1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Gauss-Legendre quadrature</title> 5<link rel="stylesheet" href="../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../quadrature.html" title="Chapter 13. Quadrature and Differentiation"> 9<link rel="prev" href="trapezoidal.html" title="Trapezoidal Quadrature"> 10<link rel="next" href="gauss_kronrod.html" title="Gauss-Kronrod Quadrature"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="trapezoidal.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../quadrature.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="gauss_kronrod.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 27<a name="math_toolkit.gauss"></a><a class="link" href="gauss.html" title="Gauss-Legendre quadrature">Gauss-Legendre quadrature</a> 28</h2></div></div></div> 29<h4> 30<a name="math_toolkit.gauss.h0"></a> 31 <span class="phrase"><a name="math_toolkit.gauss.synopsis"></a></span><a class="link" href="gauss.html#math_toolkit.gauss.synopsis">Synopsis</a> 32 </h4> 33<p> 34 <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> 35 </p> 36<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> 37 38<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 21. 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> 39<span class="keyword">struct</span> <span class="identifier">gauss</span> 40<span class="special">{</span> 41 <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> 42 <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> 43 44 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">></span> 45 <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> 46 47 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">></span> 48 <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> 49<span class="special">};</span> 50 51<span class="special">}}}</span> <span class="comment">// namespaces</span> 52</pre> 53<h4> 54<a name="math_toolkit.gauss.h1"></a> 55 <span class="phrase"><a name="math_toolkit.gauss.description"></a></span><a class="link" href="gauss.html#math_toolkit.gauss.description">description</a> 56 </h4> 57<p> 58 The <code class="computeroutput"><span class="identifier">gauss</span></code> class template performs 59 "one shot" non-adaptive Gauss-Legendre integration on some arbitrary 60 function <span class="emphasis"><em>f</em></span> using the number of evaluation points as specified 61 by <span class="emphasis"><em>Points</em></span>. 62 </p> 63<p> 64 This is intentionally a very simple quadrature routine, it obtains no estimate 65 of the error, and is not adaptive, but is very efficient in simple cases that 66 involve integrating smooth "bell like" functions and functions with 67 rapidly convergent power series. 68 </p> 69<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> 70<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> 71</pre> 72<p> 73 These functions provide direct access to the abscissa and weights used to perform 74 the quadrature: the return type depends on the <span class="emphasis"><em>Points</em></span> 75 template parameter, but is always a RandomAccessContainer type. Note that only 76 positive (or zero) abscissa and weights are stored. 77 </p> 78<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> 79<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> 80</pre> 81<p> 82 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 83 L1 norm of the returned value: if this is substantially larger than the return 84 value, then the sum was ill-conditioned. Note however, that no error estimate 85 is available. 86 </p> 87<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> 88<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> 89</pre> 90<p> 91 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 92 L1 norm of the returned value: if this is substantially larger than the return 93 value, then the sum was ill-conditioned. Note however, that no error estimate 94 is available. This function supports both finite and infinite <span class="emphasis"><em>a</em></span> 95 and <span class="emphasis"><em>b</em></span>, as long as <code class="computeroutput"><span class="identifier">a</span> 96 <span class="special"><</span> <span class="identifier">b</span></code>. 97 </p> 98<p> 99 The Gaussian quadrature routine support both real and complex-valued quadrature. 100 For example, the Lambert-W function admits the integral representation 101 </p> 102<div class="blockquote"><blockquote class="blockquote"><p> 103 <span class="serif_italic"><span class="emphasis"><em>W(z) = 1/2Π ∫<sub>-Π</sub><sup>Π</sup> ((1- 104 v cot(v) )^2 + v^2)/(z + v csc(v) exp(-v cot(v))) dv</em></span></span> 105 </p></blockquote></div> 106<p> 107 so it can be effectively computed via Gaussian quadrature using the following 108 code: 109 </p> 110<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> 111<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> 112 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cos</span><span class="special">;</span> 113 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">sin</span><span class="special">;</span> 114 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exp</span><span class="special">;</span> 115 <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> 116 <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> 117 118 <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> 119 <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> 120 <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> 121 <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> 122 <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> 123 <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> 124 <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> 125 <span class="keyword">return</span> <span class="identifier">res</span><span class="special">;</span> 126<span class="special">};</span> 127 128<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> 129<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> 130</pre> 131<h4> 132<a name="math_toolkit.gauss.h2"></a> 133 <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 134 the number of points</a> 135 </h4> 136<p> 137 Internally class <code class="computeroutput"><span class="identifier">gauss</span></code> has 138 pre-computed tables of abscissa and weights for 7, 15, 20, 25 and 30 points 139 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> 140 incurs absolutely zero set-up overhead from computing the abscissa/weight pairs. 141 When using multiprecision types with less than 100 digits of precision, then 142 there is a small initial one time cost, while the abscissa/weight pairs are 143 constructed from strings. 144 </p> 145<p> 146 However, for types with higher precision, or numbers of points other than those 147 given above, the abscissa/weight pairs are computed when first needed and then 148 cached for future use, which does incur a noticeable overhead. If this is likely 149 to be an issue, then 150 </p> 151<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 152<li class="listitem"> 153 Defining BOOST_MATH_GAUSS_NO_COMPUTE_ON_DEMAND will result in a compile-time 154 error, whenever a combination of number type and number of points is used 155 which does not have pre-computed values. 156 </li> 157<li class="listitem"> 158 There is a program <a href="../../../tools/gauss_kronrod_constants.cpp" target="_top">gauss_kronrod_constants.cpp</a> 159 which was used to provide the pre-computed values already in gauss.hpp. 160 The program can be trivially modified to generate code and constants for 161 other precisions and numbers of points. 162 </li> 163</ul></div> 164<h4> 165<a name="math_toolkit.gauss.h3"></a> 166 <span class="phrase"><a name="math_toolkit.gauss.examples"></a></span><a class="link" href="gauss.html#math_toolkit.gauss.examples">Examples</a> 167 </h4> 168<p> 169 We'll begin by integrating t<sup>2</sup> atan(t) over (0,1) using a 7 term Gauss-Legendre 170 rule, and begin by defining the function to integrate as a C++ lambda expression: 171 </p> 172<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> 173 174<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> 175</pre> 176<p> 177 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> 178 <span class="number">7</span><span class="special">>::</span><span class="identifier">integrate</span></code> method: 179 </p> 180<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> 181</pre> 182<p> 183 Which yields a value 0.2106572512 accurate to 1e-10. 184 </p> 185<p> 186 For more accurate evaluations, we'll move to a multiprecision type and use 187 a 20-point integration scheme: 188 </p> 189<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> 190 191<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> 192 193<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> 194</pre> 195<p> 196 Which yields 0.2106572512258069881080923020669, which is accurate to 5e-28. 197 </p> 198</div> 199<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 200<td align="left"></td> 201<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 202 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 203 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 204 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 205 Daryle Walker and Xiaogang Zhang<p> 206 Distributed under the Boost Software License, Version 1.0. (See accompanying 207 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>) 208 </p> 209</div></td> 210</tr></table> 211<hr> 212<div class="spirit-nav"> 213<a accesskey="p" href="trapezoidal.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../quadrature.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="gauss_kronrod.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 214</div> 215</body> 216</html> 217
