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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.gauss"></a><a class="link" href="gauss.html" title="Gauss-Legendre quadrature">Gauss-Legendre quadrature</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.gauss.h0"></a>
<span class="phrase"><a name="math_toolkit.gauss.synopsis"></a></span><a class="link" href="gauss.html#math_toolkit.gauss.synopsis">Synopsis</a>
</h4>
<p>
<code class="computeroutput"><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">quadrature</span><span class="special">/</span><span class="identifier">gauss</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></code>
</p>
<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">namespace</span> <span class="identifier">quadrature</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">Points</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <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">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">&lt;&gt;</span> <span class="special">&gt;</span>
<span class="keyword">struct</span> <span class="identifier">gauss</span>
<span class="special">{</span>
<span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">RandomAccessContainer</span><span class="special">&amp;</span> <span class="identifier">abscissa</span><span class="special">();</span>
<span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">RandomAccessContainer</span><span class="special">&amp;</span> <span class="identifier">weights</span><span class="special">();</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">&gt;</span>
<span class="keyword">static</span> <span class="identifier">value_type</span> <span class="identifier">integrate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">pL1</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">&gt;</span>
<span class="keyword">static</span> <span class="identifier">value_type</span> <span class="identifier">integrate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">pL1</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">);</span>
<span class="special">};</span>
<span class="special">}}}</span> <span class="comment">// namespaces</span>
</pre>
<h4>
<a name="math_toolkit.gauss.h1"></a>
<span class="phrase"><a name="math_toolkit.gauss.description"></a></span><a class="link" href="gauss.html#math_toolkit.gauss.description">description</a>
</h4>
<p>
The <code class="computeroutput"><span class="identifier">gauss</span></code> class template performs
"one shot" non-adaptive Gauss-Legendre integration on some arbitrary
function <span class="emphasis"><em>f</em></span> using the number of evaluation points as specified
by <span class="emphasis"><em>Points</em></span>.
</p>
<p>
This is intentionally a very simple quadrature routine, it obtains no estimate
of the error, and is not adaptive, but is very efficient in simple cases that
involve integrating smooth "bell like" functions.
</p>
<pre class="programlisting"><span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">RandomAccessContainer</span><span class="special">&amp;</span> <span class="identifier">abscissa</span><span class="special">();</span>
<span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">RandomAccessContainer</span><span class="special">&amp;</span> <span class="identifier">weights</span><span class="special">();</span>
</pre>
<p>
These functions provide direct access to the abscissa and weights used to perform
the quadrature: the return type depends on the <span class="emphasis"><em>Points</em></span>
template parameter, but is always a RandomAccessContainer type. Note that only
positive (or zero) abscissa and weights are stored.
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">&gt;</span>
<span class="keyword">static</span> <span class="identifier">value_type</span> <span class="identifier">integrate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">pL1</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">);</span>
</pre>
<p>
Integrates <span class="emphasis"><em>f</em></span> over (-1,1), and optionally sets <code class="computeroutput"><span class="special">*</span><span class="identifier">pL1</span></code> to the
L1 norm of the returned value: if this is substantially larger than the return
value, then the sum was ill-conditioned. Note however, that no error estimate
is available.
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">&gt;</span>
<span class="keyword">static</span> <span class="identifier">value_type</span> <span class="identifier">integrate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">pL1</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">);</span>
</pre>
<p>
Integrates <span class="emphasis"><em>f</em></span> over (a,b), and optionally sets <code class="computeroutput"><span class="special">*</span><span class="identifier">pL1</span></code> to the
L1 norm of the returned value: if this is substantially larger than the return
value, then the sum was ill-conditioned. Note however, that no error estimate
is available. This function supports both finite and infinite <span class="emphasis"><em>a</em></span>
and <span class="emphasis"><em>b</em></span>, as long as <code class="computeroutput"><span class="identifier">a</span>
<span class="special">&lt;</span> <span class="identifier">b</span></code>.
