1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Laguerre (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="legendre_stieltjes.html" title="Legendre-Stieltjes Polynomials"> 10<link rel="next" href="hermite.html" title="Hermite 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_stieltjes.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="hermite.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.laguerre"></a><a class="link" href="laguerre.html" title="Laguerre (and Associated) Polynomials">Laguerre (and Associated) 28 Polynomials</a> 29</h3></div></div></div> 30<h5> 31<a name="math_toolkit.sf_poly.laguerre.h0"></a> 32 <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.synopsis"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.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">laguerre</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">laguerre</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> 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">laguerre</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> 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">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</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">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</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> 49 50<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> 51<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">laguerre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</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">Ln</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Lnm1</span><span class="special">);</span> 52 53<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> 54<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">laguerre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</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">Ln</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Lnm1</span><span class="special">);</span> 55 56 57<span class="special">}}</span> <span class="comment">// namespaces</span> 58</pre> 59<h5> 60<a name="math_toolkit.sf_poly.laguerre.h1"></a> 61 <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.description"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.description">Description</a> 62 </h5> 63<p> 64 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 65 type calculation rules</em></span></a>: note than when there is a single 66 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 67 type. 68 </p> 69<p> 70 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 71 be used to control the behaviour of the function: how it handles errors, 72 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 73 documentation for more details</a>. 74 </p> 75<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> 76<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">laguerre</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> 77 78<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> 79<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">laguerre</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> 80</pre> 81<p> 82 Returns the value of the Laguerre Polynomial of order <span class="emphasis"><em>n</em></span> 83 at point <span class="emphasis"><em>x</em></span>: 84 </p> 85<div class="blockquote"><blockquote class="blockquote"><p> 86 <span class="inlinemediaobject"><img src="../../../equations/laguerre_0.svg"></span> 87 88 </p></blockquote></div> 89<p> 90 The following graph illustrates the behaviour of the first few Laguerre Polynomials: 91 </p> 92<div class="blockquote"><blockquote class="blockquote"><p> 93 <span class="inlinemediaobject"><img src="../../../graphs/laguerre.svg" align="middle"></span> 94 95 </p></blockquote></div> 96<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> 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">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> 98 99<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> 100<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">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</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> 101</pre> 102<p> 103 Returns the Associated Laguerre polynomial of degree <span class="emphasis"><em>n</em></span> 104 and order <span class="emphasis"><em>m</em></span> at point <span class="emphasis"><em>x</em></span>: 105 </p> 106<div class="blockquote"><blockquote class="blockquote"><p> 107 <span class="inlinemediaobject"><img src="../../../equations/laguerre_1.svg"></span> 108 109 </p></blockquote></div> 110<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> 111<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">laguerre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</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">Ln</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Lnm1</span><span class="special">);</span> 112</pre> 113<p> 114 Implements the three term recurrence relation for the Laguerre polynomials, 115 this function can be used to create a sequence of values evaluated at the 116 same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>n</em></span>. 