1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Legendre (and Associated) 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="../sf_poly.html" title="Polynomials"> 10<link rel="next" href="legendre_stieltjes.html" title="Legendre-Stieltjes 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="../sf_poly.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="legendre_stieltjes.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"></a><a class="link" href="legendre.html" title="Legendre (and Associated) Polynomials">Legendre (and Associated) 28 Polynomials</a> 29</h3></div></div></div> 30<h5> 31<a name="math_toolkit.sf_poly.legendre.h0"></a> 32 <span class="phrase"><a name="math_toolkit.sf_poly.legendre.synopsis"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.synopsis">Synopsis</a> 33 </h5> 34<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">legendre</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 35</pre> 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> 37 38<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 39<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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> 40 41<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 42<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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 43 44<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 45<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">legendre_p_prime</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> 46 47<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 48<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">legendre_p_prime</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 49 50<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 51<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">legendre_p_zeros</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 52 53<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 54<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">legendre_p_zeros</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">);</span> 55 56<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 57<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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> 58 59<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 60<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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 61 62<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 63<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">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> 64 65<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 66<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">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 67 68<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> 69<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">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span> 70 71<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> 72<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">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span> 73 74 75<span class="special">}}</span> <span class="comment">// namespaces</span> 76</pre> 77<p> 78 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 79 type calculation rules</em></span></a>: note than when there is a single 80 template argument the result is the same type as that argument or <code class="computeroutput"><span class="keyword">double</span></code> if the template argument is an integer 81 type. 82 </p> 83<p> 84 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 85 be used to control the behaviour of the function: how it handles errors, 86 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 87 documentation for more details</a>. 88 </p> 89<h5> 90<a name="math_toolkit.sf_poly.legendre.h1"></a> 91 <span class="phrase"><a name="math_toolkit.sf_poly.legendre.description"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.description">Description</a> 92 </h5> 93<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 94<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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> 95 96<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 97<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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 98</pre> 99<p> 100 Returns the Legendre Polynomial of the first kind: 101 </p> 102<div class="blockquote"><blockquote class="blockquote"><p> 103 <span class="inlinemediaobject"><img src="../../../equations/legendre_0.svg"></span> 104 105 </p></blockquote></div> 106<p> 107 Requires -1 <= x <= 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 108 </p> 109<p> 110 Negative orders are handled via the reflection formula: 111 </p> 112<div class="blockquote"><blockquote class="blockquote"><p> 113 P<sub>-l-1</sub>(x) = P<sub>l</sub>(x) 114 </p></blockquote></div> 115<p> 116 The following graph illustrates the behaviour of the first few Legendre Polynomials: 117 </p> 118<div class="blockquote"><blockquote class="blockquote"><p> 119 <span class="inlinemediaobject"><img src="../../../graphs/legendre_p.svg" align="middle"></span> 120 121 </p></blockquote></div> 122<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 123<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">legendre_p_prime</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> 124 125<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 126<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">legendre_p_prime</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 127</pre> 128<p> 129 Returns the derivatives of the Legendre polynomials. 