libs/math/doc/html/math_toolkit/sf_poly/hermite.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Hermite 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 2.11.0">
   8 <link rel="up" href="../sf_poly.html" title="Polynomials">
   9 <link rel="prev" href="laguerre.html" title="Laguerre (and Associated) Polynomials">
  10 <link rel="next" href="chebyshev.html" title="Chebyshev 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="laguerre.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="chebyshev.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.hermite"></a><a class="link" href="hermite.html" title="Hermite Polynomials">Hermite Polynomials</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.sf_poly.hermite.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.sf_poly.hermite.synopsis"></a></span><a class="link" href="hermite.html#math_toolkit.sf_poly.hermite.synopsis">Synopsis</a>
  32       </h5>
  33 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">hermite</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34 </pre>
  35 <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>
  36
  37 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  38 <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">hermite</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>
  39
  40 <span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  41 <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">hermite</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  42
  43 <span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
  44 <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">hermite_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">Hn</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Hnm1</span><span class="special">);</span>
  45
  46 <span class="special">}}</span> <span class="comment">// namespaces</span>
  47 </pre>
  48 <h5>
  49 <a name="math_toolkit.sf_poly.hermite.h1"></a>
  50         <span class="phrase"><a name="math_toolkit.sf_poly.hermite.description"></a></span><a class="link" href="hermite.html#math_toolkit.sf_poly.hermite.description">Description</a>
  51       </h5>
  52 <p>
  53         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
  54         type calculation rules</em></span></a>: note than when there is a single
  55         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
  56         type.
  57       </p>
  58 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  59 <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">hermite</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>
  60
  61 <span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  62 <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">hermite</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  63 </pre>
  64 <p>
  65         Returns the value of the Hermite Polynomial of order <span class="emphasis"><em>n</em></span>
  66         at point <span class="emphasis"><em>x</em></span>:
  67       </p>
  68 <div class="blockquote"><blockquote class="blockquote"><p>
  69           <span class="inlinemediaobject"><img src="../../../equations/hermite_0.svg"></span>
  70
  71         </p></blockquote></div>
  72 <p>
  73         The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  74         be used to control the behaviour of the function: how it handles errors,
  75         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
  76         documentation for more details</a>.
  77       </p>
  78 <p>
  79         The following graph illustrates the behaviour of the first few Hermite Polynomials:
  80       </p>
  81 <div class="blockquote"><blockquote class="blockquote"><p>
  82           <span class="inlinemediaobject"><img src="../../../graphs/hermite.svg" align="middle"></span>
  83
  84         </p></blockquote></div>
  85 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
  86 <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">hermite_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">Hn</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Hnm1</span><span class="special">);</span>
  87 </pre>
  88 <p>
  89         Implements the three term recurrence relation for the Hermite polynomials,
  90         this function can be used to create a sequence of values evaluated at the
  91         same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>n</em></span>.
  92       </p>
  93 <div class="blockquote"><blockquote class="blockquote"><p>
  94           <span class="inlinemediaobject"><img src="../../../equations/hermite_1.svg"></span>
  95
  96         </p></blockquote></div>
  97 <p>
  98         For example we could produce a vector of the first 10 polynomial values using:
  99       </p>
 100 <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>
 101 <span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">v</span><span class="special">;</span>
 102 <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">hermite</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">hermite</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
 103 <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">&lt;</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span>
 104    <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">hermite_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>
 105 </pre>
 106 <p>
 107         Formally the arguments are:
 108       </p>
 109 <div class="variablelist">
 110 <p class="title"><b></b></p>
 111 <dl class="variablelist">
 112 <dt><span class="term">n</span></dt>
 113 <dd><p>
 114               The degree <span class="emphasis"><em>n</em></span> of the last polynomial calculated.
 115             </p></dd>
 116 <dt><span class="term">x</span></dt>
 117 <dd><p>
 118               The abscissa value
 119             </p></dd>
 120 <dt><span class="term">Hn</span></dt>
 121 <dd><p>
 122               The value of the polynomial evaluated at degree <span class="emphasis"><em>n</em></span>.
 123             </p></dd>
 124 <dt><span class="term">Hnm1</span></dt>
 125 <dd><p>
 126               The value of the polynomial evaluated at degree <span class="emphasis"><em>n-1</em></span>.
 127             </p></dd>
 128 </dl>
 129 </div>
 130 <h5>
 131 <a name="math_toolkit.sf_poly.hermite.h2"></a>
 132         <span class="phrase"><a name="math_toolkit.sf_poly.hermite.accuracy"></a></span><a class="link" href="hermite.html#math_toolkit.sf_poly.hermite.accuracy">Accuracy</a>
 133       </h5>
 134 <p>
 135         The following table shows peak errors (in units of epsilon) for various domains
 136         of input arguments. Note that only results for the widest floating point
 137         type on the system are given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 138         zero error</a>.
