1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Spherical Bessel Functions of the First and Second Kinds</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../bessel.html" title="Bessel Functions"> 9<link rel="prev" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds"> 10<link rel="next" href="bessel_derivatives.html" title="Derivatives of the Bessel Functions"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 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 21. 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 21. 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 21. 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 21. 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<div class="blockquote"><blockquote class="blockquote"><p> 60 sph_bessel(v, x) = j<sub>v</sub>(x) 61 </p></blockquote></div> 62<div class="blockquote"><blockquote class="blockquote"><p> 63 sph_neumann(v, x) = y<sub>v</sub>(x) = n<sub>v</sub>(x) 64 </p></blockquote></div> 65<p> 66 where: 67 </p> 68<div class="blockquote"><blockquote class="blockquote"><p> 69 <span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span> 70 71 </p></blockquote></div> 72<p> 73 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 74 type calculation rules</em></span></a> for the single argument type T. 75 </p> 76<p> 77 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 78 be used to control the behaviour of the function: how it handles errors, 79 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 80 documentation for more details</a>. 81 </p> 82<p> 83 The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> 84 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>. 85 </p> 86<p> 87 The j<sub>v</sub> function is cyclic like J<sub>v</sub> but differs in its behaviour at the origin: 88 </p> 89<div class="blockquote"><blockquote class="blockquote"><p> 90 <span class="inlinemediaobject"><img src="../../../graphs/sph_bessel.svg" align="middle"></span> 91 92 </p></blockquote></div> 93<p> 94 Likewise y<sub>v</sub> is also cyclic for large x, but tends to -∞ 95for small <span class="emphasis"><em>x</em></span>: 96 </p> 97<div class="blockquote"><blockquote class="blockquote"><p> 98 <span class="inlinemediaobject"><img src="../../../graphs/sph_neumann.svg" align="middle"></span> 99 100 </p></blockquote></div> 101<h5> 102<a name="math_toolkit.bessel.sph_bessel.h2"></a> 103 <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> 104 </h5> 105<p> 106 There are two sets of test values: spot values calculated using <a href="http://functions.wolfram.com/" target="_top">functions.wolfram.com</a>, 107 and a much larger set of tests computed using a simplified version of this 108 implementation (with all the special case handling removed). 109 </p> 110<h5> 111<a name="math_toolkit.bessel.sph_bessel.h3"></a> 112 <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> 113 </h5> 114<div class="table"> 115<a name="math_toolkit.bessel.sph_bessel.table_sph_bessel"></a><p class="title"><b>Table 8.48. Error rates for sph_bessel</b></p> 116<div class="table-contents"><table class="table" summary="Error rates for sph_bessel"> 117<colgroup> 118<col> 119<col> 120<col> 121<col> 122<col> 123</colgroup> 124<thead><tr> 125<th> 126 </th> 127<th> 128 <p> 129 GNU C++ version 7.1.0<br> linux<br> long double 130 </p> 131 </th> 132<th> 133 <p> 134 GNU C++ version 7.1.0<br> linux<br> double 135 </p> 136 </th> 137<th> 138 <p> 139 Sun compiler version 0x5150<br> Sun Solaris<br> long double 140 </p> 141 </th> 142<th> 143 <p> 144 Microsoft Visual C++ version 14.1<br> Win32<br> double 145 </p> 146 </th> 147</tr></thead> 148<tbody><tr> 149<td> 150 <p> 151 Bessel j: Random Data 152 </p> 153 </td> 154<td> 155 <p> 156 <span class="blue">Max = 243ε (Mean = 