130 </p> 131<pre class="programlisting"><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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 132<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">legendre_p_zeros</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 133 134<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 135<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">legendre_p_zeros</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">);</span> 136</pre> 137<p> 138 The zeros of the Legendre polynomials are calculated by Newton's method using 139 an initial guess given by Tricomi with root bracketing provided by Szego. 140 </p> 141<p> 142 Since the Legendre polynomials are alternatively even and odd, only the non-negative 143 zeros are returned. For the odd Legendre polynomials, the first zero is always 144 zero. The rest of the zeros are returned in increasing order. 145 </p> 146<p> 147 Note that the argument to the routine is an integer, and the output is a 148 floating-point type. Hence the template argument is mandatory. The time to 149 extract a single root is linear in <code class="computeroutput"><span class="identifier">l</span></code> 150 (this is scaling to evaluate the Legendre polynomials), so recovering all 151 roots is ��(<code class="computeroutput"><span class="identifier">l</span></code><sup>2</sup>). Algorithms 152 with linear scaling <a href="https://doi.org/10.1137/06067016X" target="_top">exist</a> 153 for recovering all roots, but requires tooling not currently built into boost.math. 154 This implementation proceeds under the assumption that calculating zeros 155 of these functions will not be a bottleneck for any workflow. 156 </p> 157<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 158<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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> 159 160<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 161<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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 162</pre> 163<p> 164 Returns the associated Legendre polynomial of the first kind: 165 </p> 166<div class="blockquote"><blockquote class="blockquote"><p> 167 <span class="inlinemediaobject"><img src="../../../equations/legendre_1.svg"></span> 168 169 </p></blockquote></div> 170<p> 171 Requires -1 <= x <= 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 172 </p> 173<p> 174 Negative values of <span class="emphasis"><em>l</em></span> and <span class="emphasis"><em>m</em></span> are 175 handled via the identity relations: 176 </p> 177<div class="blockquote"><blockquote class="blockquote"><p> 178 <span class="inlinemediaobject"><img src="../../../equations/legendre_3.svg"></span> 179 180 </p></blockquote></div> 181<div class="caution"><table border="0" summary="Caution"> 182<tr> 183<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td> 184<th align="left">Caution</th> 185</tr> 186<tr><td align="left" valign="top"> 187<p> 188 The definition of the associated Legendre polynomial used here includes 189 a leading Condon-Shortley phase term of (-1)<sup>m</sup>. This matches the definition 190 given by Abramowitz and Stegun (8.6.6) and that used by <a href="http://mathworld.wolfram.com/LegendrePolynomial.html" target="_top">Mathworld</a> 191 and <a href="http://documents.wolfram.com/mathematica/functions/LegendreP" target="_top">Mathematica's 192 LegendreP function</a>. However, uses in the literature do not always 193 include this phase term, and strangely the specification for the associated 194 Legendre function in the C++ TR1 (assoc_legendre) also omits it, in spite 195 of stating that it uses Abramowitz and Stegun as the final arbiter on these 196 matters. 197 </p> 198<p> 199 See: 200 </p> 201<p> 202 <a href="http://mathworld.wolfram.com/LegendrePolynomial.html" target="_top">Weisstein, 203 Eric W. "Legendre Polynomial." From MathWorld--A Wolfram Web 204 Resource</a>. 205 </p> 206<p> 207 Abramowitz, M. and Stegun, I. A. (Eds.). "Legendre Functions" 208 and "Orthogonal Polynomials." Ch. 22 in Chs. 8 and 22 in Handbook 209 of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 210 9th printing. New York: Dover, pp. 331-339 and 771-802, 1972. 