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- <div class="section">
- <div class="titlepage"><div><div><h2 class="title" style="clear: both">
- <a name="math_toolkit.lanczos"></a><a class="link" href="lanczos.html" title="The Lanczos Approximation">The Lanczos Approximation</a>
- </h2></div></div></div>
- <h5>
- <a name="math_toolkit.lanczos.h0"></a>
- <span class="phrase"><a name="math_toolkit.lanczos.motivation"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.motivation">Motivation</a>
- </h5>
- <p>
- <span class="emphasis"><em>Why base gamma and gamma-like functions on the Lanczos approximation?</em></span>
- </p>
- <p>
- First of all I should make clear that for the gamma function over real numbers
- (as opposed to complex ones) the Lanczos approximation (See <a href="http://en.wikipedia.org/wiki/Lanczos_approximation" target="_top">Wikipedia
- or </a> <a href="http://mathworld.wolfram.com/LanczosApproximation.html" target="_top">Mathworld</a>)
- appears to offer no clear advantage over more traditional methods such as
- <a href="http://en.wikipedia.org/wiki/Stirling_approximation" target="_top">Stirling's
- approximation</a>. <a class="link" href="lanczos.html#pugh">Pugh</a> carried out an extensive
- comparison of the various methods available and discovered that they were all
- very similar in terms of complexity and relative error. However, the Lanczos
- approximation does have a couple of properties that make it worthy of further
- consideration:
- </p>
- <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
- <li class="listitem">
- The approximation has an easy to compute truncation error that holds for
- all <span class="emphasis"><em>z > 0</em></span>. In practice that means we can use the
- same approximation for all <span class="emphasis"><em>z > 0</em></span>, and be certain
- that no matter how large or small <span class="emphasis"><em>z</em></span> is, the truncation
- error will <span class="emphasis"><em>at worst</em></span> be bounded by some finite value.
- </li>
- <li class="listitem">
- The approximation has a form that is particularly amenable to analytic
- manipulation, in particular ratios of gamma or gamma-like functions are
- particularly easy to compute without resorting to logarithms.
- </li>
- </ul></div>
- <p>
- It is the combination of these two properties that make the approximation attractive:
- Stirling's approximation is highly accurate for large z, and has some of the
- same analytic properties as the Lanczos approximation, but can't easily be
- used across the whole range of z.
- </p>
- <p>
- As the simplest example, consider the ratio of two gamma functions: one could
- compute the result via lgamma:
- </p>
- <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>
- </pre>
- <p>
- However, even if lgamma is uniformly accurate to 0.5ulp, the worst case relative
- error in the above can easily be shown to be:
- </p>
- <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>
- </pre>
- <p>
- For small <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> that's not a problem,
- but to put the relationship another way: <span class="emphasis"><em>each time a and b increase
- in magnitude by a factor of 10, at least one decimal digit of precision will
- be lost.</em></span>
- </p>
- <p>
- In contrast, by analytically combining like power terms in a ratio of Lanczos
- approximation's, these errors can be virtually eliminated for small <span class="emphasis"><em>a</em></span>
- and <span class="emphasis"><em>b</em></span>, and kept under control for very large (or very
- small for that matter) <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>. Of
- course, computing large powers is itself a notoriously hard problem, but even
- so, analytic combinations of Lanczos approximations can make the difference
- between obtaining a valid result, or simply garbage. Refer to the implementation
- notes for the <a class="link" href="sf_beta/beta_function.html" title="Beta">beta</a>
- 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
- gamma</a> and <a class="link" href="sf_beta/ibeta_function.html" title="Incomplete Beta Functions">beta</a>
- functions use similar analytic combinations of power terms, to combine gamma
- and beta functions divided by large powers into single (simpler) expressions.
- </p>
- <h5>
- <a name="math_toolkit.lanczos.h1"></a>
- <span class="phrase"><a name="math_toolkit.lanczos.the_approximation"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.the_approximation">The
- Approximation</a>
- </h5>
- <p>
- The Lanczos Approximation to the Gamma Function is given by:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../equations/lanczos0.svg"></span>
- </p></blockquote></div>
- <p>
- 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>
- is an arbitrary parameter that controls the "shape" of the terms
- in the sum which is given by:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../equations/lanczos0a.svg"></span>
- </p></blockquote></div>
- <p>
- With individual coefficients defined in closed form by:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../equations/lanczos0b.svg"></span>
- </p></blockquote></div>
- <p>
- However, evaluation of the sum in that form can lead to numerical instability
- in the computation of the ratios of rising and falling factorials (effectively
- we're multiplying by a series of numbers very close to 1, so roundoff errors
- can accumulate quite rapidly).