13.3ε)</span><br> <br> 157 (<span class="emphasis"><em><cmath>:</em></span> Max = 1.91e+06ε (Mean = 1.09e+05ε)) 158 </p> 159 </td> 160<td> 161 <p> 162 <span class="blue">Max = 0.978ε (Mean = 0.0445ε)</span><br> 163 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.79e+03ε (Mean = 107ε)) 164 </p> 165 </td> 166<td> 167 <p> 168 <span class="blue">Max = 243ε (Mean = 33.7ε)</span> 169 </p> 170 </td> 171<td> 172 <p> 173 <span class="blue">Max = 245ε (Mean = 16.3ε)</span> 174 </p> 175 </td> 176</tr></tbody> 177</table></div> 178</div> 179<br class="table-break"><div class="table"> 180<a name="math_toolkit.bessel.sph_bessel.table_sph_neumann"></a><p class="title"><b>Table 8.49. Error rates for sph_neumann</b></p> 181<div class="table-contents"><table class="table" summary="Error rates for sph_neumann"> 182<colgroup> 183<col> 184<col> 185<col> 186<col> 187<col> 188</colgroup> 189<thead><tr> 190<th> 191 </th> 192<th> 193 <p> 194 GNU C++ version 7.1.0<br> linux<br> long double 195 </p> 196 </th> 197<th> 198 <p> 199 GNU C++ version 7.1.0<br> linux<br> double 200 </p> 201 </th> 202<th> 203 <p> 204 Sun compiler version 0x5150<br> Sun Solaris<br> long double 205 </p> 206 </th> 207<th> 208 <p> 209 Microsoft Visual C++ version 14.1<br> Win32<br> double 210 </p> 211 </th> 212</tr></thead> 213<tbody><tr> 214<td> 215 <p> 216 y: Random Data 217 </p> 218 </td> 219<td> 220 <p> 221 <span class="blue">Max = 234ε (Mean = 19.5ε)</span><br> <br> 222 (<span class="emphasis"><em><cmath>:</em></span> Max = 1.6e+06ε (Mean = 1.4e+05ε)) 223 </p> 224 </td> 225<td> 226 <p> 227 <span class="blue">Max = 0.995ε (Mean = 0.0665ε)</span><br> 228 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 8.5e+04ε (Mean = 5.33e+03ε)) 229 </p> 230 </td> 231<td> 232 <p> 233 <span class="blue">Max = 234ε (Mean = 19.8ε)</span> 234 </p> 235 </td> 236<td> 237 <p> 238 <span class="blue">Max = 281ε (Mean = 31.1ε)</span> 239 </p> 240 </td> 241</tr></tbody> 242</table></div> 243</div> 244<br class="table-break"><h5> 245<a name="math_toolkit.bessel.sph_bessel.h4"></a> 246 <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> 247 </h5> 248<p> 249 Other than error handling and a couple of special cases these functions are 250 implemented directly in terms of their definitions: 251 </p> 252<div class="blockquote"><blockquote class="blockquote"><p> 253 <span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span> 254 255 </p></blockquote></div> 256<p> 257 The special cases occur for: 258 </p> 259<div class="blockquote"><blockquote class="blockquote"><p> 260 j<sub>0</sub>= <a class="link" href="../sinc/sinc_pi.html" title="sinc_pi">sinc_pi</a>(x) = sin(x) 261 / x 262 </p></blockquote></div> 263<p> 264 and for small <span class="emphasis"><em>x < 1</em></span>, we can use the series: 265 </p> 266<div class="blockquote"><blockquote class="blockquote"><p> 267 <span class="inlinemediaobject"><img src="../../../equations/sbessel5.svg"></span> 268 269 </p></blockquote></div> 270<p> 271 which neatly avoids the problem of calculating 0/0 that can occur with the 272 main definition as x → 0. 273 </p> 274</div> 275<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 276<td align="left"></td> 277<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 278 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 279 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 280 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 281 Daryle Walker and Xiaogang Zhang<p> 282 Distributed under the Boost Software License, Version 1.0. (See accompanying 283 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>) 284 </p> 285</div></td> 286</tr></table> 287<hr> 288<div class="spirit-nav"> 289<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> 290</div> 291</body> 292</html> 293
