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111 lines
3.0 KiB
Plaintext
111 lines
3.0 KiB
Plaintext
[section:polygamma Polygamma]
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/polygamma.hpp>
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``
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namespace boost{ namespace math{
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template <class T>
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``__sf_result`` polygamma(int n, T z);
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template <class T, class ``__Policy``>
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``__sf_result`` polygamma(int n, T z, const ``__Policy``&);
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}} // namespaces
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[h4 Description]
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Returns the polygamma function of /x/. Polygamma is defined as the n'th
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derivative of the digamma function:
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[equation polygamma1]
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The following graphs illustrate the behaviour of the function for odd and even order:
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[graph polygamma2]
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[graph polygamma3]
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[optional_policy]
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The return type of this function is computed using the __arg_promotion_rules:
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the result is of type `double` when T is an integer type, and type T otherwise.
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[h4 Accuracy]
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The following table shows the peak errors (in units of epsilon)
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found on various platforms with various floating point types.
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Unless otherwise specified any floating point type that is narrower
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than the one shown will have __zero_error.
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[table_polygamma]
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As shown above, error rates are generally very acceptable for moderately sized
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arguments. Error rates should stay low for exact inputs, however, please note that the
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function becomes exceptionally sensitive to small changes in input for large n and negative x,
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indeed for cases where ['n!] would overflow, the function changes directly from -[infin] to
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+[infin] somewhere between each negative integer - ['these cases are not handled correctly].
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[*For these reasons results should be treated with extreme caution when /n/ is large and x negative].
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[h4 Testing]
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Testing is against Mathematica generated spot values to 35 digit precision.
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[h4 Implementation]
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For x < 0 the following reflection formula is used:
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[equation polygamma2]
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The n'th derivative of ['cot(x)] is tabulated for small /n/, and for larger n
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has the general form:
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[equation polygamma3]
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The coefficients of the cosine terms can be calculated iteratively starting
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from ['C[sub 1,0] = -1] and then using
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[equation polygamma7]
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to generate coefficients for n+1.
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Note that every other coefficient is zero, and therefore what we have are
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even or odd polynomials depending on whether n is even or odd.
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Once x is positive then we have two methods available to us, for small x
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we use the series expansion:
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[equation polygamma4]
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Note that the evaluation of zeta functions at integer values is essentially a table lookup
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as __zeta is optimized for those cases.
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For large x we use the asymptotic expansion:
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[equation polygamma5]
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For x in-between the two extremes we use the relation:
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[equation polygamma6]
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to make x large enough for the asymptotic expansion to be used.
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There are also two special cases:
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[equation polygamma8]
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[equation polygamma9]
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[endsect][/section:polygamma The Polygamma Function]
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[/
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Copyright 2014 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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