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| <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> |
| <span class="phrase"><a name="math_toolkit.zetas.zeta.synopsis"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.synopsis">Synopsis</a> |
| </h5> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">zeta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| </pre> |
| <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces</span> |
| </pre> |
| <p> |
| 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"><span class="emphasis"><em>result |
| type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, and T otherwise. |
| </p> |
| <p> |
| 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>. |
| </p> |
| <h5> |
| <a name="math_toolkit.zetas.zeta.h1"></a> |
| <span class="phrase"><a name="math_toolkit.zetas.zeta.description"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.description">Description</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html" target="_top">zeta |
| function</a> of z: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/zeta1.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/zeta1.svg" align="middle"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/zeta2.svg" align="middle"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.zetas.zeta.h2"></a> |
| <span class="phrase"><a name="math_toolkit.zetas.zeta.accuracy"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| 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>. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.zetas.zeta.errors_in_the_function_zeta_z"></a><p class="title"><b>Table 6.30. Errors In the Function zeta(z)</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In the Function zeta(z)"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| z > 0 |
| </p> |
| </th> |
| <th> |
| <p> |
| z < 0 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.99 Mean=0.1 |
| </p> |
| <p> |
| GSL Peak=8.7 Mean=1.0 |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=2.1 |
| Mean=1.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=7.1 Mean=3.0 |
| </p> |
| <p> |
| GSL Peak=137 Mean=14 |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=5084 |
| Mean=470 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| RedHat Linux IA_EM64, gcc-4.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.99 Mean=0.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=570 Mean=60 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-4.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.99 Mean=0.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=559 Mean=56 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.0 Mean=0.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1018 Mean=79 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><h5> |
| <a name="math_toolkit.zetas.zeta.h3"></a> |
| <span class="phrase"><a name="math_toolkit.zetas.zeta.testing"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.testing">Testing</a> |
| </h5> |
| <p> |
| 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. |
| </p> |
| <h5> |
| <a name="math_toolkit.zetas.zeta.h4"></a> |
| <span class="phrase"><a name="math_toolkit.zetas.zeta.implementation"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.implementation">Implementation</a> |
| </h5> |
| <p> |
| All versions of these functions first use the usual reflection formulas to |
| make their arguments positive: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/zeta3.svg"></span> |
| </p> |
| <p> |
| The generic versions of these functions are implemented using the series: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/zeta6.svg"></span> |
| </p> |
| <p> |
| 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. |
| </p> |
| <p> |
| For 0 < z < 1 the approximating form is: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/zeta4.svg"></span> |
| </p> |
| <p> |
| For a rational approximation R(1-z) and a constant C. |
| </p> |
| <p> |
| For 1 < z < 4 the approximating form is: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/zeta5.svg"></span> |
| </p> |
| <p> |
| For a rational approximation R(n-z) and a constant C and integer n. |
| </p> |
| <p> |
| For z > 4 the approximating form is: |
| </p> |
| <p> |
| ζ(z) = 1 + e<sup>R(z - n)</sup> |
| </p> |
| <p> |
| 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. |
| </p> |
| <p> |
| Finally, there are some special cases for integer arguments, there are closed |
| forms for negative or even integers: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/zeta7.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/zeta8.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/zeta9.svg"></span> |
| </p> |
| <p> |
| and for positive odd integers we simply cache pre-computed values as these |
| are of great benefit to some infinite series calculations. |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
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| <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<p> |
| 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>) |
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