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<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Riemann Zeta Function</title> <link rel="stylesheet" href="../../math.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> <link rel="home" href="../../index.html" title="Math Toolkit 2.1.0"> <link rel="up" href="../zetas.html" title="Zeta Functions"> <link rel="prev" href="../zetas.html" title="Zeta Functions"> <link rel="next" href="../expint.html" title="Exponential Integrals"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="../zetas.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../zetas.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../expint.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section"> <div class="titlepage"><div><div><h3 class="title"> <a name="math_toolkit.zetas.zeta"></a><a class="link" href="zeta.html" title="Riemann Zeta Function">Riemann Zeta Function</a> </h3></div></div></div> <h5> <a name="math_toolkit.zetas.zeta.h0"></a> <a name="math_toolkit.zetas.zeta.synopsis"></a><a class="link" href="zeta.html#math_toolkit.zetas.zeta.synopsis">Synopsis</a> </h5> <pre class="programlisting">#include <boost/math/special_functions/zeta.hpp> </pre> <pre class="programlisting">namespace boost{ namespace math{ template <class T> <a class="link" href="../result_type.html" title="Calculation of the Type of the Result">calculated-result-type</a> zeta(T z); template <class T, class <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a>> <a class="link" href="../result_type.html" title="Calculation of the Type of the Result">calculated-result-type</a> zeta(T z, const <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a>&); }} // namespaces </pre> The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result">result type calculation rules</a>: the return type is <code class="computeroutput">double</code> if T is an integer type, and T otherwise. The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy documentation for more details</a>. <h5> <a name="math_toolkit.zetas.zeta.h1"></a> <a name="math_toolkit.zetas.zeta.description"></a><a class="link" href="zeta.html#math_toolkit.zetas.zeta.description">Description</a> </h5> <pre class="programlisting">template <class T> <a class="link" href="../result_type.html" title="Calculation of the Type of the Result">calculated-result-type</a> zeta(T z); template <class T, class <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a>> <a class="link" href="../result_type.html" title="Calculation of the Type of the Result">calculated-result-type</a> zeta(T z, const <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a>&); </pre> Returns the <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html" target="_top">zeta function</a> of z: <img src="../../../equations/zeta1.png"> <img src="../../../graphs/zeta1.png" align="middle"> <img src="../../../graphs/zeta2.png" align="middle"> <h5> <a name="math_toolkit.zetas.zeta.h2"></a> <a name="math_toolkit.zetas.zeta.accuracy"></a><a class="link" href="zeta.html#math_toolkit.zetas.zeta.accuracy">Accuracy</a> </h5> The following table shows the peak errors (in units of epsilon) found on various platforms with various floating point types, along with comparisons to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a> and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries. Unless otherwise specified any floating point type that is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero error</a>. <div class="table"> <a name="math_toolkit.zetas.zeta.errors_in_the_function_zeta_z"></a>Table 6.30. Errors In the Function zeta(z) <div class="table-contents"><table class="table" summary="Errors In the Function zeta(z)"> <colgroup> <col> <col> <col> <col> </colgroup> <thead><tr> <th> Significand Size </th> <th> Platform and Compiler </th> <th> z > 0 </th> <th> z < 0 </th> </tr></thead> <tbody> <tr> <td> 53 </td> <td> Win32, Visual C++ 8 </td> <td> Peak=0.99 Mean=0.1 GSL Peak=8.7 Mean=1.0 <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=2.1 Mean=1.1 </td> <td> Peak=7.1 Mean=3.0 GSL Peak=137 Mean=14 <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=5084 Mean=470 </td> </tr> <tr> <td> 64 </td> <td> RedHat Linux IA_EM64, gcc-4.1 </td> <td> Peak=0.99 Mean=0.5 </td> <td> Peak=570 Mean=60 </td> </tr> <tr> <td> 64 </td> <td> Redhat Linux IA64, gcc-4.1 </td> <td> Peak=0.99 Mean=0.5 </td> <td> Peak=559 Mean=56 </td> </tr> <tr> <td> 113 </td> <td> HPUX IA64, aCC A.06.06 </td> <td> Peak=1.0 Mean=0.4 </td> <td> Peak=1018 Mean=79 </td> </tr> </tbody> </table></div> </div> <h5> <a name="math_toolkit.zetas.zeta.h3"></a> <a name="math_toolkit.zetas.zeta.testing"></a><a class="link" href="zeta.html#math_toolkit.zetas.zeta.testing">Testing</a> </h5> The tests for these functions come in two parts: basic sanity checks use spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Zeta" target="_top">Mathworld's online evaluator</a>, while accuracy checks use high-precision test values calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> and this implementation. Note that the generic and type-specific versions of these functions use differing implementations internally, so this gives us reasonably independent test data. Using our test data to test other "known good" implementations also provides an additional sanity check. <h5> <a name="math_toolkit.zetas.zeta.h4"></a> <a name="math_toolkit.zetas.zeta.implementation"></a><a class="link" href="zeta.html#math_toolkit.zetas.zeta.implementation">Implementation</a> </h5> All versions of these functions first use the usual reflection formulas to make their arguments positive: <img src="../../../equations/zeta3.png"> The generic versions of these functions are implemented using the series: <img src="../../../equations/zeta6.png"> When the significand (mantissa) size is recognised (currently for 53, 64 and 113-bit reals, plus single-precision 24-bit handled via promotion to double) then a series of rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised by JM</a> are used. For 0 < z < 1 the approximating form is: <img src="../../../equations/zeta4.png"> For a rational approximation R(1-z) and a constant C. For 1 < z < 4 the approximating form is: <img src="../../../equations/zeta5.png"> For a rational approximation R(n-z) and a constant C and integer n. For z > 4 the approximating form is: ζ(z) = 1 + eR(z - n) For a rational approximation R(z-n) and integer n, note that the accuracy required for R(z-n) is not full machine precision, but an absolute error of: ε/R(0). This saves us quite a few digits when dealing with large z, especially when ε is small. </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="../zetas.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../zetas.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../expint.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>