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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"><</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> | |
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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
42 | ||
43 | <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> | |
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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</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 15. 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 15. 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"><</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> | |
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"><</span><span class="keyword">double</span><span class="special">></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"><</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 6.37. 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 = 4.46ε (Mean = 1.41ε)</span> | |
180 | </p> | |
181 | </td> | |
182 | <td> | |
183 | <p> | |
184 | <span class="blue">Max = 0ε (Mean = 0ε)</span> | |
185 | </p> | |
186 | </td> | |
187 | <td> | |
188 | <p> | |
189 | <span class="blue">Max = 6.24ε (Mean = 2.07ε)</span> | |
190 | </p> | |
191 | </td> | |
192 | <td> | |
193 | <p> | |
194 | <span class="blue">Max = 6.24ε (Mean = 2.07ε)</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 © 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 Råde, 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> | |
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