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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.gauss"></a><a class="link" href="gauss.html" title="Gauss-Legendre quadrature">Gauss-Legendre quadrature</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.gauss.h0"></a>
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<span class="phrase"><a name="math_toolkit.gauss.synopsis"></a></span><a class="link" href="gauss.html#math_toolkit.gauss.synopsis">Synopsis</a>
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</h4>
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<p>
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<code class="computeroutput"><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">quadrature</span><span class="special">/</span><span class="identifier">gauss</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></code>
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</p>
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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">namespace</span> <span class="identifier">quadrature</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">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 20. 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"><></span> <span class="special">></span>
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<span class="keyword">struct</span> <span class="identifier">gauss</span>
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<span class="special">{</span>
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<span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">RandomAccessContainer</span><span class="special">&</span> <span class="identifier">abscissa</span><span class="special">();</span>
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<span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">RandomAccessContainer</span><span class="special">&</span> <span class="identifier">weights</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">F</span><span class="special">></span>
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<span class="keyword">static</span> <span class="keyword">auto</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">decltype</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">F</span><span class="special">>()(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">Real</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">F</span><span class="special">></span>
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<span class="keyword">static</span> <span class="keyword">auto</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="keyword">decltype</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">F</span><span class="special">>()(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>()))</span>
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<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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<h4>
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<a name="math_toolkit.gauss.h1"></a>
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<span class="phrase"><a name="math_toolkit.gauss.description"></a></span><a class="link" href="gauss.html#math_toolkit.gauss.description">description</a>
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</h4>
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<p>
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The <code class="computeroutput"><span class="identifier">gauss</span></code> class template performs
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"one shot" non-adaptive Gauss-Legendre integration on some arbitrary
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function <span class="emphasis"><em>f</em></span> using the number of evaluation points as specified
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by <span class="emphasis"><em>Points</em></span>.
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</p>
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<p>
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This is intentionally a very simple quadrature routine, it obtains no estimate
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of the error, and is not adaptive, but is very efficient in simple cases that
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involve integrating smooth "bell like" functions and functions with
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rapidly convergent power series.
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</p>
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<pre class="programlisting"><span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">RandomAccessContainer</span><span class="special">&</span> <span class="identifier">abscissa</span><span class="special">();</span>
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<span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">RandomAccessContainer</span><span class="special">&</span> <span class="identifier">weights</span><span class="special">();</span>
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</pre>
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<p>
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These functions provide direct access to the abscissa and weights used to perform
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the quadrature: the return type depends on the <span class="emphasis"><em>Points</em></span>
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template parameter, but is always a RandomAccessContainer type. Note that only
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positive (or zero) abscissa and weights are stored.
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</p>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">></span>
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<span class="keyword">static</span> <span class="keyword">auto</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">decltype</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">F</span><span class="special">>()(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>()))</span>
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</pre>
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<p>
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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
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L1 norm of the returned value: if this is substantially larger than the return
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value, then the sum was ill-conditioned. Note however, that no error estimate
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is available.
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</p>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">></span>
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<span class="keyword">static</span> <span class="keyword">auto</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="keyword">decltype</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">F</span><span class="special">>()(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>()))</span>
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</pre>
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<p>
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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
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L1 norm of the returned value: if this is substantially larger than the return
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value, then the sum was ill-conditioned. Note however, that no error estimate
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is available. This function supports both finite and infinite <span class="emphasis"><em>a</em></span>
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and <span class="emphasis"><em>b</em></span>, as long as <code class="computeroutput"><span class="identifier">a</span>
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<span class="special"><</span> <span class="identifier">b</span></code>.
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</p>
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<p>
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The Gaussian quadrature routine support both real and complex-valued quadrature.