211 </p> 212</td></tr> 213</table></div> 214<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 215<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">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> 216 217<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 218<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">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 219</pre> 220<p> 221 Returns the value of the Legendre polynomial that is the second solution 222 to the Legendre differential equation, for example: 223 </p> 224<div class="blockquote"><blockquote class="blockquote"><p> 225 <span class="inlinemediaobject"><img src="../../../equations/legendre_2.svg"></span> 226 227 </p></blockquote></div> 228<p> 229 Requires -1 <= x <= 1, otherwise <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> 230 is called. 231 </p> 232<p> 233 The following graph illustrates the first few Legendre functions of the second 234 kind: 235 </p> 236<div class="blockquote"><blockquote class="blockquote"><p> 237 <span class="inlinemediaobject"><img src="../../../graphs/legendre_q.svg" align="middle"></span> 238 239 </p></blockquote></div> 240<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> 241<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">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span> 242</pre> 243<p> 244 Implements the three term recurrence relation for the Legendre polynomials, 245 this function can be used to create a sequence of values evaluated at the 246 same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>l</em></span>. This 247 recurrence relation holds for Legendre Polynomials of both the first and 248 second kinds. 249 </p> 250<div class="blockquote"><blockquote class="blockquote"><p> 251 <span class="inlinemediaobject"><img src="../../../equations/legendre_4.svg"></span> 252 253 </p></blockquote></div> 254<p> 255 For example we could produce a vector of the first 10 polynomial values using: 256 </p> 257<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span> <span class="comment">// Abscissa value</span> 258<span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">;</span> 259<span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span> 260<span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span> 261<span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special"><</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span> 262 <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_next</span><span class="special">(</span><span class="identifier">l</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">],</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">-</span><span class="number">1</span><span class="special">]));</span> 263<span class="comment">// Double check values:</span> 264<span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special"><</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span> 265 <span class="identifier">assert</span><span class="special">(</span><span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">]</span> <span class="special">==</span> <span class="identifier">legendre_p</span><span class="special">(</span><span class="identifier">l</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span> 266</pre> 267<p> 268 Formally the arguments are: 269 </p> 270<div class="variablelist"> 271<p class="title"><b></b></p> 272<dl class="variablelist"> 273<dt><span class="term">l</span></dt> 274<dd><p> 275 The degree of the last polynomial calculated. 276 </p></dd> 277<dt><span class="term">x</span></dt> 278<dd><p> 279 The abscissa value 280 </p></dd> 281<dt><span class="term">Pl</span></dt> 282<dd><p> 283 The value of the polynomial evaluated at degree <span class="emphasis"><em>l</em></span>. 284 </p></dd> 285<dt><span class="term">Plm1</span></dt> 286<dd><p> 287 The value of the polynomial evaluated at degree <span class="emphasis"><em>l-1</em></span>. 288 </p></dd> 289</dl> 290</div> 291<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> 292<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">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span> 293</pre> 294<p> 295 Implements the three term recurrence relation for the Associated Legendre 296 polynomials, this function can be used to create a sequence of values evaluated 297 at the same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>l</em></span>. 298 </p> 299<div class="blockquote"><blockquote class="blockquote"><p> 300 <span class="inlinemediaobject"><img src="../../../equations/legendre_5.svg"></span> 301 302 </p></blockquote></div> 303<p> 304 For example we could produce a vector of the first m+10 polynomial values 305 using: 306 </p> 307<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span> <span class="comment">// Abscissa value</span> 308<span class="keyword">int</span> <span class="identifier">m</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span> <span class="comment">// order</span> 309<span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">;</span> 310<span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="identifier">m</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span> 311<span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="number">1</span> <span class="special">+</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span> 