<title>Trigamma</title>
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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>
<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">trigamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</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">trigamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</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 19. 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">trigamma</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 19. Policies: Controlling Precision, Error Handling etc">Policy</a><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 20. 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">trigamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</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>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
Returns the trigamma function of <span class="emphasis"><em>x</em></span>. Trigamma is defined
as the derivative of the digamma function:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/trigamma1.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/trigamma.svg" align="middle"></span>
+
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../equations/trigamma1.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/trigamma.svg" align="middle"></span>
- </p>
-<p>
- The final <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+ The final <a class="link" href="../../policy.html" title="Chapter 20. 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 19. Policies: Controlling Precision, Error Handling etc">policy
+ 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
documentation for more details</a>.
</p>
<p>
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.trigamma.table_trigamma"></a><p class="title"><b>Table 7.5. Error rates for trigamma</b></p>
+<a name="math_toolkit.sf_gamma.trigamma.table_trigamma"></a><p class="title"><b>Table 8.5. Error rates for trigamma</b></p>
<div class="table-contents"><table class="table" summary="Error rates for trigamma">
<colgroup>
<col>
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>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/trigamma__double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/trigamma__80_bit_long_double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/trigamma____float128.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/trigamma__double.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/trigamma__80_bit_long_double.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/trigamma____float128.svg" align="middle"></span>
+
+ </p></blockquote></div>
<h5>
<a name="math_toolkit.sf_gamma.trigamma.h3"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.testing"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.testing">Testing</a>
The arbitrary precision version of this function simply calls <a class="link" href="polygamma.html" title="Polygamma">polygamma</a>.
</p>
<p>
- For built in fixed precision types, negative arguments are first made positive
+ For built in fixed-precision types, negative arguments are first made positive
via:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/trigamma2.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/trigamma2.svg"></span>
+
+ </p></blockquote></div>
<p>
Then arguments in the range [0, 1) are shifted to >= 1 via:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/trigamma3.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/trigamma3.svg"></span>
+
+ </p></blockquote></div>
<p>
Then evaluation is via one of a number of rational approximations, for small
- x these are of the form:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/trigamma4.svg"></span>
- </p>
-<p>
- and for large x of the form:
+ <span class="emphasis"><em>x</em></span> these are of the form:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/trigamma4.svg"></span>
+
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../equations/trigamma5.svg"></span>
+ and for large <span class="emphasis"><em>x</em></span> of the form:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/trigamma5.svg"></span>
+
+ </p></blockquote></div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
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