117 </p> 118<div class="blockquote"><blockquote class="blockquote"><p> 119 <span class="inlinemediaobject"><img src="../../../equations/laguerre_2.svg"></span> 120 121 </p></blockquote></div> 122<p> 123 For example we could produce a vector of the first 10 polynomial values using: 124 </p> 125<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> 126<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> 127<span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="identifier">x</span><span class="special">)).</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span> 128<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> 129 <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre_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> 130</pre> 131<p> 132 Formally the arguments are: 133 </p> 134<div class="variablelist"> 135<p class="title"><b></b></p> 136<dl class="variablelist"> 137<dt><span class="term">n</span></dt> 138<dd><p> 139 The degree <span class="emphasis"><em>n</em></span> of the last polynomial calculated. 140 </p></dd> 141<dt><span class="term">x</span></dt> 142<dd><p> 143 The abscissa value 144 </p></dd> 145<dt><span class="term">Ln</span></dt> 146<dd><p> 147 The value of the polynomial evaluated at degree <span class="emphasis"><em>n</em></span>. 148 </p></dd> 149<dt><span class="term">Lnm1</span></dt> 150<dd><p> 151 The value of the polynomial evaluated at degree <span class="emphasis"><em>n-1</em></span>. 152 </p></dd> 153</dl> 154</div> 155<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> 156<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">laguerre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</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">Ln</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Lnm1</span><span class="special">);</span> 157</pre> 158<p> 159 Implements the three term recurrence relation for the Associated Laguerre 160 polynomials, this function can be used to create a sequence of values evaluated 161 at the same <span class="emphasis"><em>x</em></span>, and for rising degree <span class="emphasis"><em>n</em></span>. 162 </p> 163<div class="blockquote"><blockquote class="blockquote"><p> 164 <span class="inlinemediaobject"><img src="../../../equations/laguerre_3.svg"></span> 165 166 </p></blockquote></div> 167<p> 168 For example we could produce a vector of the first 10 polynomial values using: 169 </p> 170<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> 171<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> 172<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> 173<span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre</span><span class="special">(</span><span class="number">0</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">push_back</span><span class="special">(</span><span class="identifier">laguerre</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">x</span><span class="special">));</span> 174<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> 175 <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre_next</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> <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> 176</pre> 177<p> 178 Formally the arguments are: 179 </p> 180<div class="variablelist"> 181<p class="title"><b></b></p> 182<dl class="variablelist"> 183<dt><span class="term">n</span></dt> 184<dd><p> 185 The degree of the last polynomial calculated. 186 </p></dd> 187<dt><span class="term">m</span></dt> 188<dd><p> 189 The order of the Associated Polynomial. 190 </p></dd> 191<dt><span class="term">x</span></dt> 192<dd><p> 193 The abscissa value. 194 </p></dd> 195<dt><span class="term">Ln</span></dt> 196<dd><p> 197 The value of the polynomial evaluated at degree <span class="emphasis"><em>n</em></span>. 198 </p></dd> 199<dt><span class="term">Lnm1</span></dt> 200<dd><p> 201 The value of the polynomial evaluated at degree <span class="emphasis"><em>n-1</em></span>. 202 </p></dd> 203</dl> 204</div> 205<h5> 206<a name="math_toolkit.sf_poly.laguerre.h2"></a> 207 <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.accuracy"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.accuracy">Accuracy</a> 208 </h5> 209<p> 210 The following table shows peak errors (in units of epsilon) for various domains 211 of input arguments. Note that only results for the widest floating point 212 type on the system are given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively 213 zero error</a>. 