312<span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special"><</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span> 313 <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_next</span><span class="special">(</span><span class="identifier">l</span> <span class="special">+</span> <span class="number">10</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">],</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">-</span><span class="number">1</span><span class="special">]));</span> 314<span class="comment">// Double check values:</span> 315<span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special"><</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span> 316 <span class="identifier">assert</span><span class="special">(</span><span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">]</span> <span class="special">==</span> <span class="identifier">legendre_p</span><span class="special">(</span><span class="number">10</span> <span class="special">+</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span> 317</pre> 318<p> 319 Formally the arguments are: 320 </p> 321<div class="variablelist"> 322<p class="title"><b></b></p> 323<dl class="variablelist"> 324<dt><span class="term">l</span></dt> 325<dd><p> 326 The degree of the last polynomial calculated. 327 </p></dd> 328<dt><span class="term">m</span></dt> 329<dd><p> 330 The order of the Associated Polynomial. 331 </p></dd> 332<dt><span class="term">x</span></dt> 333<dd><p> 334 The abscissa value 335 </p></dd> 336<dt><span class="term">Pl</span></dt> 337<dd><p> 338 The value of the polynomial evaluated at degree <span class="emphasis"><em>l</em></span>. 339 </p></dd> 340<dt><span class="term">Plm1</span></dt> 341<dd><p> 342 The value of the polynomial evaluated at degree <span class="emphasis"><em>l-1</em></span>. 343 </p></dd> 344</dl> 345</div> 346<h5> 347<a name="math_toolkit.sf_poly.legendre.h2"></a> 348 <span class="phrase"><a name="math_toolkit.sf_poly.legendre.accuracy"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.accuracy">Accuracy</a> 349 </h5> 350<p> 351 The following table shows peak errors (in units of epsilon) for various domains 352 of input arguments. Note that only results for the widest floating point 353 type on the system are given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively 354 zero error</a>. 355 </p> 356<div class="table"> 357<a name="math_toolkit.sf_poly.legendre.table_legendre_p"></a><p class="title"><b>Table 8.32. Error rates for legendre_p</b></p> 358<div class="table-contents"><table class="table" summary="Error rates for legendre_p"> 359<colgroup> 360<col> 361<col> 362<col> 363<col> 364<col> 365</colgroup> 366<thead><tr> 367<th> 368 </th> 369<th> 370 <p> 371 GNU C++ version 7.1.0<br> linux<br> double 372 </p> 373 </th> 374<th> 375 <p> 376 GNU C++ version 7.1.0<br> linux<br> long double 377 </p> 378 </th> 379<th> 380 <p> 381 Sun compiler version 0x5150<br> Sun Solaris<br> long double 382 </p> 383 </th> 384<th> 385 <p> 386 Microsoft Visual C++ version 14.1<br> Win32<br> double 387 </p> 388 </th> 389</tr></thead> 390<tbody> 391<tr> 392<td> 393 <p> 394 Legendre Polynomials: Small Values 395 </p> 396 </td> 397<td> 398 <p> 399 <span class="blue">Max = 0.732ε (Mean = 0.0619ε)</span><br> 400 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 211ε (Mean = 20.4ε)) 401 </p> 402 </td> 403<td> 404 <p> 405 <span class="blue">Max = 69.2ε (Mean = 9.58ε)</span><br> <br> 406 (<span class="emphasis"><em><cmath>:</em></span> Max = 124ε (Mean = 13.2ε)) 407 </p> 408 </td> 409<td> 410 <p> 411 <span class="blue">Max = 69.2ε (Mean = 9.58ε)</span> 412 </p> 413 </td> 414<td> 415 <p> 416 <span class="blue">Max = 211ε (Mean = 20.4ε)</span> 417 </p> 418 </td> 419</tr> 420<tr> 421<td> 422 <p> 423 Legendre Polynomials: Large Values 424 </p> 425 </td> 426<td> 427 <p> 428 <span class="blue">Max = 0.632ε (Mean = 0.0693ε)</span><br> 429 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 300ε (Mean = 33.2ε)) 430 </p> 431 </td> 432<td> 433 <p> 434 <span class="blue">Max = 699ε (Mean = 59.6ε)</span><br> <br> 435 (<span class="emphasis"><em><cmath>:</em></span> Max = 343ε (Mean = 32.1ε)) 436 </p> 437 </td> 438<td> 439 <p> 440 <span class="blue">Max = 699ε (Mean = 59.6ε)</span> 441 </p> 442 </td> 443<td> 444 <p> 445 <span class="blue">Max = 300ε (Mean = 33.2ε)</span> 446 </p> 447 </td> 448</tr> 449</tbody> 450</table></div> 451</div> 452<br class="table-break"><div class="table"> 453<a name="math_toolkit.sf_poly.legendre.table_legendre_q"></a><p class="title"><b>Table 8.33. Error rates for legendre_q</b></p> 454<div class="table-contents"><table class="table" summary="Error rates for legendre_q"> 455<colgroup> 456<col> 457<col> 458<col> 459<col> 460<col> 461</colgroup> 462<thead><tr> 463<th> 464 </th> 465<th> 466 <p> 467 GNU C++ version 7.1.0<br> linux<br> double 468 </p> 469 </th> 470<th> 471 <p> 472 GNU C++ version 7.1.0<br> linux<br> long double 473 </p> 474 </th> 475<th> 476 <p> 477 Sun compiler version 0x5150<br> Sun Solaris<br> long double 478 </p> 479 </th> 480<th> 481 <p> 482 Microsoft Visual C++ version 14.1<br> Win32<br> double 