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<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">&lt;</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">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/digamma1.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/digamma.svg" align="middle"></span>
</p></blockquote></div>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;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&#160;20.&#160;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="table">
<a name="math_toolkit.sf_gamma.digamma.table_digamma"></a><p class="title"><b>Table&#160;8.4.&#160;Error rates for digamma</b></p>
<div class="table-contents"><table class="table" summary="Error rates for digamma">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
Digamma Function: Large Values
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.84&#949; (Mean = 0.71&#949;))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.18&#949; (Mean = 0.331&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.39&#949; (Mean = 0.413&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.39&#949; (Mean = 0.413&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.98&#949; (Mean = 0.369&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Digamma Function: Near the Positive Root
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.891&#949; (Mean = 0.0995&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 135&#949; (Mean = 11.9&#949;))<br>
(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 2.02e+03&#949; (Mean = 256&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.37&#949; (Mean = 0.477&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.31&#949; (Mean = 0.471&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.997&#949; (Mean = 0.527&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Digamma Function: Near Zero
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.953&#949; (Mean = 0.348&#949;))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.17&#949; (Mean = 0.564&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.984&#949; (Mean = 0.361&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.984&#949; (Mean = 0.361&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.953&#949; (Mean = 0.337&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Digamma Function: Negative Values
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 4.56e+04&#949; (Mean = 3.91e+03&#949;))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 4.6e+04&#949; (Mean = 3.94e+03&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 180&#949; (Mean = 13&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 180&#949; (Mean = 13&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 214&#949; (Mean = 16.1&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Digamma Function: Values near 0
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.866&#949; (Mean = 0.387&#949;))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 3.58e+05&#949; (Mean = 1.6e+05&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1&#949; (Mean = 0.592&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1&#949; (Mean = 0.592&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Digamma Function: Integer arguments
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.992&#949; (Mean = 0.215&#949;)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.18&#949; (Mean = 0.607&#949;))<br>
(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 4.33&#949; (Mean = 0.982&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.888&#949; (Mean = 0.403&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.888&#949; (Mean = 0.403&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.992&#949; (Mean = 0.452&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Digamma Function: Half integer arguments
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.09&#949; (Mean = 0.531&#949;))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 46.2&#949; (Mean = 7.24&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.906&#949; (Mean = 0.409&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.906&#949; (Mean = 0.409&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.78&#949; (Mean = 0.314&#949;)</span>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><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>
<p>
The following error plot are based on an exhaustive search of the functions
domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/digamma__double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/digamma__80_bit_long_double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/digamma____float128.svg" align="middle"></span>
</p></blockquote></div>
<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 &gt; BIG the asymptotic expansion:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/digamma2.svg"></span>
</p></blockquote></div>
<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 &gt; BIG, and then evaluation via the asymptotic expansion above.
As special cases integer and half integer arguments are handled via:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/digamma4.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/digamma5.svg"></span>
</p></blockquote></div>
<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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/digamma3.svg"></span>
</p></blockquote></div>
<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 &gt;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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, 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>)
</p>
</div></td>
</tr></table>
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