[section:lanczos The Lanczos Approximation][h4 Motivation]['Why base gamma and gamma-like functions on the Lanczos approximation?]First of all I should make clear that for the gamma functionover real numbers (as opposed to complex ones)the Lanczos approximation (See [@http://en.wikipedia.org/wiki/Lanczos_approximation Wikipedia or ][@http://mathworld.wolfram.com/LanczosApproximation.html Mathworld])appears to offer no clear advantage over more traditional methods such as[@http://en.wikipedia.org/wiki/Stirling_approximation Stirling's approximation].__pugh carried out an extensive comparison of the various methods availableand discovered that they were all very similar in terms of complexityand relative error.  However, the Lanczos approximation does have a couple ofproperties that make it worthy of further consideration:* The approximation has an easy to compute truncation error that holds for all /z > 0/.  In practice that means we can use the same approximation for all/z > 0/, and be certain that no matter how large or small /z/ is, the truncation error will /at worst/ be bounded by some finite value.* The approximation has a form that is particularly amenable to analyticmanipulation, in particular ratios of gamma or gamma-like functionsare particularly easy to compute without resorting to logarithms.It is the combination of these two properties that make the approximationattractive: Stirling's approximation is highly accurate for large z, andhas some of the same analytic properties as the Lanczos approximation, butcan't easily be used across the whole range of z.As the simplest example, consider the ratio of two gamma functions: one could compute the result via lgamma:   exp(lgamma(a) - lgamma(b));However, even if lgamma is uniformly accurate to 0.5ulp, the worst case relative error in the above can easily be shown to be:   Erel > a * log(a)/2 + b * log(b)/2For small /a/ and /b/ that's not a problem, but to put the relationship anotherway: ['each time a and b increase in magnitude by a factor of 10, at least onedecimal digit of precision will be lost.]  In contrast, by analytically combining like powerterms in a ratio of Lanczos approximation's, these errors can be virtually eliminatedfor small /a/ and /b/, and kept under control for very large (or very smallfor that matter) /a/ and /b/.  Of course, computing large powers is itself anotoriously hard problem, but even so, analytic combinations of Lanczos approximations can make the difference between obtaining a valid result, orsimply garbage.  Refer to the implementation notes for the __beta function foran example of this method in practice.  The incomplete [link math_toolkit.special.sf_gamma.igamma gamma_p gamma] and [link math_toolkit.special.sf_beta.ibeta_function beta] functionsuse similar analytic combinations of power terms, to combine gamma and betafunctions divided by large powers into single (simpler) expressions.[h4 The Approximation]The Lanczos Approximation to the Gamma Function is given by:[equation lanczos0]Where S[sub g](z) is an infinite sum, that is convergent for all z > 0, and /g/ is an arbitrary parameter that controls the "shape" of theterms in the sum which is given by:[equation lanczos0a]With individual coefficients defined in closed form by:[equation lanczos0b]However, evaluation of the sum in that form can lead to numerical instabilityin the computation of the ratios of rising and falling factorials (effectivelywe're multiplying by a series of numbers very close to 1, so roundoff errorscan accumulate quite rapidly).The Lanczos approximation is therefore often written in partial fraction formwith the leading constants absorbed by the coefficients in the sum:[equation lanczos1]where:[equation lanczos2]Again parameter /g/ is an arbitrarily chosen constant, and /N/ is an arbitrarily chosen number of terms to evaluate in the "Lanczos sum" part.  [note Some authorschoose 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 __godfrey, but not __pugh, so take carewhen referring to the literature in this field.][h4 Computing the Coefficients]The coefficients C0..CN-1 need to be computed from /N/ and /g/ at high precision, and then stored as part of the program.Calculation of the coefficients is performed via the method of __godfrey;let the constants be contained in a column vector P, then:P = B D C Fwhere B is an NxN matrix:[equation lanczos4]D is an NxN matrix:[equation lanczos3]C is an NxN matrix:[equation lanczos5]and F is an N element column vector:[equation lanczos6]Note than the matrices B, D and C contain all integer terms and dependonly on /N/, this product should be computed first, and then multipliedby /F/ as the last step.[h4 Choosing the Right Parameters]The trick is to choose/N/ and /g/ to give the desired level of accuracy: choosing a small value for/g/ leads to a strictly convergent series, but one which converges only slowly.Choosing a larger value of /g/ causes the terms in the series to be largeand\/or divergent for about the first /g-1/ terms, and to then suddenly converge with a "crunch".__pugh has determined the optimalvalue of /g/ for /N/ in the range /1 <= N <= 60/: unfortunately in practice choosingthese values leads to cancellation errors in the Lanczos sum as the largestterm in the (alternating) series is approximately 1000 times larger than the result.  