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  25 <div class="section">
  26 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  27 <a name="math_toolkit.lanczos"></a><a class="link" href="lanczos.html" title="The Lanczos Approximation">The Lanczos Approximation</a>
  28 </h2></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.lanczos.h0"></a>
  31       <span class="phrase"><a name="math_toolkit.lanczos.motivation"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.motivation">Motivation</a>
  32     </h5>
  33 <p>
  34       <span class="emphasis"><em>Why base gamma and gamma-like functions on the Lanczos approximation?</em></span>
  35     </p>
  36 <p>
  37       First of all I should make clear that for the gamma function over real numbers
  38       (as opposed to complex ones) the Lanczos approximation (See <a href="http://en.wikipedia.org/wiki/Lanczos_approximation" target="_top">Wikipedia
  39       or </a> <a href="http://mathworld.wolfram.com/LanczosApproximation.html" target="_top">Mathworld</a>)
  40       appears to offer no clear advantage over more traditional methods such as
  41       <a href="http://en.wikipedia.org/wiki/Stirling_approximation" target="_top">Stirling's
  42       approximation</a>. <a class="link" href="lanczos.html#pugh">Pugh</a> carried out an extensive
  43       comparison of the various methods available and discovered that they were all
  44       very similar in terms of complexity and relative error. 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 &gt; 0</em></span>. In practice that means we can use the
  52           same approximation for all <span class="emphasis"><em>z &gt; 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">&gt;</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 <p>
 109       <span class="inlinemediaobject"><img src="../../equations/lanczos0.svg"></span>
 110     </p>
 111 <p>
 112       Where S<sub>g</sub>(z) is an infinite sum, that is convergent for all z &gt; 0, and <span class="emphasis"><em>g</em></span>
 113       is an arbitrary parameter that controls the "shape" of the terms
 114       in the sum which is given by:
 115     </p>
 116 <p>
 117       <span class="inlinemediaobject"><img src="../../equations/lanczos0a.svg"></span>
 118     </p>
 119 <p>
 120       With individual coefficients defined in closed form by:
 121     </p>
 122 <p>
 123       <span class="inlinemediaobject"><img src="../../equations/lanczos0b.svg"></span>
 124     </p>
 125 <p>
 126       However, evaluation of the sum in that form can lead to numerical instability
 127       in the computation of the ratios of rising and falling factorials (effectively
 128       we're multiplying by a series of numbers very close to 1, so roundoff errors
 129       can accumulate quite rapidly).
 130     </p>
 131 <p>
 132       The Lanczos approximation is therefore often written in partial fraction form
 133       with the leading constants absorbed by the coefficients in the sum:
 134     </p>
 135 <p>
 136       <span class="inlinemediaobject"><img src="../../equations/lanczos1.svg"></span>
 137     </p>
 138 <p>
 139       where:
 140     </p>
 141 <p>
 142       <span class="inlinemediaobject"><img src="../../equations/lanczos2.svg"></span>
 143     </p>
 144 <p>
 145       Again parameter <span class="emphasis"><em>g</em></span> is an arbitrarily chosen constant, and
 146       <span class="emphasis"><em>N</em></span> is an arbitrarily chosen number of terms to evaluate
 147       in the "Lanczos sum" part.
 148     </p>
 149 <div class="note"><table border="0" summary="Note">
 150 <tr>
 151 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td>
 152 <th align="left">Note</th>
 153 </tr>
 154 <tr><td align="left" valign="top"><p>
 155         Some authors choose to define the sum from k=1 to N, and hence end up with
 156         N+1 coefficients. This happens to confuse both the following discussion and
 157         the code (since C++ deals with half open array ranges, rather than the closed
 158         range of the sum). This convention is consistent with <a class="link" href="lanczos.html#godfrey">Godfrey</a>,
 159         but not <a class="link" href="lanczos.html#pugh">Pugh</a>, so take care when referring to
 160         the literature in this field.
 161       </p></td></tr>
 162 </table></div>
 163 <h5>
 164 <a name="math_toolkit.lanczos.h2"></a>
 165       <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
 166       the Coefficients</a>
 167     </h5>
 168 <p>
 169       The coefficients C0..CN-1 need to be computed from <span class="emphasis"><em>N</em></span> and
 170       <span class="emphasis"><em>g</em></span> at high precision, and then stored as part of the program.
