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120 lines
3.3 KiB
Plaintext
120 lines
3.3 KiB
Plaintext
[section:zeta Riemann Zeta Function]
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/zeta.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`` zeta(T z);
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template <class T, class ``__Policy``>
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``__sf_result`` zeta(T z, const ``__Policy``&);
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}} // namespaces
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The return type of these functions is computed using the __arg_promotion_rules:
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the return type is `double` if T is an integer type, and T otherwise.
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[optional_policy]
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[h4 Description]
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template <class T>
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``__sf_result`` zeta(T z);
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template <class T, class ``__Policy``>
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``__sf_result`` zeta(T z, const ``__Policy``&);
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Returns the [@http://mathworld.wolfram.com/RiemannZetaFunction.html zeta function]
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of z:
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[equation zeta1]
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[graph zeta1]
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[graph zeta2]
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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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along with comparisons to the __gsl and __cephes libraries.
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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_zeta]
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[h4 Testing]
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The tests for these functions come in two parts:
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basic sanity checks use spot values calculated using
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[@http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Zeta Mathworld's online evaluator],
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while accuracy checks use high-precision test values calculated at 1000-bit precision with
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[@http://shoup.net/ntl/doc/RR.txt NTL::RR] and this implementation.
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Note that the generic and type-specific
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versions of these functions use differing implementations internally, so this
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gives us reasonably independent test data. Using our test data to test other
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"known good" implementations also provides an additional sanity check.
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[h4 Implementation]
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All versions of these functions first use the usual reflection formulas
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to make their arguments positive:
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[equation zeta3]
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The generic versions of these functions are implemented using the series:
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[equation zeta6]
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When the significand (mantissa) size is recognised
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(currently for 53, 64 and 113-bit reals, plus single-precision 24-bit handled via promotion to double)
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then a series of rational approximations [jm_rationals] are used.
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For 0 < z < 1 the approximating form is:
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[equation zeta4]
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For a rational approximation R(1-z) and a constant C.
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For 1 < z < 4 the approximating form is:
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[equation zeta5]
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For a rational approximation R(n-z) and a constant C and integer n.
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For z > 4 the approximating form is:
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[zeta](z) = 1 + e[super R(z - n)]
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For a rational approximation R(z-n) and integer n, note that the accuracy
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required for R(z-n) is not full machine precision, but an absolute error
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of: [epsilon]/R(0). This saves us quite a few digits when dealing with large
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z, especially when [epsilon] is small.
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Finally, there are some special cases for integer arguments, there are
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closed forms for negative or even integers:
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[equation zeta7]
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[equation zeta8]
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[equation zeta9]
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and for positive odd integers we simply cache pre-computed values as these are of great
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benefit to some infinite series calculations.
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[endsect]
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[/ :error_function The Error Functions]
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[/
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Copyright 2006 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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