These optimal values appear not to be useful in practice unless the evaluationcan be done with a number of guard digits /and/ the coefficients are storedat higher precision than that desired in the result.  These values are bestreserved for say, computing to float precision with double precision arithmetic. [table Optimal choices for N and g when computing with guard digits (source: Pugh)[[Significand Size] [N] [g][Max Error]][[24] [6] [5.581][9.51e-12]][[53][13][13.144565][9.2213e-23]]]The alternative described by __godfrey is to perform an exhaustivesearch of the /N/ and /g/ parameter space to determine the optimal combination fora given /p/ digit floating-point type.  Repeating this work found a good approximation for double precision arithmetic (close to the one __godfrey found), but failed to find reallygood approximations for 80 or 128-bit long doubles.  Further it was observed that the approximations obtained tended to optimised for the small valuesof z (1 < z < 200) used to test the implementation against the factorials.Computing ratios of gamma functions with large arguments were observed tosuffer from error resulting from the truncation of the Lancozos series.__pugh identified all the locations where the theoretical error of the approximation were at a minimum, but unfortunately has published only the largestof these minima.  However, he makes the observation that the minimacoincide closely with the location where the first neglected term (a[sub N]) in theLanczos series S[sub g](z) changes sign.  These locations are quite easy tolocate, albeit with considerable computer time.  These "sweet spots" needonly be computed once, tabulated, and then searched when required for anapproximation that delivers the required precision for some fixed precisiontype.Unfortunately, following this path failed to find a really good approximationfor 128-bit long doubles, and those found for 64 and 80-bit reals required anexcessive number of terms.  There are two competing issues here: high precisionrequires a large value of /g/, but avoiding cancellation errors in the evaluationrequires a small /g/.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 usingpolynomial arithmetic),and doing so changes the coefficients so that /they are all positive/.  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 erroris less than the machine epsilon for the type being tested, yielded goodapproximations for all the types tested.  The optimal values found were quiteclose to the best cases reported by __pugh (just slightly larger /N/ and slightlysmaller /g/ for a given precision than __pugh reports), and even though convertingto rational form doubles the number of stored coefficients, it should benoted that half of them are integers (and therefore require less storage space)and the approximations require a smaller /N/ than would otherwise be required,so fewer floating point operations may be required overall.The following table shows the optimal values for /N/ and /g/ when computingat fixed precision.  These should be taken as work in progress: there are novalues for 106-bit significand machines (Darwin long doubles & NTL quad_float),and further optimisation of the values of /g/ may be possible.Errors given in the tableare estimates of the error due to truncation of the Lanczos infinite seriesto /N/ terms.  They are calculated from the sum of the first five neglectedterms - and are known to be rather pessimistic estimates - although it is noticeablethat the best combinations of /N/ and /g/ occurred when the estimated truncation erroralmost exactly matches the machine epsilon for the type in question.[table Optimum value for N and g when computing at fixed precision[[Significand Size][Platform/Compiler Used][N][g][Max Truncation Error]][[24][Win32, VC++ 7.1] [6] [1.428456135094165802001953125][9.41e-007]][[53][Win32, VC++ 7.1] [13] [6.024680040776729583740234375][3.23e-016]][[64][Suse Linux 9 IA64, gcc-3.3.3] [17] [12.2252227365970611572265625][2.34e-024]][[116][HP Tru64 Unix 5.1B \/ Alpha, Compaq C++ V7.1-006] [24] [20.3209821879863739013671875][4.75e-035]]]Finally note that the Lanczos approximation can be written as followsby removing a factor of exp(g) from the denominator, and then dividing all the coefficients by exp(g):[equation lanczos7]This form is more convenient for calculating lgamma, but for the gammafunction the division by /e/ turns a possibly exact quality into aninexact value: this reduces accuracy in the common case that the input is exact, and so isn't used for the gamma function.[h4 References]# [#godfrey]Paul Godfrey, [@http://my.fit.edu/~gabdo/gamma.txt "A note on the computation of the convergentLanczos complex Gamma approximation"].# [#pugh]Glendon Ralph Pugh, [@http://bh0.physics.ubc.ca/People/matt/Doc/ThesesOthers/Phd/pugh.pdf "An Analysis of the Lanczos Gamma Approximation"], PhD Thesis November 2004.# Viktor T. Toth, [@http://www.rskey.org/gamma.htm "Calculators and the Gamma Function"].# Mathworld, [@http://mathworld.wolfram.com/LanczosApproximation.html The Lanczos Approximation].[endsect][/section:lanczos The Lanczos Approximation][/   Copyright 2006 John Maddock and Paul A. Bristow.  Distributed under the Boost Software License, Version 1.0.  (See accompanying file LICENSE_1_0.txt or copy at  http://www.boost.org/LICENSE_1_0.txt).]