1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>The Lanczos Approximation</title> 5<link rel="stylesheet" href="../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../backgrounders.html" title="Chapter 23. 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However, the Lanczos 45 approximation does have a couple of properties that make it worthy of further 46 consideration: 47 </p> 48<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 49<li class="listitem"> 50 The approximation has an easy to compute truncation error that holds for 51 all <span class="emphasis"><em>z > 0</em></span>. In practice that means we can use the 52 same approximation for all <span class="emphasis"><em>z > 0</em></span>, and be certain 53 that no matter how large or small <span class="emphasis"><em>z</em></span> is, the truncation 54 error will <span class="emphasis"><em>at worst</em></span> be bounded by some finite value. 55 </li> 56<li class="listitem"> 57 The approximation has a form that is particularly amenable to analytic 58 manipulation, in particular ratios of gamma or gamma-like functions are 59 particularly easy to compute without resorting to logarithms. 60 </li> 61</ul></div> 62<p> 63 It is the combination of these two properties that make the approximation attractive: 64 Stirling's approximation is highly accurate for large z, and has some of the 65 same analytic properties as the Lanczos approximation, but can't easily be 66 used across the whole range of z. 67 </p> 68<p> 69 As the simplest example, consider the ratio of two gamma functions: one could 70 compute the result via lgamma: 71 </p> 72<pre class="programlisting"><span class="identifier">exp</span><span class="special">(</span><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">b</span><span class="special">));</span> 73</pre> 74<p> 75 However, even if lgamma is uniformly accurate to 0.5ulp, the worst case relative 76 error in the above can easily be shown to be: 77 </p> 78<pre class="programlisting"><span class="identifier">Erel</span> <span class="special">></span> <span class="identifier">a</span> <span class="special">*</span> <span class="identifier">log</span><span class="special">(</span><span class="identifier">a</span><span class="special">)/</span><span class="number">2</span> <span class="special">+</span> <span class="identifier">b</span> <span class="special">*</span> <span class="identifier">log</span><span class="special">(</span><span class="identifier">b</span><span class="special">)/</span><span class="number">2</span> 79</pre> 80<p> 81 For small <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> that's not a problem, 82 but to put the relationship another way: <span class="emphasis"><em>each time a and b increase 83 in magnitude by a factor of 10, at least one decimal digit of precision will 84 be lost.</em></span> 85 </p> 86<p> 87 In contrast, by analytically combining like power terms in a ratio of Lanczos 88 approximation's, these errors can be virtually eliminated for small <span class="emphasis"><em>a</em></span> 89 and <span class="emphasis"><em>b</em></span>, and kept under control for very large (or very 90 small for that matter) <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>. Of 91 course, computing large powers is itself a notoriously hard problem, but even 92 so, analytic combinations of Lanczos approximations can make the difference 93 between obtaining a valid result, or simply garbage. Refer to the implementation 94 notes for the <a class="link" href="sf_beta/beta_function.html" title="Beta">beta</a> 95 function for an example of this method in practice. The incomplete <a class="link" href="sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p 96 gamma</a> and <a class="link" href="sf_beta/ibeta_function.html" title="Incomplete Beta Functions">beta</a> 97 functions use similar analytic combinations of power terms, to combine gamma 98 and beta functions divided by large powers into single (simpler) expressions. 