 171       Calculation of the coefficients is performed via the method of <a class="link" href="lanczos.html#godfrey">Godfrey</a>;
 172       let the constants be contained in a column vector P, then:
 173     </p>
 174 <p>
 175       P = D B C F
 176     </p>
 177 <p>
 178       where B is an NxN matrix:
 179     </p>
 180 <p>
 181       <span class="inlinemediaobject"><img src="../../equations/lanczos4.svg"></span>
 182     </p>
 183 <p>
 184       D is an NxN matrix:
 185     </p>
 186 <p>
 187       <span class="inlinemediaobject"><img src="../../equations/lanczos3.svg"></span>
 188     </p>
 189 <p>
 190       C is an NxN matrix:
 191     </p>
 192 <p>
 193       <span class="inlinemediaobject"><img src="../../equations/lanczos5.svg"></span>
 194     </p>
 195 <p>
 196       and F is an N element column vector:
 197     </p>
 198 <p>
 199       <span class="inlinemediaobject"><img src="../../equations/lanczos6.svg"></span>
 200     </p>
 201 <p>
 202       Note than the matrices B, D and C contain all integer terms and depend only
 203       on <span class="emphasis"><em>N</em></span>, this product should be computed first, and then
 204       multiplied by <span class="emphasis"><em>F</em></span> as the last step.
 205     </p>
 206 <h5>
 207 <a name="math_toolkit.lanczos.h3"></a>
 208       <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
 209       the Right Parameters</a>
 210     </h5>
 211 <p>
 212       The trick is to choose <span class="emphasis"><em>N</em></span> and <span class="emphasis"><em>g</em></span> to
 213       give the desired level of accuracy: choosing a small value for <span class="emphasis"><em>g</em></span>
 214       leads to a strictly convergent series, but one which converges only slowly.
 215       Choosing a larger value of <span class="emphasis"><em>g</em></span> causes the terms in the series
 216       to be large and/or divergent for about the first <span class="emphasis"><em>g-1</em></span> terms,
 217       and to then suddenly converge with a "crunch".
 218     </p>
 219 <p>
 220       <a class="link" href="lanczos.html#pugh">Pugh</a> has determined the optimal value of <span class="emphasis"><em>g</em></span>
 221       for <span class="emphasis"><em>N</em></span> in the range <span class="emphasis"><em>1 &lt;= N &lt;= 60</em></span>:
 222       unfortunately in practice choosing these values leads to cancellation errors
 223       in the Lanczos sum as the largest term in the (alternating) series is approximately
 224       1000 times larger than the result. These optimal values appear not to be useful
 225       in practice unless the evaluation can be done with a number of guard digits
 226       <span class="emphasis"><em>and</em></span> the coefficients are stored at higher precision than
 227       that desired in the result. These values are best reserved for say, computing
 228       to float precision with double precision arithmetic.
 229     </p>
 230 <div class="table">
 231 <a name="math_toolkit.lanczos.optimal_choices_for_n_and_g_when"></a><p class="title"><b>Table&#160;17.1.&#160;Optimal choices for N and g when computing with guard digits (source:
 232       Pugh)</b></p>
 233 <div class="table-contents"><table class="table" summary="Optimal choices for N and g when computing with guard digits (source:
 234       Pugh)">
 235 <colgroup>
 236 <col>
 237 <col>
 238 <col>
 239 <col>
 240 </colgroup>
 241 <thead><tr>
 242 <th>
 243               <p>
 244                 Significand Size
 245               </p>
 246             </th>
 247 <th>
 248               <p>
 249                 N
 250               </p>
 251             </th>
 252 <th>
 253               <p>
 254                 g
 255               </p>
 256             </th>
 257 <th>
 258               <p>
 259                 Max Error
 260               </p>
 261             </th>
 262 </tr></thead>
 263 <tbody>
 264 <tr>
 265 <td>
 266               <p>
 267                 24
 268               </p>
 269             </td>
 270 <td>
 271               <p>
 272                 6
 273               </p>
 274             </td>
 275 <td>
 276               <p>
 277                 5.581
 278               </p>
 279             </td>
 280 <td>
 281               <p>
 282                 9.51e-12
 283               </p>
 284             </td>
 285 </tr>
 286 <tr>
 287 <td>
 288               <p>
 289                 53
 290               </p>
 291             </td>
 292 <td>
 293               <p>
 294                 13
 295               </p>
 296             </td>
 297 <td>
 298               <p>
 299                 13.144565
 300               </p>
 301             </td>
 302 <td>
 303               <p>
 304                 9.2213e-23
 305               </p>
 306             </td>
 307 </tr>
 308 </tbody>
 309 </table></div>
 310 </div>
 311 <br class="table-break"><p>
 312       The alternative described by <a class="link" href="lanczos.html#godfrey">Godfrey</a> is to perform
 313       an exhaustive search of the <span class="emphasis"><em>N</em></span> and <span class="emphasis"><em>g</em></span>
 314       parameter space to determine the optimal combination for a given <span class="emphasis"><em>p</em></span>
 315       digit floating-point type. Repeating this work found a good approximation for
 316       double precision arithmetic (close to the one <a class="link" href="lanczos.html#godfrey">Godfrey</a>
 317       found), but failed to find really good approximations for 80 or 128-bit long
 318       doubles. Further it was observed that the approximations obtained tended to
 319       optimised for the small values of z (1 &lt; z &lt; 200) used to test the implementation
 320       against the factorials. Computing ratios of gamma functions with large arguments
 321       were observed to suffer from error resulting from the truncation of the Lancozos
 322       series.
