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497 lines
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<title>Log Gamma</title>
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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.lgamma"></a><a class="link" href="lgamma.html" title="Log Gamma">Log Gamma</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.sf_gamma.lgamma.h0"></a>
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<span class="phrase"><a name="math_toolkit.sf_gamma.lgamma.synopsis"></a></span><a class="link" href="lgamma.html#math_toolkit.sf_gamma.lgamma.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">gamma</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">lgamma</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">lgamma</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="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">lgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">sign</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">lgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">sign</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.lgamma.h1"></a>
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<span class="phrase"><a name="math_toolkit.sf_gamma.lgamma.description"></a></span><a class="link" href="lgamma.html#math_toolkit.sf_gamma.lgamma.description">Description</a>
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</h5>
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<p>
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The <a href="http://en.wikipedia.org/wiki/Gamma_function" target="_top">lgamma function</a>
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is defined by:
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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/lgamm1.svg"></span>
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</p></blockquote></div>
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<p>
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The second form of the function takes a pointer to an integer, which if non-null
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is set on output to the sign of tgamma(z).
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</p>
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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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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/lgamma.svg" align="middle"></span>
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</p></blockquote></div>
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<p>
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The return type of these functions 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> if T is an integer type, or type T
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otherwise.
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</p>
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<h5>
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<a name="math_toolkit.sf_gamma.lgamma.h2"></a>
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<span class="phrase"><a name="math_toolkit.sf_gamma.lgamma.accuracy"></a></span><a class="link" href="lgamma.html#math_toolkit.sf_gamma.lgamma.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, along with comparisons
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to various other libraries. Unless otherwise specified any floating point
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type that is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
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zero error</a>.
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</p>
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<p>
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Note that while the relative errors near the positive roots of lgamma are
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very low, the lgamma function has an infinite number of irrational roots
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for negative arguments: very close to these negative roots only a low absolute
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error can be guaranteed.
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</p>
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<div class="table">
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<a name="math_toolkit.sf_gamma.lgamma.table_lgamma"></a><p class="title"><b>Table 8.3. Error rates for lgamma</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for lgamma">
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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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factorials
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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 = 33.6ε (Mean = 2.78ε))<br> (<span class="emphasis"><em>Rmath
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3.2.3:</em></span> Max = 1.55ε (Mean = 0.592ε))
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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 = 0.991ε (Mean = 0.308ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 1.67ε (Mean = 0.487ε))<br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 1.67ε (Mean = 0.487ε))
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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 = 0.991ε (Mean = 0.383ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 1.36ε (Mean = 0.476ε))
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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 = 0.914ε (Mean = 0.175ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.958ε (Mean = 0.38ε))
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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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near 0
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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 = 5.21ε (Mean = 1.57ε))<br> (<span class="emphasis"><em>Rmath
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3.2.3:</em></span> Max = 0ε (Mean = 0ε))
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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 = 1.42ε (Mean = 0.566ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 0.964ε (Mean = 0.543ε))<br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.964ε (Mean = 0.543ε))
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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 = 1.42ε (Mean = 0.566ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.964ε (Mean = 0.543ε))
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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 = 0.964ε (Mean = 0.462ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.962ε (Mean = 0.372ε))
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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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near 1
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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 = 442ε (Mean = 88.8ε))<br> (<span class="emphasis"><em>Rmath
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3.2.3:</em></span> Max = 7.99e+04ε (Mean = 1.68e+04ε))
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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 = 0.948ε (Mean = 0.36ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 0.615ε (Mean = 0.096ε))<br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.615ε (Mean = 0.096ε))
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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 = 0.948ε (Mean = 0.36ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 1.71ε (Mean = 0.581ε))
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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 = 0.867ε (Mean = 0.468ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.906ε (Mean = 0.565ε))
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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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near 2
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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 = 1.17e+03ε (Mean = 274ε))<br> (<span class="emphasis"><em>Rmath
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3.2.3:</em></span> Max = 2.63e+05ε (Mean = 5.84e+04ε))
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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 = 0.878ε (Mean = 0.242ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 0.741ε (Mean = 0.263ε))<br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.741ε (Mean = 0.263ε))
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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 = 0.878ε (Mean = 0.242ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.598ε (Mean = 0.235ε))
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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 = 0.591ε (Mean = 0.159ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.741ε (Mean = 0.473ε))
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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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near -10
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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 = 24.9ε (Mean = 4.6ε))<br> (<span class="emphasis"><em>Rmath
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3.2.3:</em></span> Max = 4.22ε (Mean = 1.26ε))
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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 = 3.81ε (Mean = 1.01ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 0.997ε (Mean = 0.412ε))<br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.997ε (Mean = 0.412ε))
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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 = 3.81ε (Mean = 1.01ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 3.04ε (Mean = 1.01ε))
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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 = 4.22ε (Mean = 1.33ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.997ε (Mean = 0.444ε))
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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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near -55
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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 = 7.02ε (Mean = 1.47ε))<br> (<span class="emphasis"><em>Rmath
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3.2.3:</em></span> Max = 250ε (Mean = 60.9ε))
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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 = 0.821ε (Mean = 0.513ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 1.58ε (Mean = 0.672ε))<br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 1.58ε (Mean = 0.672ε))
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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 = 1.59ε (Mean = 0.587ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.821ε (Mean = 0.674ε))
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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 = 0.821ε (Mean = 0.419ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 249ε (Mean = 43.1ε))
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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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The following error plot are based on an exhaustive search of the functions
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domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
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precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
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<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
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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/lgamma__double.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/lgamma__80_bit_long_double.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/lgamma____float128.svg" align="middle"></span>
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</p></blockquote></div>
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<h5>
|
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<a name="math_toolkit.sf_gamma.lgamma.h3"></a>
|
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<span class="phrase"><a name="math_toolkit.sf_gamma.lgamma.testing"></a></span><a class="link" href="lgamma.html#math_toolkit.sf_gamma.lgamma.testing">Testing</a>
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</h5>
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<p>
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The main tests for this function involve comparisons against the logs of
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the factorials which can be independently calculated to very high accuracy.
