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24 | </div> | |
25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="math_toolkit.bessel.sph_bessel"></a><a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">Spherical Bessel Functions | |
28 | of the First and Second Kinds</a> | |
29 | </h3></div></div></div> | |
30 | <h5> | |
31 | <a name="math_toolkit.bessel.sph_bessel.h0"></a> | |
32 | <span class="phrase"><a name="math_toolkit.bessel.sph_bessel.synopsis"></a></span><a class="link" href="sph_bessel.html#math_toolkit.bessel.sph_bessel.synopsis">Synopsis</a> | |
33 | </h5> | |
34 | <p> | |
35 | <code class="computeroutput"><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">bessel</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></code> | |
36 | </p> | |
37 | <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> | |
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">sph_bessel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</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">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> <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">sph_bessel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</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> | |
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">sph_neumann</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span> | |
45 | ||
46 | <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> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
47 | <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">sph_neumann</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</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> | |
48 | </pre> | |
49 | <h5> | |
50 | <a name="math_toolkit.bessel.sph_bessel.h1"></a> | |
51 | <span class="phrase"><a name="math_toolkit.bessel.sph_bessel.description"></a></span><a class="link" href="sph_bessel.html#math_toolkit.bessel.sph_bessel.description">Description</a> | |
52 | </h5> | |
53 | <p> | |
54 | The functions <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_bessel</a> | |
55 | and <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_neumann</a> return | |
56 | the result of the Spherical Bessel functions of the first and second kinds | |
57 | respectively: | |
58 | </p> | |
59 | <p> | |
60 | sph_bessel(v, x) = j<sub>v</sub>(x) | |
61 | </p> | |
62 | <p> | |
63 | sph_neumann(v, x) = y<sub>v</sub>(x) = n<sub>v</sub>(x) | |
64 | </p> | |
65 | <p> | |
66 | where: | |
67 | </p> | |
68 | <p> | |
69 | <span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span> | |
70 | </p> | |
71 | <p> | |
72 | 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 | |
73 | type calculation rules</em></span></a> for the single argument type T. | |
74 | </p> | |
75 | <p> | |
76 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
77 | be used to control the behaviour of the function: how it handles errors, | |
78 | 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 | |
79 | documentation for more details</a>. | |
80 | </p> | |
81 | <p> | |
82 | The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> | |
83 | whenever the result is undefined or complex: this occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span> <span class="number">0</span></code>. | |
84 | </p> | |
85 | <p> | |
86 | The j<sub>v</sub>   function is cyclic like J<sub>v</sub>   but differs in its behaviour at the origin: | |
87 | </p> | |
88 | <p> | |
89 | <span class="inlinemediaobject"><img src="../../../graphs/sph_bessel.svg" align="middle"></span> | |
90 | </p> | |
91 | <p> | |
92 | Likewise y<sub>v</sub>   is also cyclic for large x, but tends to -∞   | |
93 | for small <span class="emphasis"><em>x</em></span>: | |
94 | </p> | |
95 | <p> | |
96 | <span class="inlinemediaobject"><img src="../../../graphs/sph_neumann.svg" align="middle"></span> | |
97 | </p> | |
98 | <h5> | |
99 | <a name="math_toolkit.bessel.sph_bessel.h2"></a> | |
100 | <span class="phrase"><a name="math_toolkit.bessel.sph_bessel.testing"></a></span><a class="link" href="sph_bessel.html#math_toolkit.bessel.sph_bessel.testing">Testing</a> | |
101 | </h5> | |
102 | <p> | |
103 | There are two sets of test values: spot values calculated using <a href="http://functions.wolfram.com/" target="_top">functions.wolfram.com</a>, | |
104 | and a much larger set of tests computed using a simplified version of this | |
105 | implementation (with all the special case handling removed). | |
106 | </p> | |
107 | <h5> | |
108 | <a name="math_toolkit.bessel.sph_bessel.h3"></a> | |
109 | <span class="phrase"><a name="math_toolkit.bessel.sph_bessel.accuracy"></a></span><a class="link" href="sph_bessel.html#math_toolkit.bessel.sph_bessel.accuracy">Accuracy</a> | |
110 | </h5> | |
111 | <div class="table"> | |
112 | <a name="math_toolkit.bessel.sph_bessel.table_sph_bessel"></a><p class="title"><b>Table 6.48. Error rates for sph_bessel</b></p> | |
113 | <div class="table-contents"><table class="table" summary="Error rates for sph_bessel"> | |
114 | <colgroup> | |
115 | <col> | |