99 </p> 100<h5> 101<a name="math_toolkit.lanczos.h1"></a> 102 <span class="phrase"><a name="math_toolkit.lanczos.the_approximation"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.the_approximation">The 103 Approximation</a> 104 </h5> 105<p> 106 The Lanczos Approximation to the Gamma Function is given by: 107 </p> 108<div class="blockquote"><blockquote class="blockquote"><p> 109 <span class="inlinemediaobject"><img src="../../equations/lanczos0.svg"></span> 110 111 </p></blockquote></div> 112<p> 113 Where S<sub>g</sub>(z) is an infinite sum, that is convergent for all z > 0, and <span class="emphasis"><em>g</em></span> 114 is an arbitrary parameter that controls the "shape" of the terms 115 in the sum which is given by: 116 </p> 117<div class="blockquote"><blockquote class="blockquote"><p> 118 <span class="inlinemediaobject"><img src="../../equations/lanczos0a.svg"></span> 119 120 </p></blockquote></div> 121<p> 122 With individual coefficients defined in closed form by: 123 </p> 124<div class="blockquote"><blockquote class="blockquote"><p> 125 <span class="inlinemediaobject"><img src="../../equations/lanczos0b.svg"></span> 126 127 </p></blockquote></div> 128<p> 129 However, evaluation of the sum in that form can lead to numerical instability 130 in the computation of the ratios of rising and falling factorials (effectively 131 we're multiplying by a series of numbers very close to 1, so roundoff errors 132 can accumulate quite rapidly). 133 </p> 134<p> 135 The Lanczos approximation is therefore often written in partial fraction form 136 with the leading constants absorbed by the coefficients in the sum: 137 </p> 138<div class="blockquote"><blockquote class="blockquote"><p> 139 <span class="inlinemediaobject"><img src="../../equations/lanczos1.svg"></span> 140 141 </p></blockquote></div> 142<p> 143 where: 144 </p> 145<div class="blockquote"><blockquote class="blockquote"><p> 146 <span class="inlinemediaobject"><img src="../../equations/lanczos2.svg"></span> 147 148 </p></blockquote></div> 149<p> 150 Again parameter <span class="emphasis"><em>g</em></span> is an arbitrarily chosen constant, and 151 <span class="emphasis"><em>N</em></span> is an arbitrarily chosen number of terms to evaluate 152 in the "Lanczos sum" part. 153 </p> 154<div class="note"><table border="0" summary="Note"> 155<tr> 156<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td> 157<th align="left">Note</th> 158</tr> 159<tr><td align="left" valign="top"><p> 160 Some authors choose to define the sum from k=1 to N, and hence end up with 161 N+1 coefficients. This happens to confuse both the following discussion and 162 the code (since C++ deals with half open array ranges, rather than the closed 163 range of the sum). This convention is consistent with <a class="link" href="lanczos.html#godfrey">Godfrey</a>, 164 but not <a class="link" href="lanczos.html#pugh">Pugh</a>, so take care when referring to 165 the literature in this field. 166 </p></td></tr> 167</table></div> 168<h5> 169<a name="math_toolkit.lanczos.h2"></a> 170 <span class="phrase"><a name="math_toolkit.lanczos.computing_the_coefficients"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.computing_the_coefficients">Computing 171 the Coefficients</a> 172 </h5> 173<p> 174 The coefficients C0..CN-1 need to be computed from <span class="emphasis"><em>N</em></span> and 175 <span class="emphasis"><em>g</em></span> at high precision, and then stored as part of the program. 176 Calculation of the coefficients is performed via the method of <a class="link" href="lanczos.html#godfrey">Godfrey</a>; 177 let the constants be contained in a column vector P, then: 178 </p> 179<p> 180 P = D B C F 181 </p> 182<p> 183 where B is an NxN matrix: 184 </p> 185<div class="blockquote"><blockquote class="blockquote"><p> 186 <span class="inlinemediaobject"><img src="../../equations/lanczos4.svg"></span> 187 188 </p></blockquote></div> 189<p> 190 D is an NxN matrix: 191 </p> 192<div class="blockquote"><blockquote class="blockquote"><p> 193 <span class="inlinemediaobject"><img src="../../equations/lanczos3.svg"></span> 194 195 </p></blockquote></div> 196<p> 197 C is an NxN matrix: 198 </p> 199<div class="blockquote"><blockquote class="blockquote"><p> 200 <span class="inlinemediaobject"><img src="../../equations/lanczos5.svg"></span> 