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<title>Riemann Zeta Function</title>
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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 20. 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 20. 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 20. 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 20. 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 20. 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 20. 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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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/zeta1.svg"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/zeta1.svg" align="middle"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/zeta2.svg" align="middle"></span>
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</p></blockquote></div>
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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 8.76. 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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GNU C++ version 7.1.0<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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GNU C++ version 7.1.0<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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Sun compiler version 0x5150<br> Sun Solaris<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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Microsoft Visual C++ version 14.1<br> Win32<br> 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 = 0.846ε (Mean = 0.0833ε)</span><br>
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<br> (<span class="emphasis"><em><cmath>:</em></span> Max = 5.45ε (Mean = 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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 8.69ε (Mean = 1.03ε))
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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>
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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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</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 = 2.93ε)</span><br> <br>
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(<span class="emphasis"><em><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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 137ε (Mean = 13.8ε))
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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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<td>
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<p>
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<span class="blue">Max = 6.84ε (Mean = 3.12ε)</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 = 0.995ε (Mean = 0.5ε)</span><br> <br>
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(<span class="emphasis"><em><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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 7.73ε (Mean = 4.07ε))
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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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<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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</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 = 0.998ε (Mean = 0.508ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 8.53e+06ε (Mean = 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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 0.991ε (Mean = 0.28ε))
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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>
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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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</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 = 9ε (Mean = 3.06ε)</span><br> <br>
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(<span class="emphasis"><em><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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 3.75ε (Mean = 1.1ε))
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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 = 28ε (Mean = 5.62ε)</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 = 3ε)</span>
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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</div>
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<br class="table-break"><p>
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The following error plot are based on an exhaustive search of the functions
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domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
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precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
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<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/zeta__double.svg" align="middle"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/zeta__80_bit_long_double.svg" align="middle"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/zeta____float128.svg" align="middle"></span>
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</p></blockquote></div>
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<h5>
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<a name="math_toolkit.zetas.zeta.h3"></a>
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<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>
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</h5>
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<p>
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The tests for these functions come in two parts: basic sanity checks use
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spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Zeta" target="_top">Mathworld's
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online evaluator</a>, while accuracy checks use high-precision test values
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calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
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and this implementation. Note that the generic and type-specific versions
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of these functions use differing implementations internally, so this gives
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us reasonably independent test data. Using our test data to test other "known
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good" implementations also provides an additional sanity check.
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</p>
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<h5>
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<a name="math_toolkit.zetas.zeta.h4"></a>
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<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>
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</h5>
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<p>
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All versions of these functions first use the usual reflection formulas to
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make their arguments positive:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/zeta3.svg"></span>
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</p></blockquote></div>
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<p>
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The generic versions of these functions are implemented using the series:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/zeta6.svg"></span>
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</p></blockquote></div>
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<p>
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When the significand (mantissa) size is recognised (currently for 53, 64
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and 113-bit reals, plus single-precision 24-bit handled via promotion to
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double) then a series of rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
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by JM</a> are used.
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</p>
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<p>
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For 0 < z < 1 the approximating form is:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/zeta4.svg"></span>
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</p></blockquote></div>
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<p>
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For a rational approximation <span class="emphasis"><em>R(1-z)</em></span> and a constant
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<span class="emphasis"><em>C</em></span>:
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</p>
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<p>
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For 1 < z < 4 the approximating form is:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/zeta5.svg"></span>
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</p></blockquote></div>
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<p>
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For a rational approximation <span class="emphasis"><em>R(n-z)</em></span> and a constant
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<span class="emphasis"><em>C</em></span> and integer <span class="emphasis"><em>n</em></span>:
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</p>
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<p>
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For z > 4 the approximating form is:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">ζ(z) = 1 + e<sup>R(z - n)</sup></span>
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</p></blockquote></div>
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<p>
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For a rational approximation <span class="emphasis"><em>R(z-n)</em></span> and integer <span class="emphasis"><em>n</em></span>,
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note that the accuracy required for <span class="emphasis"><em>R(z-n)</em></span> is not full
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machine-precision, but an absolute error of: /ε<span class="emphasis"><em>R(0)</em></span>.
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This saves us quite a few digits when dealing with large <span class="emphasis"><em>z</em></span>,
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especially when ε is small.
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</p>
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<p>
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Finally, there are some special cases for integer arguments, there are closed
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forms for negative or even integers:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/zeta7.svg"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/zeta8.svg"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/zeta9.svg"></span>
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</p></blockquote></div>
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<p>
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|
and for positive odd integers we simply cache pre-computed values as these
|
|
are of great benefit to some infinite series calculations.
|
|
</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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|
<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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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</p>
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</div></td>
|
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</tr></table>
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