| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Digamma</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="lgamma.html" title="Log Gamma"> |
| <link rel="next" href="trigamma.html" title="Trigamma"> |
| </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="lgamma.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="trigamma.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.digamma"></a><a class="link" href="digamma.html" title="Digamma">Digamma</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.sf_gamma.digamma.h0"></a> |
| <span class="phrase"><a name="math_toolkit.sf_gamma.digamma.synopsis"></a></span><a class="link" href="digamma.html#math_toolkit.sf_gamma.digamma.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">digamma</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">digamma</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">digamma</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.digamma.h1"></a> |
| <span class="phrase"><a name="math_toolkit.sf_gamma.digamma.description"></a></span><a class="link" href="digamma.html#math_toolkit.sf_gamma.digamma.description">Description</a> |
| </h5> |
| <p> |
| Returns the digamma or psi function of <span class="emphasis"><em>x</em></span>. Digamma is |
| defined as the logarithmic derivative of the gamma function: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/digamma1.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/digamma.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.digamma.h2"></a> |
| <span class="phrase"><a name="math_toolkit.sf_gamma.digamma.accuracy"></a></span><a class="link" href="digamma.html#math_toolkit.sf_gamma.digamma.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> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| Random Positive Values |
| </p> |
| </th> |
| <th> |
| <p> |
| Values Near The Positive Root |
| </p> |
| </th> |
| <th> |
| <p> |
| Values Near Zero |
| </p> |
| </th> |
| <th> |
| <p> |
| Negative Values |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32 Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.98 Mean=0.36 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.99 Mean=0.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.95 Mean=0.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=214 Mean=16 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Linux IA32 / GCC |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.4 Mean=0.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.3 Mean=0.45 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.98 Mean=0.35 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=180 Mean=13 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Linux IA64 / GCC |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.92 Mean=0.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.3 Mean=0.45 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.98 Mean=0.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=180 Mean=13 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.9 Mean=0.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.1 Mean=0.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.99 Mean=0.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=64 Mean=6 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <p> |
| As shown above, error rates for positive arguments are generally very low. |
| For negative arguments there are an infinite number of irrational roots: |
| relative errors very close to these can be arbitrarily large, although absolute |
| error will remain very low. |
| </p> |
| <h5> |
| <a name="math_toolkit.sf_gamma.digamma.h3"></a> |
| <span class="phrase"><a name="math_toolkit.sf_gamma.digamma.testing"></a></span><a class="link" href="digamma.html#math_toolkit.sf_gamma.digamma.testing">Testing</a> |
| </h5> |
| <p> |
| There are two sets of tests: spot values are computed using the online calculator |
| at functions.wolfram.com, while random test values are generated using the |
| high-precision reference implementation (a differentiated <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a> see below). |
| </p> |
| <h5> |
| <a name="math_toolkit.sf_gamma.digamma.h4"></a> |
| <span class="phrase"><a name="math_toolkit.sf_gamma.digamma.implementation"></a></span><a class="link" href="digamma.html#math_toolkit.sf_gamma.digamma.implementation">Implementation</a> |
| </h5> |
| <p> |
| The implementation is divided up into the following domains: |
| </p> |
| <p> |
| For Negative arguments the reflection formula: |
| </p> |
| <pre class="programlisting"><span class="identifier">digamma</span><span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">x</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">digamma</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pi</span><span class="special">/</span><span class="identifier">tan</span><span class="special">(</span><span class="identifier">pi</span><span class="special">*</span><span class="identifier">x</span><span class="special">);</span> |
| </pre> |
| <p> |
| is used to make <span class="emphasis"><em>x</em></span> positive. |
| </p> |
| <p> |
| For arguments in the range [0,1] the recurrence relation: |
| </p> |
| <pre class="programlisting"><span class="identifier">digamma</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">digamma</span><span class="special">(</span><span class="identifier">x</span><span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="special">-</span> <span class="number">1</span><span class="special">/</span><span class="identifier">x</span> |
