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diff --git a/libs/math/doc/html/math_toolkit/sf_gamma/polygamma.html b/libs/math/doc/html/math_toolkit/sf_gamma/polygamma.html new file mode 100644 index 000000000..de3526403 --- /dev/null +++ b/libs/math/doc/html/math_toolkit/sf_gamma/polygamma.html @@ -0,0 +1,285 @@ +<html> +<head> +<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> +<title>Polygamma</title> +<link rel="stylesheet" href="../../math.css" type="text/css"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> +<link rel="up" href="../sf_gamma.html" title="Gamma Functions"> +<link rel="prev" href="trigamma.html" title="Trigamma"> +<link rel="next" href="gamma_ratios.html" title="Ratios of Gamma Functions"> +</head> +<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> +<table cellpadding="2" width="100%"><tr> +<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> +<td align="center"><a href="../../../../../../index.html">Home</a></td> +<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> +<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> +<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> +<td align="center"><a href="../../../../../../more/index.htm">More</a></td> +</tr></table> +<hr> +<div class="spirit-nav"> +<a accesskey="p" href="trigamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="gamma_ratios.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +<div class="section"> +<div class="titlepage"><div><div><h3 class="title"> +<a name="math_toolkit.sf_gamma.polygamma"></a><a class="link" href="polygamma.html" title="Polygamma">Polygamma</a> +</h3></div></div></div> +<h5> +<a name="math_toolkit.sf_gamma.polygamma.h0"></a> + <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> + </h5> +<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">polygamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> +</pre> +<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> + +<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> +<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> +<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> + +<span class="special">}}</span> <span class="comment">// namespaces</span> +</pre> +<h5> +<a name="math_toolkit.sf_gamma.polygamma.h1"></a> + <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> + </h5> +<p> + Returns the polygamma function of <span class="emphasis"><em>x</em></span>. Polygamma is defined + as the n'th derivative of the digamma function: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma1.svg"></span> + </p> +<p> + The following graphs illustrate the behaviour of the function for odd and + even order: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../graphs/polygamma2.svg" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../graphs/polygamma3.svg" align="middle"></span> + </p> +<p> + The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can + be used to control the behaviour of the function: how it handles errors, + what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy + documentation for more details</a>. + </p> +<p> + 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 + 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 + T otherwise. + </p> +<h5> +<a name="math_toolkit.sf_gamma.polygamma.h2"></a> + <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> + </h5> +<p> + The following table shows the peak errors (in units of epsilon) found on + various platforms with various floating point types. Unless otherwise specified + any floating point type that is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero error</a>. + </p> +<div class="informaltable"><table class="table"> +<colgroup> +<col> +<col> +<col> +<col> +</colgroup> +<thead><tr> +<th> + <p> + Significand Size + </p> + </th> +<th> + <p> + Platform and Compiler + </p> + </th> +<th> + <p> + Small-medium positive arguments + </p> + </th> +<th> + <p> + Small-medium negative x + </p> + </th> +</tr></thead> +<tbody> +<tr> +<td> + <p> + 53 + </p> + </td> +<td> + <p> + Win32 Visual C++ 12 + </p> + </td> +<td> + <p> + Peak=5.0 Mean=1 + </p> + </td> +<td> + <p> + Peak=1200 Mean=65 + </p> + </td> +</tr> +<tr> +<td> + <p> + 64 + </p> + </td> +<td> + <p> + Win64 Mingw GCC + </p> + </td> +<td> + <p> + Peak=16 Mean=3 + </p> + </td> +<td> + <p> + Peak=33 Mean=3 + </p> + </td> +</tr> +<tr> +<td> + <p> + 113 + </p> + </td> +<td> + <p> + Win64 Mingw GCC __float128 + </p> + </td> +<td> + <p> + Peak=6.5 Mean=1 + </p> + </td> +<td> + <p> + Peak=30 Mean=4 + </p> + </td> +</tr> +</tbody> +</table></div> +<p> + As shown above, error rates are generally very acceptable for moderately + sized arguments. Error rates should stay low for exact inputs, however, please + note that the function becomes exceptionally sensitive to small changes in + input for large n and negative x, indeed for cases where <span class="emphasis"><em>n!</em></span> + would overflow, the function changes directly from -∞ to +∞ somewhere between + each negative integer - <span class="emphasis"><em>these cases are not handled correctly</em></span>. + </p> +<p> + <span class="bold"><strong>For these reasons results should be treated with extreme + caution when <span class="emphasis"><em>n</em></span> is large and x negative</strong></span>. + </p> +<h5> +<a name="math_toolkit.sf_gamma.polygamma.h3"></a> + <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> + </h5> +<p> + Testing is against Mathematica generated spot values to 35 digit precision. + </p> +<h5> +<a name="math_toolkit.sf_gamma.polygamma.h4"></a> + <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> + </h5> +<p> + For x < 0 the following reflection formula is used: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma2.svg"></span> + </p> +<p> + The n'th derivative of <span class="emphasis"><em>cot(x)</em></span> is tabulated for small + <span class="emphasis"><em>n</em></span>, and for larger n has the general form: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma3.svg"></span> + </p> +<p> + The coefficients of the cosine terms can be calculated iteratively starting + from <span class="emphasis"><em>C<sub>1,0</sub> = -1</em></span> and then using + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma7.svg"></span> + </p> +<p> + to generate coefficients for n+1. + </p> +<p> + Note that every other coefficient is zero, and therefore what we have are + even or odd polynomials depending on whether n is even or odd. + </p> +<p> + Once x is positive then we have two methods available to us, for small x + we use the series expansion: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma4.svg"></span> + </p> +<p> + Note that the evaluation of zeta functions at integer values is essentially + a table lookup as <a class="link" href="../zetas/zeta.html" title="Riemann Zeta Function">zeta</a> is + optimized for those cases. + </p> +<p> + For large x we use the asymptotic expansion: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma5.svg"></span> + </p> +<p> + For x in-between the two extremes we use the relation: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma6.svg"></span> + </p> +<p> + to make x large enough for the asymptotic expansion to be used. + </p> +<p> + There are also two special cases: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma8.svg"></span> + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma9.svg"></span> + </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, 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="trigamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="gamma_ratios.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +</body> +</html> |