116 | <col> | |
117 | <col> | |
118 | <col> | |
119 | <col> | |
120 | </colgroup> | |
121 | <thead><tr> | |
122 | <th> | |
123 | </th> | |
124 | <th> | |
125 | <p> | |
126 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
127 | </p> | |
128 | </th> | |
129 | <th> | |
130 | <p> | |
131 | GNU C++ version 5.1.0<br> linux<br> long double | |
132 | </p> | |
133 | </th> | |
134 | <th> | |
135 | <p> | |
136 | GNU C++ version 5.1.0<br> linux<br> double | |
137 | </p> | |
138 | </th> | |
139 | <th> | |
140 | <p> | |
141 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
142 | </p> | |
143 | </th> | |
144 | </tr></thead> | |
145 | <tbody><tr> | |
146 | <td> | |
147 | <p> | |
148 | Bessel j: Random Data | |
149 | </p> | |
150 | </td> | |
151 | <td> | |
152 | <p> | |
153 | <span class="blue">Max = 245ε (Mean = 16.3ε)</span> | |
154 | </p> | |
155 | </td> | |
156 | <td> | |
157 | <p> | |
158 | <span class="blue">Max = 243ε (Mean = 13.3ε)</span><br> <br> | |
159 | (<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 1.91e+06ε (Mean = | |
160 | 1.09e+05ε)) | |
161 | </p> | |
162 | </td> | |
163 | <td> | |
164 | <p> | |
165 | <span class="blue">Max = 0.978ε (Mean = 0.0539ε)</span><br> | |
166 | <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.79e+03ε (Mean = 107ε)) | |
167 | </p> | |
168 | </td> | |
169 | <td> | |
170 | <p> | |
171 | <span class="blue">Max = 243ε (Mean = 33.7ε)</span> | |
172 | </p> | |
173 | </td> | |
174 | </tr></tbody> | |
175 | </table></div> | |
176 | </div> | |
177 | <br class="table-break"><div class="table"> | |
178 | <a name="math_toolkit.bessel.sph_bessel.table_sph_neumann"></a><p class="title"><b>Table 6.49. Error rates for sph_neumann</b></p> | |
179 | <div class="table-contents"><table class="table" summary="Error rates for sph_neumann"> | |
180 | <colgroup> | |
181 | <col> | |
182 | <col> | |
183 | <col> | |
184 | <col> | |
185 | <col> | |
186 | </colgroup> | |
187 | <thead><tr> | |
188 | <th> | |
189 | </th> | |
190 | <th> | |
191 | <p> | |
192 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
193 | </p> | |
194 | </th> | |
195 | <th> | |
196 | <p> | |
197 | GNU C++ version 5.1.0<br> linux<br> double | |
198 | </p> | |
199 | </th> | |
200 | <th> | |
201 | <p> | |
202 | GNU C++ version 5.1.0<br> linux<br> long double | |
203 | </p> | |
204 | </th> | |
205 | <th> | |
206 | <p> | |
207 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
208 | </p> | |
209 | </th> | |
210 | </tr></thead> | |
211 | <tbody><tr> | |
212 | <td> | |
213 | <p> | |
214 | y: Random Data | |
215 | </p> | |
216 | </td> | |
217 | <td> | |
218 | <p> | |
219 | <span class="blue">Max = 281ε (Mean = 31.1ε)</span> | |
220 | </p> | |
221 | </td> | |
222 | <td> | |
223 | <p> | |
224 | <span class="blue">Max = 0.995ε (Mean = 0.0665ε)</span><br> | |
225 | <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 8.5e+04ε (Mean = 5.33e+03ε)) | |
226 | </p> | |
227 | </td> | |
228 | <td> | |
229 | <p> | |
230 | <span class="blue">Max = 234ε (Mean = 19.5ε)</span><br> <br> | |
231 | (<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 1.6e+06ε (Mean = 1.4e+05ε)) | |
232 | </p> | |
233 | </td> | |
234 | <td> | |
235 | <p> | |
236 | <span class="blue">Max = 234ε (Mean = 19.8ε)</span> | |
237 | </p> | |
238 | </td> | |
239 | </tr></tbody> | |
240 | </table></div> | |
241 | </div> | |
242 | <br class="table-break"><h5> | |
243 | <a name="math_toolkit.bessel.sph_bessel.h4"></a> | |
244 | <span class="phrase"><a name="math_toolkit.bessel.sph_bessel.implementation"></a></span><a class="link" href="sph_bessel.html#math_toolkit.bessel.sph_bessel.implementation">Implementation</a> | |
245 | </h5> | |
246 | <p> | |
247 | Other than error handling and a couple of special cases these functions are | |
248 | implemented directly in terms of their definitions: | |
249 | </p> | |
250 | <p> | |
251 | <span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span> | |
252 | </p> | |
253 | <p> | |
254 | The special cases occur for: | |
255 | </p> | |
256 | <p> | |
257 | j<sub>0</sub>  = <a class="link" href="../sinc/sinc_pi.html" title="sinc_pi">sinc_pi</a>(x) = sin(x) | |
258 | / x | |
259 | </p> | |
260 | <p> | |
261 | and for small <span class="emphasis"><em>x < 1</em></span>, we can use the series: | |
262 | </p> | |
263 | <p> | |
264 | <span class="inlinemediaobject"><img src="../../../equations/sbessel5.svg"></span> | |
265 | </p> | |
266 | <p> | |
267 | which neatly avoids the problem of calculating 0/0 that can occur with the | |
268 | main definition as x → 0. | |
269 | </p> | |
270 | </div> | |
271 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
272 | <td align="left"></td> | |
273 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
274 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
275 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
276 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
277 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
278 | 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>) | |
279 | </p> | |
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