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  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;15.&#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;15.&#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;15.&#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;15.&#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 <p>
  69         <span class="inlinemediaobject"><img src="../../../equations/hermite_0.svg"></span>
  70       </p>
  71 <p>
  72         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  73         be used to control the behaviour of the function: how it handles errors,
  74         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
  75         documentation for more details</a>.
  76       </p>
  77 <p>
  78         The following graph illustrates the behaviour of the first few Hermite Polynomials:
  79       </p>
  80 <p>
  81         <span class="inlinemediaobject"><img src="../../../graphs/hermite.svg" align="middle"></span>
  82       </p>
  83 <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>
  84 <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>
  85 </pre>
  86 <p>
  87         Implements the three term recurrence relation for the Hermite polynomials,
  88         this function can be used to create a sequence of values evaluated at the
  89         same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>n</em></span>.
  90       </p>
  91 <p>
  92         <span class="inlinemediaobject"><img src="../../../equations/hermite_1.svg"></span>
  93       </p>
  94 <p>
  95         For example we could produce a vector of the first 10 polynomial values using:
  96       </p>
  97 <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>
  98 <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>
  99 <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>
 100 <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>
 101    <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>
 102 </pre>
 103 <p>
 104         Formally the arguments are:
 105       </p>
 106 <div class="variablelist">
 107 <p class="title"><b></b></p>
 108 <dl class="variablelist">
 109 <dt><span class="term">n</span></dt>
 110 <dd><p>
 111               The degree <span class="emphasis"><em>n</em></span> of the last polynomial calculated.
 112             </p></dd>
 113 <dt><span class="term">x</span></dt>
 114 <dd><p>
 115               The abscissa value
 116             </p></dd>
 117 <dt><span class="term">Hn</span></dt>
 118 <dd><p>
 119               The value of the polynomial evaluated at degree <span class="emphasis"><em>n</em></span>.
 120             </p></dd>
 121 <dt><span class="term">Hnm1</span></dt>
 122 <dd><p>
 123               The value of the polynomial evaluated at degree <span class="emphasis"><em>n-1</em></span>.
 124             </p></dd>
 125 </dl>
 126 </div>
 127 <h5>
 128 <a name="math_toolkit.sf_poly.hermite.h2"></a>
 129         <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>
 130       </h5>
 131 <p>
 132         The following table shows peak errors (in units of epsilon) for various domains
 133         of input arguments. Note that only results for the widest floating point
 134         type on the system are given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 135         zero error</a>.
 136       </p>
 137 <div class="table">
 138 <a name="math_toolkit.sf_poly.hermite.table_hermite"></a><p class="title"><b>Table&#160;6.37.&#160;Error rates for hermite</b></p>
 139 <div class="table-contents"><table class="table" summary="Error rates for hermite">
 140 <colgroup>
 141 <col>
 142 <col>
 143 <col>
 144 <col>
 145 <col>
 146 </colgroup>
 147 <thead><tr>
 148 <th>
 149               </th>
 150 <th>
 151                 <p>
 152                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 153                 </p>
 154               </th>
 155 <th>
 156                 <p>
 157                   GNU C++ version 5.1.0<br> linux<br> double
 158                 </p>
 159               </th>
 160 <th>
 161                 <p>
 162                   GNU C++ version 5.1.0<br> linux<br> long double
 163                 </p>
 164               </th>
 165 <th>
 166                 <p>
 167                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 168                 </p>
 169               </th>
 170 </tr></thead>
 171 <tbody><tr>
 172 <td>
 173                 <p>
 174                   Hermite Polynomials
 175                 </p>
 176               </td>
 177 <td>
 178                 <p>
 179                   <span class="blue">Max = 4.46&#949; (Mean = 1.41&#949;)</span>
 180                 </p>
 181               </td>
 182 <td>
 183                 <p>
 184                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 185                 </p>
 186               </td>
 187 <td>
 188                 <p>
 189                   <span class="blue">Max = 6.24&#949; (Mean = 2.07&#949;)</span>
 190                 </p>
 191               </td>
 192 <td>
 193                 <p>
 194                   <span class="blue">Max = 6.24&#949; (Mean = 2.07&#949;)</span>
 195                 </p>
 196               </td>
 197 </tr></tbody>
 198 </table></div>
 199 </div>
 200 <br class="table-break"><p>
 201         Note that the worst errors occur when the degree increases, values greater
 202         than ~120 are very unlikely to produce sensible results, especially in the
 203         associated polynomial case when the order is also large. Further the relative
 204         errors are likely to grow arbitrarily large when the function is very close
 205         to a root.
 206       </p>
 207 <h5>
 208 <a name="math_toolkit.sf_poly.hermite.h3"></a>
 209         <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>
 210       </h5>
 211 <p>
 212         A mixture of spot tests of values calculated using functions.wolfram.com,
 213         and randomly generated test data are used: the test data was computed using
 214         <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> at 1000-bit
 215         precision.
 216       </p>
 217 <h5>
 218 <a name="math_toolkit.sf_poly.hermite.h4"></a>
 219         <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>
 220       </h5>
 221 <p>
 222         These functions are implemented using the stable three term recurrence relations.
 223         These relations guarantee low absolute error but cannot guarantee low relative
 224         error near one of the roots of the polynomials.
 225       </p>
 226 </div>
 227 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 228 <td align="left"></td>
 229 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 230       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 231       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 232       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 233         Distributed under the Boost Software License, Version 1.0. (See accompanying
 234         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>)
 235       </p>
 236 </div></td>
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