201 202 </p></blockquote></div> 203<p> 204 and F is an N element column vector: 205 </p> 206<div class="blockquote"><blockquote class="blockquote"><p> 207 <span class="inlinemediaobject"><img src="../../equations/lanczos6.svg"></span> 208 209 </p></blockquote></div> 210<p> 211 Note than the matrices B, D and C contain all integer terms and depend only 212 on <span class="emphasis"><em>N</em></span>, this product should be computed first, and then 213 multiplied by <span class="emphasis"><em>F</em></span> as the last step. 214 </p> 215<h5> 216<a name="math_toolkit.lanczos.h3"></a> 217 <span class="phrase"><a name="math_toolkit.lanczos.choosing_the_right_parameters"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.choosing_the_right_parameters">Choosing 218 the Right Parameters</a> 219 </h5> 220<p> 221 The trick is to choose <span class="emphasis"><em>N</em></span> and <span class="emphasis"><em>g</em></span> to 222 give the desired level of accuracy: choosing a small value for <span class="emphasis"><em>g</em></span> 223 leads to a strictly convergent series, but one which converges only slowly. 224 Choosing a larger value of <span class="emphasis"><em>g</em></span> causes the terms in the series 225 to be large and/or divergent for about the first <span class="emphasis"><em>g-1</em></span> terms, 226 and to then suddenly converge with a "crunch". 227 </p> 228<p> 229 <a class="link" href="lanczos.html#pugh">Pugh</a> has determined the optimal value of <span class="emphasis"><em>g</em></span> 230 for <span class="emphasis"><em>N</em></span> in the range <span class="emphasis"><em>1 <= N <= 60</em></span>: 231 unfortunately in practice choosing these values leads to cancellation errors 232 in the Lanczos sum as the largest term in the (alternating) series is approximately 233 1000 times larger than the result. These optimal values appear not to be useful 234 in practice unless the evaluation can be done with a number of guard digits 235 <span class="emphasis"><em>and</em></span> the coefficients are stored at higher precision than 236 that desired in the result. These values are best reserved for say, computing 237 to float precision with double precision arithmetic. 238 </p> 239<div class="table"> 240<a name="math_toolkit.lanczos.optimal_choices_for_n_and_g_when"></a><p class="title"><b>Table 23.1. Optimal choices for N and g when computing with guard digits (source: 241 Pugh)</b></p> 242<div class="table-contents"><table class="table" summary="Optimal choices for N and g when computing with guard digits (source: 243 Pugh)"> 244<colgroup> 245<col> 246<col> 247<col> 248<col> 249</colgroup> 250<thead><tr> 251<th> 252 <p> 253 Significand Size 254 </p> 255 </th> 256<th> 257 <p> 258 N 259 </p> 260 </th> 261<th> 262 <p> 263 g 264 </p> 265 </th> 266<th> 267 <p> 268 Max Error 269 </p> 270 </th> 271</tr></thead> 272<tbody> 273<tr> 274<td> 275 <p> 276 24 277 </p> 278 </td> 279<td> 280 <p> 281 6 282 </p> 283 </td> 284<td> 285 <p> 286 5.581 287 </p> 288 </td> 289<td> 290 <p> 291 9.51e-12 292 </p> 293 </td> 294</tr> 295<tr> 296<td> 297 <p> 298 53 299 </p> 300 </td> 301<td> 302 <p> 303 13 304 </p> 305 </td> 306<td> 307 <p> 308 13.144565 309 </p> 310 </td> 311<td> 312 <p> 313 9.2213e-23 314 </p> 315 </td> 316</tr> 317</tbody> 318</table></div> 319</div> 320<br class="table-break"><p> 321 The alternative described by <a class="link" href="lanczos.html#godfrey">Godfrey</a> is to perform 322 an exhaustive search of the <span class="emphasis"><em>N</em></span> and <span class="emphasis"><em>g</em></span> 323 parameter space to determine the optimal combination for a given <span class="emphasis"><em>p</em></span> 324 digit floating-point type. Repeating this work found a good approximation for 325 double precision arithmetic (close to the one <a class="link" href="lanczos.html#godfrey">Godfrey</a> 326 found), but failed to find really good approximations for 80 or 128-bit long 327 doubles. Further it was observed that the approximations obtained tended to 328 optimised for the small values of z (1 < z < 200) used to test the implementation 329 against the factorials. Computing ratios of gamma functions with large arguments 330 were observed to suffer from error resulting from the truncation of the Lancozos 331 series. 