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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.sf_gamma.polygamma"></a><a class="link" href="polygamma.html" title="Polygamma">Polygamma</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.sf_gamma.polygamma.h0"></a>
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<span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.synopsis"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.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">polygamma</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">polygamma</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</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">polygamma</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</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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<h5>
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<a name="math_toolkit.sf_gamma.polygamma.h1"></a>
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<span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.description"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.description">Description</a>
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</h5>
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<p>
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Returns the polygamma function of <span class="emphasis"><em>x</em></span>. Polygamma is defined
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as the n'th derivative of the digamma function:
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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/polygamma1.svg"></span>
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</p></blockquote></div>
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<p>
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The following graphs illustrate the behaviour of the function for odd and
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even order:
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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/polygamma2.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/polygamma3.svg" align="middle"></span>
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</p></blockquote></div>
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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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<p>
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The return type of this function 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 result is of type <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and type
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T otherwise.
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</p>
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<h5>
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<a name="math_toolkit.sf_gamma.polygamma.h2"></a>
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<span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.accuracy"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.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. Unless otherwise specified
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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>.
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</p>
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<div class="table">
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<a name="math_toolkit.sf_gamma.polygamma.table_polygamma"></a><p class="title"><b>Table 8.6. Error rates for polygamma</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for polygamma">
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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> 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> 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 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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Mathematica Data
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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 = 0.824ε (Mean = 0.0574ε)</span><br>
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<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 62.9ε (Mean = 12.8ε))<br>
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(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 108ε (Mean = 15.2ε))
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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 = 7.38ε (Mean = 1.84ε)</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 = 34.3ε (Mean = 7.65ε)</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 = 9.32ε (Mean = 1.95ε)</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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Mathematica Data - large 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 = 0.998ε (Mean = 0.0592ε)</span><br>
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<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 244ε (Mean = 32.8ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_GSL_2_1_Mathematica_Data_large_arguments">And
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other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
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<span class="red">Max = 1.71e+56ε (Mean = 1.01e+55ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_Rmath_3_2_3_Mathematica_Data_large_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 = 2.23ε (Mean = 0.323ε)</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 = 11.1ε (Mean = 0.848ε)</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 = 150ε (Mean = 13.9ε)</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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Mathematica Data - negative 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 = 0.516ε (Mean = 0.022ε)</span><br> <br>
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(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 36.6ε (Mean = 3.04ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_GSL_2_1_Mathematica_Data_negative_arguments">And
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other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
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Max = 0ε (Mean = 0ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_Rmath_3_2_3_Mathematica_Data_negative_arguments">And
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other failures.</a>)
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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 = 269ε (Mean = 87.7ε)</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 = 269ε (Mean = 88.4ε)</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 = 497ε (Mean = 129ε)</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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Mathematica Data - large negative 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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 1.79ε (Mean = 0.197ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_GSL_2_1_Mathematica_Data_large_negative_arguments">And
|
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other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
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Max = 0ε (Mean = 0ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_Rmath_3_2_3_Mathematica_Data_large_negative_arguments">And
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other failures.</a>)
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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 = 155ε (Mean = 96.4ε)</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 = 155ε (Mean = 96.4ε)</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 = 162ε (Mean = 101ε)</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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Mathematica Data - small 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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 15.2ε (Mean = 5.03ε))<br> (<span class="emphasis"><em>Rmath
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3.2.3:</em></span> Max = 106ε (Mean = 20ε))
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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 = 3.33ε (Mean = 0.75ε)</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 = 3.33ε (Mean = 0.75ε)</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 = 3ε (Mean = 0.496ε)</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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Mathematica Data - Large orders and other bug cases
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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 = 151ε (Mean = 39.3ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_GSL_2_1_Mathematica_Data_Large_orders_and_other_bug_cases">And
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other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</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_7_1_0_linux_double_polygamma_Rmath_3_2_3_Mathematica_Data_Large_orders_and_other_bug_cases">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 = 54.5ε (Mean = 13.3ε)</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 = 145ε (Mean = 55.9ε)</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 = 200ε (Mean = 57.2ε)</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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As shown above, error rates are generally very acceptable for moderately
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sized arguments. Error rates should stay low for exact inputs, however, please
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note that the function becomes exceptionally sensitive to small changes in
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input for large n and negative x, indeed for cases where <span class="emphasis"><em>n!</em></span>
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would overflow, the function changes directly from -∞ to +∞ somewhere between
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each negative integer - <span class="emphasis"><em>these cases are not handled correctly</em></span>.
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</p>
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<p>
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<span class="bold"><strong>For these reasons results should be treated with extreme
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caution when <span class="emphasis"><em>n</em></span> is large and x negative</strong></span>.
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</p>
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<h5>
|
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<a name="math_toolkit.sf_gamma.polygamma.h3"></a>
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<span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.testing"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.testing">Testing</a>
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</h5>
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<p>
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Testing is against Mathematica generated spot values to 35 digit precision.
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</p>
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<h5>
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<a name="math_toolkit.sf_gamma.polygamma.h4"></a>
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<span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.implementation"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.implementation">Implementation</a>
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</h5>
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<p>
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For x < 0 the following reflection formula is used:
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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/polygamma2.svg"></span>
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|
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</p></blockquote></div>
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<p>
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The n'th derivative of <span class="emphasis"><em>cot(x)</em></span> is tabulated for small
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<span class="emphasis"><em>n</em></span>, and for larger n has the general form:
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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/polygamma3.svg"></span>
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|
|
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</p></blockquote></div>
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<p>
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The coefficients of the cosine terms can be calculated iteratively starting
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from <span class="emphasis"><em>C<sub>1,0</sub> = -1</em></span> and then using
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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/polygamma7.svg"></span>
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</p></blockquote></div>
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<p>
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to generate coefficients for n+1.
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</p>
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<p>
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Note that every other coefficient is zero, and therefore what we have are
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even or odd polynomials depending on whether n is even or odd.
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</p>
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<p>
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Once x is positive then we have two methods available to us, for small x
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we use the series expansion:
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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/polygamma4.svg"></span>
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</p></blockquote></div>
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<p>
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Note that the evaluation of zeta functions at integer values is essentially
|
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a table lookup as <a class="link" href="../zetas/zeta.html" title="Riemann Zeta Function">zeta</a> is
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optimized for those cases.
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</p>
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<p>
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For large x we use the asymptotic expansion:
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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/polygamma5.svg"></span>
|
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</p></blockquote></div>
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<p>
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For x in-between the two extremes we use the relation:
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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/polygamma6.svg"></span>
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</p></blockquote></div>
|
|
<p>
|
|
to make x large enough for the asymptotic expansion to be used.
|
|
</p>
|
|
<p>
|
|
There are also two special cases:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/polygamma8.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/polygamma9.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
</div>
|
|
<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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