1[section:polygamma Polygamma] 2 3[h4 Synopsis] 4 5`` 6#include <boost/math/special_functions/polygamma.hpp> 7`` 8 9 namespace boost{ namespace math{ 10 11 template <class T> 12 ``__sf_result`` polygamma(int n, T z); 13 14 template <class T, class ``__Policy``> 15 ``__sf_result`` polygamma(int n, T z, const ``__Policy``&); 16 17 }} // namespaces 18 19[h4 Description] 20 21Returns the polygamma function of /x/. Polygamma is defined as the n'th 22derivative of the digamma function: 23 24[equation polygamma1] 25 26The following graphs illustrate the behaviour of the function for odd and even order: 27 28[graph polygamma2] 29[graph polygamma3] 30 31[optional_policy] 32 33The return type of this function is computed using the __arg_promotion_rules: 34the result is of type `double` when T is an integer type, and type T otherwise. 35 36[h4 Accuracy] 37 38The following table shows the peak errors (in units of epsilon) 39found on various platforms with various floating point types. 40Unless otherwise specified any floating point type that is narrower 41than the one shown will have __zero_error. 42 43[table_polygamma] 44 45As shown above, error rates are generally very acceptable for moderately sized 46arguments. Error rates should stay low for exact inputs, however, please note that the 47function becomes exceptionally sensitive to small changes in input for large n and negative x, 48indeed for cases where ['n!] would overflow, the function changes directly from -[infin] to 49+[infin] somewhere between each negative integer - ['these cases are not handled correctly]. 50 51[*For these reasons results should be treated with extreme caution when /n/ is large and x negative]. 52 53[h4 Testing] 54 55Testing is against Mathematica generated spot values to 35 digit precision. 56 57[h4 Implementation] 58 59For x < 0 the following reflection formula is used: 60 61[equation polygamma2] 62 63The n'th derivative of ['cot(x)] is tabulated for small /n/, and for larger n 64has the general form: 65 66[equation polygamma3] 67 68The coefficients of the cosine terms can be calculated iteratively starting 69from ['C[sub 1,0] = -1] and then using 70 71[equation polygamma7] 72 73to generate coefficients for n+1. 74 75Note that every other coefficient is zero, and therefore what we have are 76even or odd polynomials depending on whether n is even or odd. 77 78Once x is positive then we have two methods available to us, for small x 79we use the series expansion: 80 81[equation polygamma4] 82 83Note that the evaluation of zeta functions at integer values is essentially a table lookup 84as __zeta is optimized for those cases. 85 86For large x we use the asymptotic expansion: 87 88[equation polygamma5] 89 90For x in-between the two extremes we use the relation: 91 92[equation polygamma6] 93 94to make x large enough for the asymptotic expansion to be used. 95 96There are also two special cases: 97 98[equation polygamma8] 99 100[equation polygamma9] 101 102[endsect] [/section:polygamma The Polygamma Function] 103 104[/ 105 Copyright 2014 John Maddock and Paul A. Bristow. 106 Distributed under the Boost Software License, Version 1.0. 107 (See accompanying file LICENSE_1_0.txt or copy at 108 http://www.boost.org/LICENSE_1_0.txt). 109] 110 111