332 </p> 333<p> 334 <a class="link" href="lanczos.html#pugh">Pugh</a> identified all the locations where the theoretical 335 error of the approximation were at a minimum, but unfortunately has published 336 only the largest of these minima. However, he makes the observation that the 337 minima coincide closely with the location where the first neglected term (a<sub>N</sub>) 338 in the Lanczos series S<sub>g</sub>(z) changes sign. These locations are quite easy to 339 locate, albeit with considerable computer time. These "sweet spots" 340 need only be computed once, tabulated, and then searched when required for 341 an approximation that delivers the required precision for some fixed precision 342 type. 343 </p> 344<p> 345 Unfortunately, following this path failed to find a really good approximation 346 for 128-bit long doubles, and those found for 64 and 80-bit reals required 347 an excessive number of terms. There are two competing issues here: high precision 348 requires a large value of <span class="emphasis"><em>g</em></span>, but avoiding cancellation 349 errors in the evaluation requires a small <span class="emphasis"><em>g</em></span>. 350 </p> 351<p> 352 At this point note that the Lanczos sum can be converted into rational form 353 (a ratio of two polynomials, obtained from the partial-fraction form using 354 polynomial arithmetic), and doing so changes the coefficients so that <span class="emphasis"><em>they 355 are all positive</em></span>. That means that the sum in rational form can be 356 evaluated without cancellation error, albeit with double the number of coefficients 357 for a given N. Repeating the search of the "sweet spots", this time 358 evaluating the Lanczos sum in rational form, and testing only those "sweet 359 spots" whose theoretical error is less than the machine epsilon for the 360 type being tested, yielded good approximations for all the types tested. The 361 optimal values found were quite close to the best cases reported by <a class="link" href="lanczos.html#pugh">Pugh</a> 362 (just slightly larger <span class="emphasis"><em>N</em></span> and slightly smaller <span class="emphasis"><em>g</em></span> 363 for a given precision than <a class="link" href="lanczos.html#pugh">Pugh</a> reports), and even 364 though converting to rational form doubles the number of stored coefficients, 365 it should be noted that half of them are integers (and therefore require less 366 storage space) and the approximations require a smaller <span class="emphasis"><em>N</em></span> 367 than would otherwise be required, so fewer floating point operations may be 368 required overall. 369 </p> 370<p> 371 The following table shows the optimal values for <span class="emphasis"><em>N</em></span> and 372 <span class="emphasis"><em>g</em></span> when computing at fixed precision. These should be taken 373 as work in progress: there are no values for 106-bit significand machines (Darwin 374 long doubles & NTL quad_float), and further optimisation of the values 375 of <span class="emphasis"><em>g</em></span> may be possible. Errors given in the table are estimates 376 of the error due to truncation of the Lanczos infinite series to <span class="emphasis"><em>N</em></span> 377 terms. They are calculated from the sum of the first five neglected terms - 378 and are known to be rather pessimistic estimates - although it is noticeable 379 that the best combinations of <span class="emphasis"><em>N</em></span> and <span class="emphasis"><em>g</em></span> 380 occurred when the estimated truncation error almost exactly matches the machine 381 epsilon for the type in question. 