</p>
<h4>
<a name="math_toolkit.gauss.h2"></a>
<span class="phrase"><a name="math_toolkit.gauss.choosing_the_number_of_points"></a></span><a class="link" href="gauss.html#math_toolkit.gauss.choosing_the_number_of_points">Choosing
the number of points</a>
</h4>
<p>
Internally class <code class="computeroutput"><span class="identifier">gauss</span></code> has
pre-computed tables of abscissa and weights for 7, 15, 20, 25 and 30 points
at up to 100-decimal digit precision. That means that using for example, <code class="computeroutput"><span class="identifier">gauss</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">30</span><span class="special">&gt;::</span><span class="identifier">integrate</span></code>
incurs absolutely zero set-up overhead from computing the abscissa/weight pairs.
When using multiprecision types with less than 100 digits of precision, then
there is a small initial one time cost, while the abscissa/weight pairs are
constructed from strings.
</p>
<p>
However, for types with higher precision, or numbers of points other than those
given above, the abscissa/weight pairs are computed when first needed and then
cached for future use, which does incur a noticeable overhead. If this is likely
to be an issue, then
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Defining BOOST_MATH_GAUSS_NO_COMPUTE_ON_DEMAND will result in a compile-time
error, whenever a combination of number type and number of points is used
which does not have pre-computed values.
</li>
<li class="listitem">
There is a program <a href="../../../tools/gauss_kronrod_constants.cpp" target="_top">gauss_kronrod_constants.cpp</a>
which was used to provide the pre-computed values already in gauss.hpp.
The program can be trivially modified to generate code and constants for
other precisions and numbers of points.
</li>
</ul></div>
<h4>
<a name="math_toolkit.gauss.h3"></a>
<span class="phrase"><a name="math_toolkit.gauss.examples"></a></span><a class="link" href="gauss.html#math_toolkit.gauss.examples">Examples</a>
</h4>
<p>
We'll begin by integrating t<sup>2</sup> atan(t) over (0,1) using a 7 term Gauss-Legendre
rule, and begin by defining the function to integrate as a C++ lambda expression:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quadrature</span><span class="special">;</span>
<span class="keyword">auto</span> <span class="identifier">f</span> <span class="special">=</span> <span class="special">[](</span><span class="keyword">const</span> <span class="keyword">double</span><span class="special">&amp;</span> <span class="identifier">t</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">t</span> <span class="special">*</span> <span class="identifier">t</span> <span class="special">*</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">atan</span><span class="special">(</span><span class="identifier">t</span><span class="special">);</span> <span class="special">};</span>
</pre>
<p>
Integration is simply a matter of calling the <code class="computeroutput"><span class="identifier">gauss</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span>
<span class="number">7</span><span class="special">&gt;::</span><span class="identifier">integrate</span></code> method:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">Q</span> <span class="special">=</span> <span class="identifier">gauss</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">7</span><span class="special">&gt;::</span><span class="identifier">integrate</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
Which yields a value 0.2106572512 accurate to 1e-10.
</p>
<p>
For more accurate evaluations, we'll move to a multiprecision type and use
a 20-point integration scheme:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_bin_float_quad</span><span class="special">;</span>
<span class="keyword">auto</span> <span class="identifier">f2</span> <span class="special">=</span> <span class="special">[](</span><span class="keyword">const</span> <span class="identifier">cpp_bin_float_quad</span><span class="special">&amp;</span> <span class="identifier">t</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">t</span> <span class="special">*</span> <span class="identifier">t</span> <span class="special">*</span> <span class="identifier">atan</span><span class="special">(</span><span class="identifier">t</span><span class="special">);</span> <span class="special">};</span>
<span class="identifier">cpp_bin_float_quad</span> <span class="identifier">Q2</span> <span class="special">=</span> <span class="identifier">gauss</span><span class="special">&lt;</span><span class="identifier">cpp_bin_float_quad</span><span class="special">,</span> <span class="number">20</span><span class="special">&gt;::</span><span class="identifier">integrate</span><span class="special">(</span><span class="identifier">f2</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
Which yields 0.2106572512258069881080923020669, which is accurate to 5e-28.
</p>
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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