- </p>
- <p>
- The Lanczos approximation is therefore often written in partial fraction form
- with the leading constants absorbed by the coefficients in the sum:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../equations/lanczos1.svg"></span>
- </p></blockquote></div>
- <p>
- where:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../equations/lanczos2.svg"></span>
- </p></blockquote></div>
- <p>
- Again parameter <span class="emphasis"><em>g</em></span> is an arbitrarily chosen constant, and
- <span class="emphasis"><em>N</em></span> is an arbitrarily chosen number of terms to evaluate
- in the "Lanczos sum" part.
- </p>
- <div class="note"><table border="0" summary="Note">
- <tr>
- <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td>
- <th align="left">Note</th>
- </tr>
- <tr><td align="left" valign="top"><p>
- Some authors choose to define the sum from k=1 to N, and hence end up with
- N+1 coefficients. This happens to confuse both the following discussion and
- the code (since C++ deals with half open array ranges, rather than the closed
- range of the sum). This convention is consistent with <a class="link" href="lanczos.html#godfrey">Godfrey</a>,
- but not <a class="link" href="lanczos.html#pugh">Pugh</a>, so take care when referring to
- the literature in this field.
- </p></td></tr>
- </table></div>
- <h5>
- <a name="math_toolkit.lanczos.h2"></a>
- <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
- the Coefficients</a>
- </h5>
- <p>
- The coefficients C0..CN-1 need to be computed from <span class="emphasis"><em>N</em></span> and
- <span class="emphasis"><em>g</em></span> at high precision, and then stored as part of the program.
- Calculation of the coefficients is performed via the method of <a class="link" href="lanczos.html#godfrey">Godfrey</a>;
- let the constants be contained in a column vector P, then:
- </p>
- <p>
- P = D B C F
- </p>
- <p>
- where B is an NxN matrix:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../equations/lanczos4.svg"></span>
- </p></blockquote></div>
- <p>
- D is an NxN matrix:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../equations/lanczos3.svg"></span>
- </p></blockquote></div>
- <p>
- C is an NxN matrix:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../equations/lanczos5.svg"></span>
- </p></blockquote></div>
- <p>
- and F is an N element column vector:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../equations/lanczos6.svg"></span>
- </p></blockquote></div>
- <p>
- Note than the matrices B, D and C contain all integer terms and depend only
- on <span class="emphasis"><em>N</em></span>, this product should be computed first, and then
- multiplied by <span class="emphasis"><em>F</em></span> as the last step.
- </p>
- <h5>
- <a name="math_toolkit.lanczos.h3"></a>
- <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
- the Right Parameters</a>
- </h5>
- <p>
- The trick is to choose <span class="emphasis"><em>N</em></span> and <span class="emphasis"><em>g</em></span> to
- give the desired level of accuracy: choosing a small value for <span class="emphasis"><em>g</em></span>
- leads to a strictly convergent series, but one which converges only slowly.
- Choosing a larger value of <span class="emphasis"><em>g</em></span> causes the terms in the series
- to be large and/or divergent for about the first <span class="emphasis"><em>g-1</em></span> terms,
- and to then suddenly converge with a "crunch".
- </p>
- <p>
- <a class="link" href="lanczos.html#pugh">Pugh</a> has determined the optimal value of <span class="emphasis"><em>g</em></span>
- for <span class="emphasis"><em>N</em></span> in the range <span class="emphasis"><em>1 <= N <= 60</em></span>:
- unfortunately in practice choosing these values leads to cancellation errors
- in the Lanczos sum as the largest term in the (alternating) series is approximately
- 1000 times larger than the result. These optimal values appear not to be useful
- in practice unless the evaluation can be done with a number of guard digits
- <span class="emphasis"><em>and</em></span> the coefficients are stored at higher precision than
- that desired in the result. These values are best reserved for say, computing
- to float precision with double precision arithmetic.
