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| 4 | <title>Spherical Bessel Functions of the First and Second Kinds</title> | |
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| 22 | <div class="spirit-nav"> | |
| 23 | <a accesskey="p" href="mbessel.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../bessel.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="bessel_derivatives.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.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> | |
| 280 | </div></td> | |
| 281 | </tr></table> | |
| 282 | <hr> | |
| 283 | <div class="spirit-nav"> | |
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