| </pre> |
| <p> |
| is used to shift the evaluation to [1,2]. |
| </p> |
| <p> |
| For arguments in the range [1,2] a rational approximation <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised |
| by JM</a> is used (see below). |
| </p> |
| <p> |
| For arguments in the range [2,BIG] the recurrence relation: |
| </p> |
| <pre class="programlisting"><span class="identifier">digamma</span><span class="special">(</span><span class="identifier">x</span><span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">digamma</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span><span class="special">/</span><span class="identifier">x</span><span class="special">;</span> |
| </pre> |
| <p> |
| is used to shift the evaluation to the range [1,2]. |
| </p> |
| <p> |
| For arguments > BIG the asymptotic expansion: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/digamma2.svg"></span> |
| </p> |
| <p> |
| can be used. However, this expansion is divergent after a few terms: exactly |
| how many terms depends on the size of <span class="emphasis"><em>x</em></span>. Therefore the |
| value of <span class="emphasis"><em>BIG</em></span> must be chosen so that the series can be |
| truncated at a term that is too small to have any effect on the result when |
| evaluated at <span class="emphasis"><em>BIG</em></span>. Choosing BIG=10 for up to 80-bit reals, |
| and BIG=20 for 128-bit reals allows the series to truncated after a suitably |
| small number of terms and evaluated as a polynomial in <code class="computeroutput"><span class="number">1</span><span class="special">/(</span><span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span><span class="special">)</span></code>. |
| </p> |
| <p> |
| The arbitrary precision version of this function uses recurrence relations |
| until x > BIG, and then evaluation via the asymptotic expansion above. |
| As special cases integer and half integer arguments are handled via: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/digamma4.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/digamma5.svg"></span> |
| </p> |
| <p> |
| The rational approximation <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised |
| by JM</a> in the range [1,2] is derived as follows. |
| </p> |
| <p> |
| First a high precision approximation to digamma was constructed using a 60-term |
| differentiated <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>, |
| the form used is: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/digamma3.svg"></span> |
| </p> |
| <p> |
| Where P(x) and Q(x) are the polynomials from the rational form of the Lanczos |
| sum, and P'(x) and Q'(x) are their first derivatives. The Lanzos part of |
| this approximation has a theoretical precision of ~100 decimal digits. However, |
| cancellation in the above sum will reduce that to around <code class="computeroutput"><span class="number">99</span><span class="special">-(</span><span class="number">1</span><span class="special">/</span><span class="identifier">y</span><span class="special">)</span></code> digits |
| if <span class="emphasis"><em>y</em></span> is the result. This approximation was used to calculate |
| the positive root of digamma, and was found to agree with the value used |
| by Cody to 25 digits (See Math. Comp. 27, 123-127 (1973) by Cody, Strecok |
| and Thacher) and with the value used by Morris to 35 digits (See TOMS Algorithm |
| 708). |
| </p> |
| <p> |
| Likewise a few spot tests agreed with values calculated using functions.wolfram.com |
| to >40 digits. That's sufficiently precise to insure that the approximation |
| below is accurate to double precision. Achieving 128-bit long double precision |
| requires that the location of the root is known to ~70 digits, and it's not |
| clear whether the value calculated by this method meets that requirement: |
| the difficulty lies in independently verifying the value obtained. |
| </p> |
| <p> |
| The rational approximation <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised |
| by JM</a> was optimised for absolute error using the form: |
| </p> |
| <pre class="programlisting"><span class="identifier">digamma</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">X0</span><span class="special">)(</span><span class="identifier">Y</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">x</span> <span class="special">-</span> <span class="number">1</span><span class="special">));</span> |
| </pre> |
| <p> |
| Where X0 is the positive root of digamma, Y is a constant, and R(x - 1) is |
| the rational approximation. Note that since X0 is irrational, we need twice |
| as many digits in X0 as in x in order to avoid cancellation error during |
| the subtraction (this assumes that <span class="emphasis"><em>x</em></span> is an exact value, |
| if it's not then all bets are off). That means that even when x is the value |
| of the root rounded to the nearest representable value, the result of digamma(x) |
| <span class="emphasis"><em><span class="bold"><strong>will not be zero</strong></span></em></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="lgamma.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="trigamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| </body> |
| </html> |