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</p>
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<p>
|
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Random tests in key problem areas are also used.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.sf_gamma.lgamma.h4"></a>
|
|
<span class="phrase"><a name="math_toolkit.sf_gamma.lgamma.implementation"></a></span><a class="link" href="lgamma.html#math_toolkit.sf_gamma.lgamma.implementation">Implementation</a>
|
|
</h5>
|
|
<p>
|
|
The generic version of this function is implemented using Sterling's approximation
|
|
for large arguments:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/gamma6.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
For small arguments, the logarithm of tgamma is used.
|
|
</p>
|
|
<p>
|
|
For negative <span class="emphasis"><em>z</em></span> the logarithm version of the reflection
|
|
formula is used:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/lgamm3.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
For types of known precision, the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
|
|
approximation</a> is used, a traits class <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lanczos</span><span class="special">::</span><span class="identifier">lanczos_traits</span></code>
|
|
maps type T to an appropriate approximation. The logarithmic version of the
|
|
<a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a> is:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/lgamm4.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
Where L<sub>e,g</sub> is the Lanczos sum, scaled by e<sup>g</sup>.
|
|
</p>
|
|
<p>
|
|
As before the reflection formula is used for <span class="emphasis"><em>z < 0</em></span>.
|
|
</p>
|
|
<p>
|
|
When z is very near 1 or 2, then the logarithmic version of the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
|
|
approximation</a> suffers very badly from cancellation error: indeed for
|
|
values sufficiently close to 1 or 2, arbitrarily large relative errors can
|
|
be obtained (even though the absolute error is tiny).
|
|
</p>
|
|
<p>
|
|
For types with up to 113 bits of precision (up to and including 128-bit long
|
|
doubles), root-preserving rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
|
|
by JM</a> are used over the intervals [1,2] and [2,3]. Over the interval
|
|
[2,3] the approximation form used is:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">z</span><span class="special">-</span><span class="number">2</span><span class="special">)(</span><span class="identifier">z</span><span class="special">+</span><span class="number">1</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">z</span><span class="special">-</span><span class="number">2</span><span class="special">));</span>
|
|
</pre>
|
|
<p>
|
|
Where Y is a constant, and R(z-2) is the rational approximation: optimised
|
|
so that its absolute error is tiny compared to Y. In addition, small values
|
|
of z greater than 3 can handled by argument reduction using the recurrence
|
|
relation:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">log</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span>
|
|
</pre>
|
|
<p>
|
|
Over the interval [1,2] two approximations have to be used, one for small
|
|
z uses:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">z</span><span class="special">-</span><span class="number">1</span><span class="special">)(</span><span class="identifier">z</span><span class="special">-</span><span class="number">2</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">z</span><span class="special">-</span><span class="number">1</span><span class="special">));</span>
|
|
</pre>
|
|
<p>
|
|
Once again Y is a constant, and R(z-1) is optimised for low absolute error
|
|
compared to Y. For z > 1.5 the above form wouldn't converge to a minimax
|
|
solution but this similar form does:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="special">(</span><span class="number">2</span><span class="special">-</span><span class="identifier">z</span><span class="special">)(</span><span class="number">1</span><span class="special">-</span><span class="identifier">z</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="number">2</span><span class="special">-</span><span class="identifier">z</span><span class="special">));</span>
|
|
</pre>
|
|
<p>
|
|
Finally for z < 1 the recurrence relation can be used to move to z >
|
|
1:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">log</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span>
|
|
</pre>
|
|
<p>
|
|
Note that while this involves a subtraction, it appears not to suffer from
|
|
cancellation error: as z decreases from 1 the <code class="computeroutput"><span class="special">-</span><span class="identifier">log</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span></code> term grows positive much more rapidly than
|
|
the <code class="computeroutput"><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">+</span><span class="number">1</span><span class="special">)</span></code> term becomes negative. So in this specific
|
|
case, significant digits are preserved, rather than cancelled.
|
|
</p>
|
|
<p>
|
|
For other types which do have a <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
|
|
approximation</a> defined for them the current solution is as follows:
|
|
imagine we balance the two terms in the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
|
|
approximation</a> by dividing the power term by its value at <span class="emphasis"><em>z
|
|
= 1</em></span>, and then multiplying the Lanczos coefficients by the same
|
|
value. Now each term will take the value 1 at <span class="emphasis"><em>z = 1</em></span>
|
|
and we can rearrange the power terms in terms of log1p. Likewise if we subtract
|
|
1 from the Lanczos sum part (algebraically, by subtracting the value of each
|
|
term at <span class="emphasis"><em>z = 1</em></span>), we obtain a new summation that can be
|
|
also be fed into log1p. Crucially, all of the terms tend to zero, as <span class="emphasis"><em>z
|
|
-> 1</em></span>:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/lgamm5.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
The C<sub>k</sub> terms in the above are the same as in the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
|
|
approximation</a>.
|
|
</p>
|
|
<p>
|
|
A similar rearrangement can be performed at <span class="emphasis"><em>z = 2</em></span>:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/lgamm6.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>
|
|
<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>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
|
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