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- <div class="titlepage"><div><div><h3 class="title">
- <a name="math_toolkit.sf_poly.hermite"></a><a class="link" href="hermite.html" title="Hermite Polynomials">Hermite Polynomials</a>
- </h3></div></div></div>
- <h5>
- <a name="math_toolkit.sf_poly.hermite.h0"></a>
- <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>
- </h5>
- <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">hermite</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
- </pre>
- <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>
- <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
- <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">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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
- <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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
- <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>
- <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>
- <span class="special">}}</span> <span class="comment">// namespaces</span>
- </pre>
- <h5>
- <a name="math_toolkit.sf_poly.hermite.h1"></a>
- <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>
- </h5>
- <p>
- 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
- type calculation rules</em></span></a>: note than when there is a single
- 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
- type.
- </p>
- <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>
- <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">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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
- <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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
- </pre>
- <p>
- Returns the value of the Hermite Polynomial of order <span class="emphasis"><em>n</em></span>
- at point <span class="emphasis"><em>x</em></span>:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/hermite_0.svg"></span>
- </p></blockquote></div>
- <p>
- The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
- be used to control the behaviour of the function: how it handles errors,
- what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy
- documentation for more details</a>.
- </p>
- <p>
- The following graph illustrates the behaviour of the first few Hermite Polynomials:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../graphs/hermite.svg" align="middle"></span>
- </p></blockquote></div>
- <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>
- <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>
- </pre>
- <p>
- Implements the three term recurrence relation for the Hermite polynomials,
- this function can be used to create a sequence of values evaluated at the
- same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>n</em></span>.
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/hermite_1.svg"></span>
- </p></blockquote></div>
- <p>
- For example we could produce a vector of the first 10 polynomial values using:
- </p>
- <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>
- <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>
- <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>
- <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>
- <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>
- </pre>
- <p>
- Formally the arguments are:
- </p>
- <div class="variablelist">
- <p class="title"><b></b></p>
- <dl class="variablelist">
- <dt><span class="term">n</span></dt>
- <dd><p>
- The degree <span class="emphasis"><em>n</em></span> of the last polynomial calculated.
- </p></dd>
- <dt><span class="term">x</span></dt>
- <dd><p>
- The abscissa value
- </p></dd>
- <dt><span class="term">Hn</span></dt>
- <dd><p>
- The value of the polynomial evaluated at degree <span class="emphasis"><em>n</em></span>.
- </p></dd>
- <dt><span class="term">Hnm1</span></dt>
- <dd><p>
- The value of the polynomial evaluated at degree <span class="emphasis"><em>n-1</em></span>.
- </p></dd>
- </dl>
- </div>
- <h5>
- <a name="math_toolkit.sf_poly.hermite.h2"></a>
- <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>
- </h5>
- <p>
- The following table shows peak errors (in units of epsilon) for various domains
- of input arguments. Note that only results for the widest floating point
- type on the system are given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
- zero error</a>.
- </p>
- <div class="table">
- <a name="math_toolkit.sf_poly.hermite.table_hermite"></a><p class="title"><b>Table 8.37. Error rates for hermite</b></p>
- <div class="table-contents"><table class="table" summary="Error rates for hermite">
- <colgroup>
- <col>
- <col>
- <col>
- <col>
- <col>
- </colgroup>
- <thead><tr>
- <th>
- </th>
- <th>
- <p>
- GNU C++ version 7.1.0<br> linux<br> double
- </p>
- </th>
- <th>
- <p>
- GNU C++ version 7.1.0<br> linux<br> long double
- </p>
- </th>
- <th>
- <p>
- Sun compiler version 0x5150<br> Sun Solaris<br> long double
- </p>
- </th>
- <th>
- <p>
- Microsoft Visual C++ version 14.1<br> Win32<br> double
- </p>
- </th>
- </tr></thead>
- <tbody><tr>
- <td>
- <p>
- Hermite Polynomials
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 6.24ε (Mean = 2.07ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 6.24ε (Mean = 2.07ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 4.46ε (Mean = 1.41ε)</span>
- </p>
- </td>
- </tr></tbody>
- </table></div>
- </div>
- <br class="table-break"><p>
- Note that the worst errors occur when the degree increases, values greater
- than ~120 are very unlikely to produce sensible results, especially in the
- associated polynomial case when the order is also large. Further the relative
- errors are likely to grow arbitrarily large when the function is very close
- to a root.
- </p>
- <h5>
- <a name="math_toolkit.sf_poly.hermite.h3"></a>
- <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>
- </h5>
- <p>
- A mixture of spot tests of values calculated using functions.wolfram.com,
- and randomly generated test data are used: the test data was computed using
- <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> at 1000-bit
- precision.
- </p>
- <h5>
- <a name="math_toolkit.sf_poly.hermite.h4"></a>
- <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>
- </h5>
- <p>
- These functions are implemented using the stable three term recurrence relations.
- These relations guarantee low absolute error but cannot guarantee low relative
- error near one of the roots of the polynomials.
- </p>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
- Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
- Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
- Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
- Daryle Walker and Xiaogang Zhang<p>
- Distributed under the Boost Software License, Version 1.0. (See accompanying
- 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>)
- </p>
- </div></td>
- </tr></table>
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