 139       </p>
 140 <div class="table">
 141 <a name="math_toolkit.sf_poly.hermite.table_hermite"></a><p class="title"><b>Table&#160;8.37.&#160;Error rates for hermite</b></p>
 142 <div class="table-contents"><table class="table" summary="Error rates for hermite">
 143 <colgroup>
 144 <col>
 145 <col>
 146 <col>
 147 <col>
 148 <col>
 149 </colgroup>
 150 <thead><tr>
 151 <th>
 152               </th>
 153 <th>
 154                 <p>
 155                   GNU C++ version 7.1.0<br> linux<br> double
 156                 </p>
 157               </th>
 158 <th>
 159                 <p>
 160                   GNU C++ version 7.1.0<br> linux<br> long double
 161                 </p>
 162               </th>
 163 <th>
 164                 <p>
 165                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 166                 </p>
 167               </th>
 168 <th>
 169                 <p>
 170                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 171                 </p>
 172               </th>
 173 </tr></thead>
 174 <tbody><tr>
 175 <td>
 176                 <p>
 177                   Hermite Polynomials
 178                 </p>
 179               </td>
 180 <td>
 181                 <p>
 182                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 183                 </p>
 184               </td>
 185 <td>
 186                 <p>
 187                   <span class="blue">Max = 6.24&#949; (Mean = 2.07&#949;)</span>
 188                 </p>
 189               </td>
 190 <td>
 191                 <p>
 192                   <span class="blue">Max = 6.24&#949; (Mean = 2.07&#949;)</span>
 193                 </p>
 194               </td>
 195 <td>
 196                 <p>
 197                   <span class="blue">Max = 4.46&#949; (Mean = 1.41&#949;)</span>
 198                 </p>
 199               </td>
 200 </tr></tbody>
 201 </table></div>
 202 </div>
 203 <br class="table-break"><p>
 204         Note that the worst errors occur when the degree increases, values greater
 205         than ~120 are very unlikely to produce sensible results, especially in the
 206         associated polynomial case when the order is also large. Further the relative
 207         errors are likely to grow arbitrarily large when the function is very close
 208         to a root.
 209       </p>
 210 <h5>
 211 <a name="math_toolkit.sf_poly.hermite.h3"></a>
 212         <span class="phrase"><a name="math_toolkit.sf_poly.hermite.testing"></a></span><a class="link" href="hermite.html#math_toolkit.sf_poly.hermite.testing">Testing</a>
 213       </h5>
 214 <p>
 215         A mixture of spot tests of values calculated using functions.wolfram.com,
 216         and randomly generated test data are used: the test data was computed using
 217         <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> at 1000-bit
 218         precision.
 219       </p>
 220 <h5>
 221 <a name="math_toolkit.sf_poly.hermite.h4"></a>
 222         <span class="phrase"><a name="math_toolkit.sf_poly.hermite.implementation"></a></span><a class="link" href="hermite.html#math_toolkit.sf_poly.hermite.implementation">Implementation</a>
 223       </h5>
 224 <p>
 225         These functions are implemented using the stable three term recurrence relations.
 226         These relations guarantee low absolute error but cannot guarantee low relative
 227         error near one of the roots of the polynomials.
 228       </p>
 229 </div>
 230 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 231 <td align="left"></td>
 232 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 233       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 234       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 235       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 236       Daryle Walker and Xiaogang Zhang<p>
 237         Distributed under the Boost Software License, Version 1.0. (See accompanying
 238         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>)
 239       </p>
 240 </div></td>
 241 </tr></table>
 242 <hr>
 243 <div class="spirit-nav">
 244 <a accesskey="p" href="laguerre.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="chebyshev.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
 245 </div>
 246 </body>
 247 </html>