 323     </p>
 324 <p>
 325       <a class="link" href="lanczos.html#pugh">Pugh</a> identified all the locations where the theoretical
 326       error of the approximation were at a minimum, but unfortunately has published
 327       only the largest of these minima. However, he makes the observation that the
 328       minima coincide closely with the location where the first neglected term (a<sub>N</sub>)
 329       in the Lanczos series S<sub>g</sub>(z) changes sign. These locations are quite easy to
 330       locate, albeit with considerable computer time. These "sweet spots"
 331       need only be computed once, tabulated, and then searched when required for
 332       an approximation that delivers the required precision for some fixed precision
 333       type.
 334     </p>
 335 <p>
 336       Unfortunately, following this path failed to find a really good approximation
 337       for 128-bit long doubles, and those found for 64 and 80-bit reals required
 338       an excessive number of terms. There are two competing issues here: high precision
 339       requires a large value of <span class="emphasis"><em>g</em></span>, but avoiding cancellation
 340       errors in the evaluation requires a small <span class="emphasis"><em>g</em></span>.
 341     </p>
 342 <p>
 343       At this point note that the Lanczos sum can be converted into rational form
 344       (a ratio of two polynomials, obtained from the partial-fraction form using
 345       polynomial arithmetic), and doing so changes the coefficients so that <span class="emphasis"><em>they
 346       are all positive</em></span>. That means that the sum in rational form can be
 347       evaluated without cancellation error, albeit with double the number of coefficients
 348       for a given N. Repeating the search of the "sweet spots", this time
 349       evaluating the Lanczos sum in rational form, and testing only those "sweet
 350       spots" whose theoretical error is less than the machine epsilon for the
 351       type being tested, yielded good approximations for all the types tested. The
 352       optimal values found were quite close to the best cases reported by <a class="link" href="lanczos.html#pugh">Pugh</a>
 353       (just slightly larger <span class="emphasis"><em>N</em></span> and slightly smaller <span class="emphasis"><em>g</em></span>
 354       for a given precision than <a class="link" href="lanczos.html#pugh">Pugh</a> reports), and even
 355       though converting to rational form doubles the number of stored coefficients,
 356       it should be noted that half of them are integers (and therefore require less
 357       storage space) and the approximations require a smaller <span class="emphasis"><em>N</em></span>
 358       than would otherwise be required, so fewer floating point operations may be
 359       required overall.
 360     </p>
 361 <p>
 362       The following table shows the optimal values for <span class="emphasis"><em>N</em></span> and
 363       <span class="emphasis"><em>g</em></span> when computing at fixed precision. These should be taken
 364       as work in progress: there are no values for 106-bit significand machines (Darwin
 365       long doubles &amp; NTL quad_float), and further optimisation of the values
 366       of <span class="emphasis"><em>g</em></span> may be possible. Errors given in the table are estimates
 367       of the error due to truncation of the Lanczos infinite series to <span class="emphasis"><em>N</em></span>
 368       terms. They are calculated from the sum of the first five neglected terms -
 369       and are known to be rather pessimistic estimates - although it is noticeable
 370       that the best combinations of <span class="emphasis"><em>N</em></span> and <span class="emphasis"><em>g</em></span>
 371       occurred when the estimated truncation error almost exactly matches the machine
 372       epsilon for the type in question.
