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<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>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.zetas.zeta"></a><a class="link" href="zeta.html" title="Riemann Zeta Function">Riemann Zeta Function</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.zetas.zeta.h0"></a>
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<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>
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</h5>
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<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>
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</pre>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<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>
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<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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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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
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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.
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
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documentation for more details</a>.
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</p>
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<h5>
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<a name="math_toolkit.zetas.zeta.h1"></a>
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<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>
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</h5>
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<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>
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<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>
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<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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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</pre>
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<p>
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Returns the <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html" target="_top">zeta
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function</a> of z:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/zeta1.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../graphs/zeta1.svg" align="middle"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../graphs/zeta2.svg" align="middle"></span>
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</p>
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<h5>
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<a name="math_toolkit.zetas.zeta.h2"></a>
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<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>
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</h5>
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<p>
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The following table shows the peak errors (in units of epsilon) found on
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various platforms with various floating point types, along with comparisons
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to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a> and
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<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries. Unless
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otherwise specified any floating point type that is narrower than the one
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shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
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zero error</a>.
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</p>
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<div class="table">
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<a name="math_toolkit.zetas.zeta.table_zeta"></a><p class="title"><b>Table 6.73. Error rates for zeta</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for zeta">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 12.0<br> Win32<br> double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 5.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 5.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5130<br> Sun Solaris<br> long double
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Zeta: Random values greater than 1
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.836ε (Mean = 0.093ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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1.16:</em></span> Max = 8.69ε (Mean = 1.03ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
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<span class="red">Max = 4.49e+33ε (Mean = 6.85e+32ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_zeta_Cephes_Zeta_Random_values_greater_than_1">And
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other failures.</a>)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.846ε (Mean = 0.0833ε)</span><br>
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<br> (<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 5.45ε (Mean
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= 1ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.846ε (Mean = 0.0743ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Zeta: Random values less than 1
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 7.03ε (Mean = 2.98ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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1.16:</em></span> Max = 137ε (Mean = 13.8ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
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<span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_zeta_Cephes_Zeta_Random_values_less_than_1">And
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other failures.</a>)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 7.03ε (Mean = 2.71ε)</span><br> <br>
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(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 538ε (Mean = 59.3ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 70.1ε (Mean = 17.1ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Zeta: Values close to and greater than 1
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.994ε (Mean = 0.421ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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1.16:</em></span> Max = 7.73ε (Mean = 4.07ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
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<span class="red">Max = 6.77e+15ε (Mean = 1.52e+15ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_zeta_Cephes_Zeta_Values_close_to_and_greater_than_1">And
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other failures.</a>)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.995ε (Mean = 0.5ε)</span><br> <br>
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(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 1.9e+06ε (Mean = 5.11e+05ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.995ε (Mean = 0.5ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Zeta: Values close to and less than 1
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.991ε (Mean = 0.375ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
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1.16:</em></span> Max = 0.991ε (Mean = 0.28ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
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<span class="red">Max = 8.66e+15ε (Mean = 1.9e+15ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_zeta_Cephes_Zeta_Values_close_to_and_less_than_1">And
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other failures.</a>)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.998ε (Mean = 0.508ε)</span><br> <br>
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(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 8.53e+06ε (Mean =
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1.87e+06ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.998ε (Mean = 0.568ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Zeta: Integer arguments
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 6.5ε (Mean = 2.17ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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1.16:</em></span> Max = 3.75ε (Mean = 1.1ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
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<span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_zeta_Cephes_Zeta_Integer_arguments">And
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other failures.</a>)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 9ε (Mean = 3.06ε)</span><br> <br>
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(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 70.3ε (Mean = 17.4ε))
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</p>
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</td>
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<td>
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<p>
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|
|
<span class="blue">Max = 21ε (Mean = 7.13ε)</span>
|
||
|
|
</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>
|
||
|
|
<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, Jeremy Murphy, 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>)
|
||
|
|
</p>
|
||
|
|
</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>
|