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80 lines
1.9 KiB
Plaintext
80 lines
1.9 KiB
Plaintext
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[section:trigamma Trigamma]
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/trigamma.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`` trigamma(T z);
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template <class T, class ``__Policy``>
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``__sf_result`` trigamma(T z, const ``__Policy``&);
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}} // namespaces
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[h4 Description]
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Returns the trigamma function of /x/. Trigamma is defined as the
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derivative of the digamma function:
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[equation trigamma1]
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[graph trigamma]
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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_trigamma]
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As shown above, error rates are generally very low for built in types.
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For multiprecision types, error rates are typically in the order of a
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few epsilon.
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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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The arbitrary precision version of this function simply calls __polygamma.
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For built in fixed precision types, negative arguments are first made positive via:
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[equation trigamma2]
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Then arguments in the range \[0, 1) are shifted to >= 1 via:
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[equation trigamma3]
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Then evaluation is via one of a number of rational approximations, for small x these are
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of the form:
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[equation trigamma4]
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and for large x of the form:
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[equation trigamma5]
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[endsect][/section:digamma The Trigamma 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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