 373     </p>
 374 <div class="table">
 375 <a name="math_toolkit.lanczos.optimum_value_for_n_and_g_when_c"></a><p class="title"><b>Table&#160;17.2.&#160;Optimum value for N and g when computing at fixed precision</b></p>
 376 <div class="table-contents"><table class="table" summary="Optimum value for N and g when computing at fixed precision">
 377 <colgroup>
 378 <col>
 379 <col>
 380 <col>
 381 <col>
 382 <col>
 383 </colgroup>
 384 <thead><tr>
 385 <th>
 386               <p>
 387                 Significand Size
 388               </p>
 389             </th>
 390 <th>
 391               <p>
 392                 Platform/Compiler Used
 393               </p>
 394             </th>
 395 <th>
 396               <p>
 397                 N
 398               </p>
 399             </th>
 400 <th>
 401               <p>
 402                 g
 403               </p>
 404             </th>
 405 <th>
 406               <p>
 407                 Max Truncation Error
 408               </p>
 409             </th>
 410 </tr></thead>
 411 <tbody>
 412 <tr>
 413 <td>
 414               <p>
 415                 24
 416               </p>
 417             </td>
 418 <td>
 419               <p>
 420                 Win32, VC++ 7.1
 421               </p>
 422             </td>
 423 <td>
 424               <p>
 425                 6
 426               </p>
 427             </td>
 428 <td>
 429               <p>
 430                 1.428456135094165802001953125
 431               </p>
 432             </td>
 433 <td>
 434               <p>
 435                 9.41e-007
 436               </p>
 437             </td>
 438 </tr>
 439 <tr>
 440 <td>
 441               <p>
 442                 53
 443               </p>
 444             </td>
 445 <td>
 446               <p>
 447                 Win32, VC++ 7.1
 448               </p>
 449             </td>
 450 <td>
 451               <p>
 452                 13
 453               </p>
 454             </td>
 455 <td>
 456               <p>
 457                 6.024680040776729583740234375
 458               </p>
 459             </td>
 460 <td>
 461               <p>
 462                 3.23e-016
 463               </p>
 464             </td>
 465 </tr>
 466 <tr>
 467 <td>
 468               <p>
 469                 64
 470               </p>
 471             </td>
 472 <td>
 473               <p>
 474                 Suse Linux 9 IA64, gcc-3.3.3
 475               </p>
 476             </td>
 477 <td>
 478               <p>
 479                 17
 480               </p>
 481             </td>
 482 <td>
 483               <p>
 484                 12.2252227365970611572265625
 485               </p>
 486             </td>
 487 <td>
 488               <p>
 489                 2.34e-024
 490               </p>
 491             </td>
 492 </tr>
 493 <tr>
 494 <td>
 495               <p>
 496                 116
 497               </p>
 498             </td>
 499 <td>
 500               <p>
 501                 HP Tru64 Unix 5.1B / Alpha, Compaq C++ V7.1-006
 502               </p>
 503             </td>
 504 <td>
 505               <p>
 506                 24
 507               </p>
 508             </td>
 509 <td>
 510               <p>
 511                 20.3209821879863739013671875
 512               </p>
 513             </td>
 514 <td>
 515               <p>
 516                 4.75e-035
 517               </p>
 518             </td>
 519 </tr>
 520 </tbody>
 521 </table></div>
 522 </div>
 523 <br class="table-break"><p>
 524       Finally note that the Lanczos approximation can be written as follows by removing
 525       a factor of exp(g) from the denominator, and then dividing all the coefficients
 526       by exp(g):
 527     </p>
 528 <p>
 529       <span class="inlinemediaobject"><img src="../../equations/lanczos7.svg"></span>
 530     </p>
 531 <p>
 532       This form is more convenient for calculating lgamma, but for the gamma function
 533       the division by <span class="emphasis"><em>e</em></span> turns a possibly exact quality into
 534       an inexact value: this reduces accuracy in the common case that the input is
 535       exact, and so isn't used for the gamma function.
 536     </p>
 537 <h5>
 538 <a name="math_toolkit.lanczos.h4"></a>
 539       <span class="phrase"><a name="math_toolkit.lanczos.references"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.references">References</a>
 540     </h5>
 541 <div class="orderedlist"><ol class="orderedlist" type="1">
 542 <li class="listitem">
 543           <a name="godfrey"></a>Paul Godfrey, <a href="http://my.fit.edu/~gabdo/gamma.txt" target="_top">"A
 544           note on the computation of the convergent Lanczos complex Gamma approximation"</a>.
 545         </li>
 546 <li class="listitem">
 547           <a name="pugh"></a>Glendon Ralph Pugh, <a href="http://bh0.physics.ubc.ca/People/matt/Doc/ThesesOthers/Phd/pugh.pdf" target="_top">"An
 548           Analysis of the Lanczos Gamma Approximation"</a>, PhD Thesis November
 549           2004.
 550         </li>
 551 <li class="listitem">
 552           Viktor T. Toth, <a href="http://www.rskey.org/gamma.htm" target="_top">"Calculators
 553           and the Gamma Function"</a>.
 554         </li>
 555 <li class="listitem">
 556           Mathworld, <a href="http://mathworld.wolfram.com/LanczosApproximation.html" target="_top">The
 557           Lanczos Approximation</a>.
 558         </li>
 559 </ol></div>
 560 </div>
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 563 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 564       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 565       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 566       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 567         Distributed under the Boost Software License, Version 1.0. (See accompanying
 568         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>)
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