- </p>
- <div class="table">
- <a name="math_toolkit.lanczos.optimal_choices_for_n_and_g_when"></a><p class="title"><b>Table 22.1. Optimal choices for N and g when computing with guard digits (source:
- Pugh)</b></p>
- <div class="table-contents"><table class="table" summary="Optimal choices for N and g when computing with guard digits (source:
- Pugh)">
- <colgroup>
- <col>
- <col>
- <col>
- <col>
- </colgroup>
- <thead><tr>
- <th>
- <p>
- Significand Size
- </p>
- </th>
- <th>
- <p>
- N
- </p>
- </th>
- <th>
- <p>
- g
- </p>
- </th>
- <th>
- <p>
- Max Error
- </p>
- </th>
- </tr></thead>
- <tbody>
- <tr>
- <td>
- <p>
- 24
- </p>
- </td>
- <td>
- <p>
- 6
- </p>
- </td>
- <td>
- <p>
- 5.581
- </p>
- </td>
- <td>
- <p>
- 9.51e-12
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- 53
- </p>
- </td>
- <td>
- <p>
- 13
- </p>
- </td>
- <td>
- <p>
- 13.144565
- </p>
- </td>
- <td>
- <p>
- 9.2213e-23
- </p>
- </td>
- </tr>
- </tbody>
- </table></div>
- </div>
- <br class="table-break"><p>
- The alternative described by <a class="link" href="lanczos.html#godfrey">Godfrey</a> is to perform
- an exhaustive search of the <span class="emphasis"><em>N</em></span> and <span class="emphasis"><em>g</em></span>
- parameter space to determine the optimal combination for a given <span class="emphasis"><em>p</em></span>
- digit floating-point type. Repeating this work found a good approximation for
- double precision arithmetic (close to the one <a class="link" href="lanczos.html#godfrey">Godfrey</a>
- found), but failed to find really good approximations for 80 or 128-bit long
- doubles. Further it was observed that the approximations obtained tended to
- optimised for the small values of z (1 < z < 200) used to test the implementation
- against the factorials. Computing ratios of gamma functions with large arguments
- were observed to suffer from error resulting from the truncation of the Lancozos
- series.
- </p>
- <p>
- <a class="link" href="lanczos.html#pugh">Pugh</a> identified all the locations where the theoretical
- error of the approximation were at a minimum, but unfortunately has published
- only the largest of these minima. However, he makes the observation that the
- minima coincide closely with the location where the first neglected term (a<sub>N</sub>)
- in the Lanczos series S<sub>g</sub>(z) changes sign. These locations are quite easy to
- locate, albeit with considerable computer time. These "sweet spots"
- need only be computed once, tabulated, and then searched when required for
- an approximation that delivers the required precision for some fixed precision
- type.
- </p>
- <p>
- Unfortunately, following this path failed to find a really good approximation
- for 128-bit long doubles, and those found for 64 and 80-bit reals required
- an excessive number of terms. There are two competing issues here: high precision
- requires a large value of <span class="emphasis"><em>g</em></span>, but avoiding cancellation
- errors in the evaluation requires a small <span class="emphasis"><em>g</em></span>.
- </p>
- <p>
- At this point note that the Lanczos sum can be converted into rational form
- (a ratio of two polynomials, obtained from the partial-fraction form using
- polynomial arithmetic), and doing so changes the coefficients so that <span class="emphasis"><em>they
- are all positive</em></span>. That means that the sum in rational form can be
- evaluated without cancellation error, albeit with double the number of coefficients
- for a given N. Repeating the search of the "sweet spots", this time
- evaluating the Lanczos sum in rational form, and testing only those "sweet
- spots" whose theoretical error is less than the machine epsilon for the
- type being tested, yielded good approximations for all the types tested. The
- optimal values found were quite close to the best cases reported by <a class="link" href="lanczos.html#pugh">Pugh</a>
- (just slightly larger <span class="emphasis"><em>N</em></span> and slightly smaller <span class="emphasis"><em>g</em></span>
- for a given precision than <a class="link" href="lanczos.html#pugh">Pugh</a> reports), and even
- though converting to rational form doubles the number of stored coefficients,
- it should be noted that half of them are integers (and therefore require less
- storage space) and the approximations require a smaller <span class="emphasis"><em>N</em></span>
- than would otherwise be required, so fewer floating point operations may be
- required overall.
- </p>
- <p>
- The following table shows the optimal values for <span class="emphasis"><em>N</em></span> and
- <span class="emphasis"><em>g</em></span> when computing at fixed precision. These should be taken
- as work in progress: there are no values for 106-bit significand machines (Darwin
- long doubles & NTL quad_float), and further optimisation of the values
- of <span class="emphasis"><em>g</em></span> may be possible. Errors given in the table are estimates
- of the error due to truncation of the Lanczos infinite series to <span class="emphasis"><em>N</em></span>
- terms. They are calculated from the sum of the first five neglected terms -
- and are known to be rather pessimistic estimates - although it is noticeable
- that the best combinations of <span class="emphasis"><em>N</em></span> and <span class="emphasis"><em>g</em></span>
- occurred when the estimated truncation error almost exactly matches the machine
- epsilon for the type in question.