214 </p> 215<div class="table"> 216<a name="math_toolkit.sf_poly.laguerre.table_laguerre_n_x_"></a><p class="title"><b>Table 8.35. Error rates for laguerre(n, x)</b></p> 217<div class="table-contents"><table class="table" summary="Error rates for laguerre(n, x)"> 218<colgroup> 219<col> 220<col> 221<col> 222<col> 223<col> 224</colgroup> 225<thead><tr> 226<th> 227 </th> 228<th> 229 <p> 230 GNU C++ version 7.1.0<br> linux<br> double 231 </p> 232 </th> 233<th> 234 <p> 235 GNU C++ version 7.1.0<br> linux<br> long double 236 </p> 237 </th> 238<th> 239 <p> 240 Sun compiler version 0x5150<br> Sun Solaris<br> long double 241 </p> 242 </th> 243<th> 244 <p> 245 Microsoft Visual C++ version 14.1<br> Win32<br> double 246 </p> 247 </th> 248</tr></thead> 249<tbody><tr> 250<td> 251 <p> 252 Laguerre Polynomials 253 </p> 254 </td> 255<td> 256 <p> 257 <span class="blue">Max = 6.82ε (Mean = 0.408ε)</span><br> <br> 258 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 3.1e+03ε (Mean = 185ε)) 259 </p> 260 </td> 261<td> 262 <p> 263 <span class="blue">Max = 1.39e+04ε (Mean = 828ε)</span><br> 264 <br> (<span class="emphasis"><em><cmath>:</em></span> Max = 4.2e+03ε (Mean 265 = 251ε)) 266 </p> 267 </td> 268<td> 269 <p> 270 <span class="blue">Max = 1.39e+04ε (Mean = 828ε)</span> 271 </p> 272 </td> 273<td> 274 <p> 275 <span class="blue">Max = 3.1e+03ε (Mean = 185ε)</span> 276 </p> 277 </td> 278</tr></tbody> 279</table></div> 280</div> 281<br class="table-break"><div class="table"> 282<a name="math_toolkit.sf_poly.laguerre.table_laguerre_n_m_x_"></a><p class="title"><b>Table 8.36. Error rates for laguerre(n, m, x)</b></p> 283<div class="table-contents"><table class="table" summary="Error rates for laguerre(n, m, x)"> 284<colgroup> 285<col> 286<col> 287<col> 288<col> 289<col> 290</colgroup> 291<thead><tr> 292<th> 293 </th> 294<th> 295 <p> 296 GNU C++ version 7.1.0<br> linux<br> double 297 </p> 298 </th> 299<th> 300 <p> 301 GNU C++ version 7.1.0<br> linux<br> long double 302 </p> 303 </th> 304<th> 305 <p> 306 Sun compiler version 0x5150<br> Sun Solaris<br> long double 307 </p> 308 </th> 309<th> 310 <p> 311 Microsoft Visual C++ version 14.1<br> Win32<br> double 312 </p> 313 </th> 314</tr></thead> 315<tbody><tr> 316<td> 317 <p> 318 Associated Laguerre Polynomials 319 </p> 320 </td> 321<td> 322 <p> 323 <span class="blue">Max = 0.84ε (Mean = 0.0358ε)</span><br> <br> 324 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 434ε (Mean = 10.7ε)) 325 </p> 326 </td> 327<td> 328 <p> 329 <span class="blue">Max = 167ε (Mean = 6.38ε)</span><br> <br> 330 (<span class="emphasis"><em><cmath>:</em></span> Max = 206ε (Mean = 6.86ε)) 331 </p> 332 </td> 333<td> 334 <p> 335 <span class="blue">Max = 167ε (Mean = 6.38ε)</span> 336 </p> 337 </td> 338<td> 339 <p> 340 <span class="blue">Max = 434ε (Mean = 11.1ε)</span> 341 </p> 342 </td> 343</tr></tbody> 344</table></div> 345</div> 346<br class="table-break"><p> 347 Note that the worst errors occur when the degree increases, values greater 348 than ~120 are very unlikely to produce sensible results, especially in the 349 associated polynomial case when the order is also large. Further the relative 350 errors are likely to grow arbitrarily large when the function is very close 351 to a root. 352 </p> 353<h5> 354<a name="math_toolkit.sf_poly.laguerre.h3"></a> 355 <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.testing"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.testing">Testing</a> 356 </h5> 357<p> 358 A mixture of spot tests of values calculated using functions.wolfram.com, 359 and randomly generated test data are used: the test data was computed using 360 <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> at 1000-bit 361 precision. 362 </p> 363<h5> 364<a name="math_toolkit.sf_poly.laguerre.h4"></a> 365 <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.implementation"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.implementation">Implementation</a> 366 </h5> 367<p> 368 These functions are implemented using the stable three term recurrence relations. 369 These relations guarantee low absolute error but cannot guarantee low relative 370 error near one of the roots of the polynomials. 371 </p> 372</div> 373<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 374<td align="left"></td> 375<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 376 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 377 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 378 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 379 Daryle Walker and Xiaogang Zhang<p> 380 Distributed under the Boost Software License, Version 1.0. (See accompanying 381 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>) 382 </p> 383</div></td> 384</tr></table> 385<hr> 386<div class="spirit-nav"> 387<a accesskey="p" href="legendre_stieltjes.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="hermite.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 388</div> 389</body> 390</html> 391