483 </p> 484 </th> 485</tr></thead> 486<tbody> 487<tr> 488<td> 489 <p> 490 Legendre Polynomials: Small Values 491 </p> 492 </td> 493<td> 494 <p> 495 <span class="blue">Max = 0.612ε (Mean = 0.0517ε)</span><br> 496 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 46.4ε (Mean = 7.46ε)) 497 </p> 498 </td> 499<td> 500 <p> 501 <span class="blue">Max = 50.9ε (Mean = 9ε)</span> 502 </p> 503 </td> 504<td> 505 <p> 506 <span class="blue">Max = 50.9ε (Mean = 8.98ε)</span> 507 </p> 508 </td> 509<td> 510 <p> 511 <span class="blue">Max = 46.4ε (Mean = 7.32ε)</span> 512 </p> 513 </td> 514</tr> 515<tr> 516<td> 517 <p> 518 Legendre Polynomials: Large Values 519 </p> 520 </td> 521<td> 522 <p> 523 <span class="blue">Max = 2.49ε (Mean = 0.202ε)</span><br> <br> 524 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 4.6e+03ε (Mean = 366ε)) 525 </p> 526 </td> 527<td> 528 <p> 529 <span class="blue">Max = 5.98e+03ε (Mean = 478ε)</span> 530 </p> 531 </td> 532<td> 533 <p> 534 <span class="blue">Max = 5.98e+03ε (Mean = 478ε)</span> 535 </p> 536 </td> 537<td> 538 <p> 539 <span class="blue">Max = 4.6e+03ε (Mean = 366ε)</span> 540 </p> 541 </td> 542</tr> 543</tbody> 544</table></div> 545</div> 546<br class="table-break"><div class="table"> 547<a name="math_toolkit.sf_poly.legendre.table_legendre_p_associated_"></a><p class="title"><b>Table 8.34. Error rates for legendre_p (associated)</b></p> 548<div class="table-contents"><table class="table" summary="Error rates for legendre_p (associated)"> 549<colgroup> 550<col> 551<col> 552<col> 553<col> 554<col> 555</colgroup> 556<thead><tr> 557<th> 558 </th> 559<th> 560 <p> 561 GNU C++ version 7.1.0<br> linux<br> double 562 </p> 563 </th> 564<th> 565 <p> 566 GNU C++ version 7.1.0<br> linux<br> long double 567 </p> 568 </th> 569<th> 570 <p> 571 Sun compiler version 0x5150<br> Sun Solaris<br> long double 572 </p> 573 </th> 574<th> 575 <p> 576 Microsoft Visual C++ version 14.1<br> Win32<br> double 577 </p> 578 </th> 579</tr></thead> 580<tbody><tr> 581<td> 582 <p> 583 Associated Legendre Polynomials: Small Values 584 </p> 585 </td> 586<td> 587 <p> 588 <span class="blue">Max = 0.999ε (Mean = 0.05ε)</span><br> <br> 589 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 121ε (Mean = 6.75ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_legendre_p_associated__GSL_2_1_Associated_Legendre_Polynomials_Small_Values">And 590 other failures.</a>) 591 </p> 592 </td> 593<td> 594 <p> 595 <span class="blue">Max = 175ε (Mean = 9.88ε)</span><br> <br> 596 (<span class="emphasis"><em><cmath>:</em></span> Max = 175ε (Mean = 9.36ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_legendre_p_associated___cmath__Associated_Legendre_Polynomials_Small_Values">And 597 other failures.</a>) 598 </p> 599 </td> 600<td> 601 <p> 602 <span class="blue">Max = 77.7ε (Mean = 5.59ε)</span> 603 </p> 604 </td> 605<td> 606 <p> 607 <span class="blue">Max = 121ε (Mean = 7.14ε)</span> 608 </p> 609 </td> 610</tr></tbody> 611</table></div> 612</div> 613<br class="table-break"><p> 614 Note that the worst errors occur when the order increases, values greater 615 than ~120 are very unlikely to produce sensible results, especially in the 616 associated polynomial case when the degree is also large. Further the relative 617 errors are likely to grow arbitrarily large when the function is very close 618 to a root. 619 </p> 620<h5> 621<a name="math_toolkit.sf_poly.legendre.h3"></a> 622 <span class="phrase"><a name="math_toolkit.sf_poly.legendre.testing"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.testing">Testing</a> 623 </h5> 624<p> 625 A mixture of spot tests of values calculated using functions.wolfram.com, 626 and randomly generated test data are used: the test data was computed using 627 <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> at 1000-bit 628 precision. 629 </p> 630<h5> 631<a name="math_toolkit.sf_poly.legendre.h4"></a> 632 <span class="phrase"><a name="math_toolkit.sf_poly.legendre.implementation"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.implementation">Implementation</a> 633 </h5> 634<p> 635 These functions are implemented using the stable three term recurrence relations. 636 These relations guarantee low absolute error but cannot guarantee low relative 637 error near one of the roots of the polynomials. 638 </p> 639</div> 640<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 641<td align="left"></td> 642<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 643 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 644 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 645 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 646 Daryle Walker and Xiaogang Zhang<p> 647 Distributed under the Boost Software License, Version 1.0. (See accompanying 648 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>) 649 </p> 650</div></td> 651</tr></table> 652<hr> 653<div class="spirit-nav"> 654<a accesskey="p" href="../sf_poly.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="legendre_stieltjes.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 655</div> 656</body> 657</html> 658
