1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Legendre-Stieltjes Polynomials</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="../sf_poly.html" title="Polynomials"> 9<link rel="prev" href="legendre.html" title="Legendre (and Associated) Polynomials"> 10<link rel="next" href="laguerre.html" title="Laguerre (and Associated) Polynomials"> 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="legendre.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_poly.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="laguerre.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="math_toolkit.sf_poly.legendre_stieltjes"></a><a class="link" href="legendre_stieltjes.html" title="Legendre-Stieltjes Polynomials">Legendre-Stieltjes 28 Polynomials</a> 29</h3></div></div></div> 30<h5> 31<a name="math_toolkit.sf_poly.legendre_stieltjes.h0"></a> 32 <span class="phrase"><a name="math_toolkit.sf_poly.legendre_stieltjes.synopsis"></a></span><a class="link" href="legendre_stieltjes.html#math_toolkit.sf_poly.legendre_stieltjes.synopsis">Synopsis</a> 33 </h5> 34<pre class="programlisting"> 35#include <boost/math/special_functions/legendre_stieltjes.hpp> 36 37 38<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> 39 40<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 41<span class="keyword">class</span> <span class="identifier">legendre_stieltjes</span> 42<span class="special">{</span> 43<span class="keyword">public</span><span class="special">:</span> 44 <span class="identifier">legendre_stieltjes</span><span class="special">(</span><span class="identifier">size_t</span> <span class="identifier">m</span><span class="special">);</span> 45 46 <span class="identifier">Real</span> <span class="identifier">norm_sq</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> 47 48 <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span> 49 50 <span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span> 51 52 <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="identifier">zeros</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> 53<span class="special">}</span> 54 55<span class="special">}}</span> 56</pre> 57<h5> 58<a name="math_toolkit.sf_poly.legendre_stieltjes.h1"></a> 59 <span class="phrase"><a name="math_toolkit.sf_poly.legendre_stieltjes.description"></a></span><a class="link" href="legendre_stieltjes.html#math_toolkit.sf_poly.legendre_stieltjes.description">Description</a> 60 </h5> 61<p> 62 The Legendre-Stieltjes polynomials are a family of polynomials used to generate 63 Gauss-Konrod quadrature formulas. Gauss-Konrod quadratures are algorithms 64 which extend a Gaussian quadrature in such a way that all abscissas are reused 65 when computed a higher-order estimate of the integral, allowing efficient 66 calculation of an error estimate. The Legendre-Stieltjes polynomials assist 67 with this task because their zeros <span class="emphasis"><em>interlace</em></span> the zeros 68 of the Legendre polynomials, meaning that between any two zeros of a Legendre 69 polynomial of degree n, there exists a zero of the Legendre-Stieltjes polynomial 70 of degree n+1. 71 </p> 72<p> 73 The Legendre-Stieltjes polynomials <span class="emphasis"><em>E<sub>n+1</sub></em></span> are defined 74 by the property that they have <span class="emphasis"><em>n</em></span> vanishing moments against 75 the oscillatory measure <span class="emphasis"><em>P<sub>n</sub></em></span>, i.e., 76 </p> 77<div class="blockquote"><blockquote class="blockquote"><p> 78 <span class="serif_italic">∫ <sub>-1</sub><sup>1</sup> E<sub>n+1</sub>(x)P<sub>n</sub>(x) x<sup>k</sup>dx = 0</span> 79 </p></blockquote></div> 80<p> 81 for <span class="emphasis"><em>k = 0, 1, ..., n</em></span>. 82 </p> 83<p> 84 The first few are 85 </p> 86<div class="blockquote"><blockquote class="blockquote"><p> 87 <span class="serif_italic">E<sub>1</sub>(x) = P<sub>1</sub>(x)</span> 88 </p></blockquote></div> 89<div class="blockquote"><blockquote class="blockquote"><p> 90 <span class="serif_italic">E<sub>2</sub>(x) = P<sub>2</sub>(x) - 2P<sub>0</sub>(x)/5</span> 91 </p></blockquote></div> 92<div class="blockquote"><blockquote class="blockquote"><p> 93 <span class="serif_italic">E<sub>3</sub>(x) = P<sub>3</sub>(x) - 9P<sub>1</sub>(x)/14</span> 94 </p></blockquote></div> 95<div class="blockquote"><blockquote class="blockquote"><p> 96 <span class="serif_italic">E<sub>4</sub>(x) = P<sub>4</sub>(x) - 20P<sub>2</sub>(x)/27 + 14P<sub>0</sub>(x)/891</span> 97 </p></blockquote></div> 98<div class="blockquote"><blockquote class="blockquote"><p> 99 <span class="serif_italic">E<sub>5</sub>(x) = P<sub>5</sub>(x) - 35P<sub>3</sub>(x)/44 + 135P<sub>1</sub>(x)/12584</span> 100 </p></blockquote></div> 101<p> 102 where <span class="emphasis"><em>P<sub>i</sub></em></span> are the Legendre polynomials. The scaling follows 103 <a href="http://www.ams.org/journals/mcom/1968-22-104/S0025-5718-68-99866-9/S0025-5718-68-99866-9.pdf" target="_top">Patterson</a>, 104 who expanded the Legendre-Stieltjes polynomials in a Legendre series and 105 took the coefficient of the highest-order Legendre polynomial in the series 106 to be unity. 107 </p> 108<p> 109 The Legendre-Stieltjes polynomials do not satisfy three-term recurrence relations 110 or have a particularly simple representation. Hence the constructor call 111 determines what, in fact, the polynomial is. Once the constructor comes back, 112 the polynomial can be evaluated via the Legendre series. 113 </p> 114<p> 115 Example usage: 116 </p> 117<pre class="programlisting"><span class="comment">// Call to the constructor determines the coefficients in the Legendre expansion</span> 118<span class="identifier">legendre_stieltjes</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">E</span><span class="special">(</span><span class="number">12</span><span class="special">);</span> 119<span class="comment">// Evaluate the polynomial at a point:</span> 120<span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">E</span><span class="special">(</span><span class="number">0.3</span><span class="special">);</span> 121<span class="comment">// Evaluate the derivative at a point:</span> 122<span class="keyword">double</span> <span class="identifier">x_p</span> <span class="special">=</span> <span class="identifier">E</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">0.3</span><span class="special">);</span> 123<span class="comment">// Use the norm_sq to change between scalings, if desired:</span> 124<span class="keyword">double</span> <span class="identifier">norm</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">E</span><span class="special">.</span><span class="identifier">norm_sq</span><span class="special">());</span> 125</pre> 126</div> 127<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 128<td align="left"></td> 129<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 130 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 131 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 132 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 133 Daryle Walker and Xiaogang Zhang<p> 134 Distributed under the Boost Software License, Version 1.0. (See accompanying 135 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>) 136 </p> 137</div></td> 138</tr></table> 139<hr> 140<div class="spirit-nav"> 141<a accesskey="p" href="legendre.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_poly.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="laguerre.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 142</div> 143</body> 144</html> 145