382 </p> 383<div class="table"> 384<a name="math_toolkit.lanczos.optimum_value_for_n_and_g_when_c"></a><p class="title"><b>Table 23.2. Optimum value for N and g when computing at fixed precision</b></p> 385<div class="table-contents"><table class="table" summary="Optimum value for N and g when computing at fixed precision"> 386<colgroup> 387<col> 388<col> 389<col> 390<col> 391<col> 392</colgroup> 393<thead><tr> 394<th> 395 <p> 396 Significand Size 397 </p> 398 </th> 399<th> 400 <p> 401 Platform/Compiler Used 402 </p> 403 </th> 404<th> 405 <p> 406 N 407 </p> 408 </th> 409<th> 410 <p> 411 g 412 </p> 413 </th> 414<th> 415 <p> 416 Max Truncation Error 417 </p> 418 </th> 419</tr></thead> 420<tbody> 421<tr> 422<td> 423 <p> 424 24 425 </p> 426 </td> 427<td> 428 <p> 429 Win32, VC++ 7.1 430 </p> 431 </td> 432<td> 433 <p> 434 6 435 </p> 436 </td> 437<td> 438 <p> 439 1.428456135094165802001953125 440 </p> 441 </td> 442<td> 443 <p> 444 9.41e-007 445 </p> 446 </td> 447</tr> 448<tr> 449<td> 450 <p> 451 53 452 </p> 453 </td> 454<td> 455 <p> 456 Win32, VC++ 7.1 457 </p> 458 </td> 459<td> 460 <p> 461 13 462 </p> 463 </td> 464<td> 465 <p> 466 6.024680040776729583740234375 467 </p> 468 </td> 469<td> 470 <p> 471 3.23e-016 472 </p> 473 </td> 474</tr> 475<tr> 476<td> 477 <p> 478 64 479 </p> 480 </td> 481<td> 482 <p> 483 Suse Linux 9 IA64, gcc-3.3.3 484 </p> 485 </td> 486<td> 487 <p> 488 17 489 </p> 490 </td> 491<td> 492 <p> 493 12.2252227365970611572265625 494 </p> 495 </td> 496<td> 497 <p> 498 2.34e-024 499 </p> 500 </td> 501</tr> 502<tr> 503<td> 504 <p> 505 116 506 </p> 507 </td> 508<td> 509 <p> 510 HP Tru64 Unix 5.1B / Alpha, Compaq C++ V7.1-006 511 </p> 512 </td> 513<td> 514 <p> 515 24 516 </p> 517 </td> 518<td> 519 <p> 520 20.3209821879863739013671875 521 </p> 522 </td> 523<td> 524 <p> 525 4.75e-035 526 </p> 527 </td> 528</tr> 529</tbody> 530</table></div> 531</div> 532<br class="table-break"><p> 533 Finally note that the Lanczos approximation can be written as follows by removing 534 a factor of exp(g) from the denominator, and then dividing all the coefficients 535 by exp(g): 536 </p> 537<div class="blockquote"><blockquote class="blockquote"><p> 538 <span class="inlinemediaobject"><img src="../../equations/lanczos7.svg"></span> 539 540 </p></blockquote></div> 541<p> 542 This form is more convenient for calculating lgamma, but for the gamma function 543 the division by <span class="emphasis"><em>e</em></span> turns a possibly exact quality into 544 an inexact value: this reduces accuracy in the common case that the input is 545 exact, and so isn't used for the gamma function. 546 </p> 547<h5> 548<a name="math_toolkit.lanczos.h4"></a> 549 <span class="phrase"><a name="math_toolkit.lanczos.references"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.references">References</a> 550 </h5> 551<div class="orderedlist"><ol class="orderedlist" type="1"> 552<li class="listitem"> 553 <a name="godfrey"></a>Paul Godfrey, <a href="http://my.fit.edu/~gabdo/gamma.txt" target="_top">"A 554 note on the computation of the convergent Lanczos complex Gamma approximation"</a>. 555 </li> 556<li class="listitem"> 557 <a name="pugh"></a>Glendon Ralph Pugh, <a href="http://bh0.physics.ubc.ca/People/matt/Doc/ThesesOthers/Phd/pugh.pdf" target="_top">"An 558 Analysis of the Lanczos Gamma Approximation"</a>, PhD Thesis November 559 2004. 560 </li> 561<li class="listitem"> 562 Viktor T. Toth, <a href="http://www.rskey.org/gamma.htm" target="_top">"Calculators 563 and the Gamma Function"</a>. 564 </li> 565<li class="listitem"> 566 Mathworld, <a href="http://mathworld.wolfram.com/LanczosApproximation.html" target="_top">The 567 Lanczos Approximation</a>. 568 </li> 569</ol></div> 570</div> 571<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 572<td align="left"></td> 573<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 574 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 575 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 576 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 577 Daryle Walker and Xiaogang Zhang<p> 578 Distributed under the Boost Software License, Version 1.0. (See accompanying 579 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>) 580 </p> 581</div></td> 582</tr></table> 583<hr> 584<div class="spirit-nav"> 585<a accesskey="p" href="relative_error.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../backgrounders.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="remez.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 586</div> 587</body> 588</html> 589