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For example, the Lambert-W function admits the integral representation
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic"><span class="emphasis"><em>W(z) = 1/2Π ∫<sub>-Π</sub><sup>Π</sup> ((1-
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v cot(v) )^2 + v^2)/(z + v csc(v) exp(-v cot(v))) dv</em></span></span>
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</p></blockquote></div>
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<p>
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so it can be effectively computed via Gaussian quadrature using the following
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code:
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</p>
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<pre class="programlisting"><span class="identifier">Complex</span> <span class="identifier">z</span><span class="special">{</span><span class="number">2</span><span class="special">,</span> <span class="number">3</span><span class="special">};</span>
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<span class="keyword">auto</span> <span class="identifier">lw</span> <span class="special">=</span> <span class="special">[&</span><span class="identifier">z</span><span class="special">](</span><span class="identifier">Real</span> <span class="identifier">v</span><span class="special">)-></span><span class="identifier">Complex</span> <span class="special">{</span>
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<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cos</span><span class="special">;</span>
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<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">sin</span><span class="special">;</span>
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<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exp</span><span class="special">;</span>
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<span class="identifier">Real</span> <span class="identifier">sinv</span> <span class="special">=</span> <span class="identifier">sin</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
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<span class="identifier">Real</span> <span class="identifier">cosv</span> <span class="special">=</span> <span class="identifier">cos</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
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<span class="identifier">Real</span> <span class="identifier">cotv</span> <span class="special">=</span> <span class="identifier">cosv</span><span class="special">/</span><span class="identifier">sinv</span><span class="special">;</span>
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<span class="identifier">Real</span> <span class="identifier">cscv</span> <span class="special">=</span> <span class="number">1</span><span class="special">/</span><span class="identifier">sinv</span><span class="special">;</span>
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<span class="identifier">Real</span> <span class="identifier">t</span> <span class="special">=</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">v</span><span class="special">*</span><span class="identifier">cotv</span><span class="special">)*(</span><span class="number">1</span><span class="special">-</span><span class="identifier">v</span><span class="special">*</span><span class="identifier">cotv</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">v</span><span class="special">*</span><span class="identifier">v</span><span class="special">;</span>
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<span class="identifier">Real</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">v</span><span class="special">*</span><span class="identifier">cscv</span><span class="special">*</span><span class="identifier">exp</span><span class="special">(-</span><span class="identifier">v</span><span class="special">*</span><span class="identifier">cotv</span><span class="special">);</span>
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<span class="identifier">Complex</span> <span class="identifier">den</span> <span class="special">=</span> <span class="identifier">z</span> <span class="special">+</span> <span class="identifier">x</span><span class="special">;</span>
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<span class="identifier">Complex</span> <span class="identifier">num</span> <span class="special">=</span> <span class="identifier">t</span><span class="special">*(</span><span class="identifier">z</span><span class="special">/</span><span class="identifier">pi</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>());</span>
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<span class="identifier">Complex</span> <span class="identifier">res</span> <span class="special">=</span> <span class="identifier">num</span><span class="special">/</span><span class="identifier">den</span><span class="special">;</span>
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<span class="keyword">return</span> <span class="identifier">res</span><span class="special">;</span>
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<span class="special">};</span>
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<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">Real</span><span class="special">,</span> <span class="number">30</span><span class="special">></span> <span class="identifier">integrator</span><span class="special">;</span>
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<span class="identifier">Complex</span> <span class="identifier">W</span> <span class="special">=</span> <span class="identifier">integrator</span><span class="special">.</span><span class="identifier">integrate</span><span class="special">(</span><span class="identifier">lw</span><span class="special">,</span> <span class="special">(</span><span class="identifier">Real</span><span class="special">)</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">pi</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>());</span>
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</pre>
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<h4>
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<a name="math_toolkit.gauss.h2"></a>
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<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
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the number of points</a>
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</h4>
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<p>
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Internally class <code class="computeroutput"><span class="identifier">gauss</span></code> has
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pre-computed tables of abscissa and weights for 7, 15, 20, 25 and 30 points
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at up to 100-decimal digit precision. That means that using for example, <code class="computeroutput"><span class="identifier">gauss</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">30</span><span class="special">>::</span><span class="identifier">integrate</span></code>
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incurs absolutely zero set-up overhead from computing the abscissa/weight pairs.
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When using multiprecision types with less than 100 digits of precision, then
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there is a small initial one time cost, while the abscissa/weight pairs are
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constructed from strings.
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</p>
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<p>
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However, for types with higher precision, or numbers of points other than those
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given above, the abscissa/weight pairs are computed when first needed and then
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cached for future use, which does incur a noticeable overhead. If this is likely
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to be an issue, then
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</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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Defining BOOST_MATH_GAUSS_NO_COMPUTE_ON_DEMAND will result in a compile-time
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error, whenever a combination of number type and number of points is used
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which does not have pre-computed values.
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</li>
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<li class="listitem">
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There is a program <a href="../../../tools/gauss_kronrod_constants.cpp" target="_top">gauss_kronrod_constants.cpp</a>
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which was used to provide the pre-computed values already in gauss.hpp.
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The program can be trivially modified to generate code and constants for
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other precisions and numbers of points.
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</li>
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</ul></div>
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<h4>
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<a name="math_toolkit.gauss.h3"></a>
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<span class="phrase"><a name="math_toolkit.gauss.examples"></a></span><a class="link" href="gauss.html#math_toolkit.gauss.examples">Examples</a>
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</h4>
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<p>
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We'll begin by integrating t<sup>2</sup> atan(t) over (0,1) using a 7 term Gauss-Legendre
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rule, and begin by defining the function to integrate as a C++ lambda expression:
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</p>
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<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>
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<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">&</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>
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</pre>
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<p>
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Integration is simply a matter of calling the <code class="computeroutput"><span class="identifier">gauss</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span>
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<span class="number">7</span><span class="special">>::</span><span class="identifier">integrate</span></code> method:
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</p>
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<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"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">7</span><span class="special">>::</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>
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</pre>
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<p>
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Which yields a value 0.2106572512 accurate to 1e-10.
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</p>
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<p>
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For more accurate evaluations, we'll move to a multiprecision type and use
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a 20-point integration scheme:
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</p>
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<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>
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<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">&</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>
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<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"><</span><span class="identifier">cpp_bin_float_quad</span><span class="special">,</span> <span class="number">20</span><span class="special">>::</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>
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</pre>
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<p>
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|
Which yields 0.2106572512258069881080923020669, which is accurate to 5e-28.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
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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>)
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</p>
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