- </p>
- <div class="table">
- <a name="math_toolkit.lanczos.optimum_value_for_n_and_g_when_c"></a><p class="title"><b>Table 22.2. Optimum value for N and g when computing at fixed precision</b></p>
- <div class="table-contents"><table class="table" summary="Optimum value for N and g when computing at fixed precision">
- <colgroup>
- <col>
- <col>
- <col>
- <col>
- <col>
- </colgroup>
- <thead><tr>
- <th>
- <p>
- Significand Size
- </p>
- </th>
- <th>
- <p>
- Platform/Compiler Used
- </p>
- </th>
- <th>
- <p>
- N
- </p>
- </th>
- <th>
- <p>
- g
- </p>
- </th>
- <th>
- <p>
- Max Truncation Error
- </p>
- </th>
- </tr></thead>
- <tbody>
- <tr>
- <td>
- <p>
- 24
- </p>
- </td>
- <td>
- <p>
- Win32, VC++ 7.1
- </p>
- </td>
- <td>
- <p>
- 6
- </p>
- </td>
- <td>
- <p>
- 1.428456135094165802001953125
- </p>
- </td>
- <td>
- <p>
- 9.41e-007
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- 53
- </p>
- </td>
- <td>
- <p>
- Win32, VC++ 7.1
- </p>
- </td>
- <td>
- <p>
- 13
- </p>
- </td>
- <td>
- <p>
- 6.024680040776729583740234375
- </p>
- </td>
- <td>
- <p>
- 3.23e-016
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- 64
- </p>
- </td>
- <td>
- <p>
- Suse Linux 9 IA64, gcc-3.3.3
- </p>
- </td>
- <td>
- <p>
- 17
- </p>
- </td>
- <td>
- <p>
- 12.2252227365970611572265625
- </p>
- </td>
- <td>
- <p>
- 2.34e-024
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- 116
- </p>
- </td>
- <td>
- <p>
- HP Tru64 Unix 5.1B / Alpha, Compaq C++ V7.1-006
- </p>
- </td>
- <td>
- <p>
- 24
- </p>
- </td>
- <td>
- <p>
- 20.3209821879863739013671875
- </p>
- </td>
- <td>
- <p>
- 4.75e-035
- </p>
- </td>
- </tr>
- </tbody>
- </table></div>
- </div>
- <br class="table-break"><p>
- Finally note that the Lanczos approximation can be written as follows by removing
- a factor of exp(g) from the denominator, and then dividing all the coefficients
- by exp(g):
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../equations/lanczos7.svg"></span>
- </p></blockquote></div>
- <p>
- This form is more convenient for calculating lgamma, but for the gamma function
- the division by <span class="emphasis"><em>e</em></span> turns a possibly exact quality into
- an inexact value: this reduces accuracy in the common case that the input is
- exact, and so isn't used for the gamma function.
- </p>
- <h5>
- <a name="math_toolkit.lanczos.h4"></a>
- <span class="phrase"><a name="math_toolkit.lanczos.references"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.references">References</a>
- </h5>
- <div class="orderedlist"><ol class="orderedlist" type="1">
- <li class="listitem">
- <a name="godfrey"></a>Paul Godfrey, <a href="http://my.fit.edu/~gabdo/gamma.txt" target="_top">"A
- note on the computation of the convergent Lanczos complex Gamma approximation"</a>.
- </li>
- <li class="listitem">
- <a name="pugh"></a>Glendon Ralph Pugh, <a href="http://bh0.physics.ubc.ca/People/matt/Doc/ThesesOthers/Phd/pugh.pdf" target="_top">"An
- Analysis of the Lanczos Gamma Approximation"</a>, PhD Thesis November
- 2004.
- </li>
- <li class="listitem">
- Viktor T. Toth, <a href="http://www.rskey.org/gamma.htm" target="_top">"Calculators
- and the Gamma Function"</a>.
- </li>
- <li class="listitem">
- Mathworld, <a href="http://mathworld.wolfram.com/LanczosApproximation.html" target="_top">The
- Lanczos Approximation</a>.
- </li>
- </ol></div>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
- Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
- Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
- Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
- Daryle Walker and Xiaogang Zhang<p>
- Distributed under the Boost Software License, Version 1.0. (See accompanying
- 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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