approximation to minimise the error 434 before you begin the Remez steps. 435 </li> 436<li class="listitem"> 437 If the approximation won't converge or is ill-conditioned from one starting 438 location, try starting from a different location. 439 </li> 440<li class="listitem"> 441 If a rational function won't converge, one can minimise a polynomial (which 442 presents no problems), then rotate one term from the numerator to the denominator 443 and minimise again. In theory one can continue moving terms one at a time 444 from numerator to denominator, and then re-minimising, retaining the last 445 set of control points at each stage. 446 </li> 447<li class="listitem"> 448 Try using a smaller interval. It may also be possible to optimise over 449 one (small) interval, rescale the control points over a larger interval, 450 and then re-minimise. 451 </li> 452<li class="listitem"> 453 Keep absissa values small: use a change of variable to keep the abscissa 454 over, say [0, b], for some smallish value <span class="emphasis"><em>b</em></span>. 455 </li> 456</ul></div> 457<a name="document_to_test_formatting.remez.references"></a><h5> 458<a name="id771857"></a> 459 <a class="link" href="remez.html#document_to_test_formatting.remez.references">References</a> 460 </h5> 461<p> 462 The original references for the Remez Method and it's extension to rational 463 functions are unfortunately in Russian: 464 </p> 465<p> 466 Remez, E.Ya., <span class="emphasis"><em>Fundamentals of numerical methods for Chebyshev approximations</em></span>, 467 "Naukova Dumka", Kiev, 1969. 468 </p> 469<p> 470 Remez, E.Ya., Gavrilyuk, V.T., <span class="emphasis"><em>Computer development of certain approaches 471 to the approximate construction of solutions of Chebyshev problems nonlinearly 472 depending on parameters</em></span>, Ukr. Mat. Zh. 12 (1960), 324-338. 473 </p> 474<p> 475 Gavrilyuk, V.T., <span class="emphasis"><em>Generalization of the first polynomial algorithm 476 of E.Ya.Remez for the problem of constructing rational-fractional Chebyshev 477 approximations</em></span>, Ukr. Mat. Zh. 16 (1961), 575-585. 478 </p> 479<p> 480 Some English language sources include: 481 </p> 482<p> 483 Fraser, W., Hart, J.F., <span class="emphasis"><em>On the computation of rational approximations 484 to continuous functions</em></span>, Comm. of the ACM 5 (1962), 401-403, 414. 485 </p> 486<p> 487 Ralston, A., <span class="emphasis"><em>Rational Chebyshev approximation by Remes' algorithms</em></span>, 488 Numer.Math. 7 (1965), no. 4, 322-330. 489 </p> 490<p> 491 A. Ralston, <span class="emphasis"><em>Rational Chebyshev approximation, Mathematical Methods 492 for Digital Computers v. 2</em></span> (Ralston A., Wilf H., eds.), Wiley, New 493 York, 1967, pp. 264-284. 494 </p> 495<p> 496 Hart, J.F. e.a., <span class="emphasis"><em>Computer approximations</em></span>, Wiley, New York 497 a.o., 1968. 498 </p> 499<p> 500 Cody, W.J., Fraser, W., Hart, J.F., <span class="emphasis"><em>Rational Chebyshev approximation 501 using linear equations</em></span>, Numer.Math. 12 (1968), 242-251. 502 </p> 503<p> 504 Cody, W.J., <span class="emphasis"><em>A survey of practical rational and polynomial approximation 505 of functions</em></span>, SIAM Review 12 (1970), no. 3, 400-423. 506 </p> 507<p> 508 Barrar, R.B., Loeb, H.J., <span class="emphasis"><em>On the Remez algorithm for non-linear families</em></span>, 509 Numer.Math. 15 (1970), 382-391. 510 </p> 511<p> 512 Dunham, Ch.B., <span class="emphasis"><em>Convergence of the Fraser-Hart algorithm for rational 513 Chebyshev approximation</em></span>, Math. Comp. 29 (1975), no. 132, 1078-1082. 514 </p> 515<p> 516 G. L. Litvinov, <span class="emphasis"><em>Approximate construction of rational approximations 517 and the effect of error autocorrection</em></span>, Russian Journal of Mathematical 518 Physics, vol.1, No. 3, 1994. 519 </p> 520</div> 521<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 522<td align="left"></td> 523<td align="right"><div class="copyright-footer">Copyright © 2007 John Maddock, Joel de Guzman, Eric Niebler and Matias 524 Capeletto<p> 525 Distributed under the Boost Software License, Version 1.0. (See accompanying 526 file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) 527 </p> 528</div></td> 529</tr></table> 530<hr> 531<div class="spirit-nav"> 532<a accesskey="p" href="test.html"><img src="../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../index.html"><img src="../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="array.html"><img src="../../../../doc/src/images/next.png" alt="Next"></a> 533</div> 534</body> 535</html>