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  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">&lt;</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">&gt;</span></code>
  36       </p>
  37 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</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">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  42
  43 <span class="keyword">template</span> <span class="special">&lt;</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">&gt;</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">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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&#160;20.&#160;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&#160;20.&#160;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">&lt;</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 -&#8734;
  95 for 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&#160;8.48.&#160;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 = 243&#949; (Mean = 13.3&#949;)</span><br> <br>
 157                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.91e+06&#949; (Mean = 1.09e+05&#949;))
 158                 </p>
 159               </td>
 160 <td>
 161                 <p>
 162                   <span class="blue">Max = 0.978&#949; (Mean = 0.0445&#949;)</span><br>
 163                   <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.79e+03&#949; (Mean = 107&#949;))
 164                 </p>
 165               </td>
 166 <td>
 167                 <p>
 168                   <span class="blue">Max = 243&#949; (Mean = 33.7&#949;)</span>
 169                 </p>
 170               </td>
 171 <td>
 172                 <p>
 173                   <span class="blue">Max = 245&#949; (Mean = 16.3&#949;)</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&#160;8.49.&#160;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 = 234&#949; (Mean = 19.5&#949;)</span><br> <br>
 222                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.6e+06&#949; (Mean = 1.4e+05&#949;))
 223                 </p>
 224               </td>
 225 <td>
 226                 <p>
 227                   <span class="blue">Max = 0.995&#949; (Mean = 0.0665&#949;)</span><br>
 228                   <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 8.5e+04&#949; (Mean = 5.33e+03&#949;))
 229                 </p>
 230               </td>
 231 <td>
 232                 <p>
 233                   <span class="blue">Max = 234&#949; (Mean = 19.8&#949;)</span>
 234                 </p>
 235               </td>
 236 <td>
 237                 <p>
 238                   <span class="blue">Max = 281&#949; (Mean = 31.1&#949;)</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 &lt; 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 &#8594; 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 &#169; 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       R&#229;de, 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">
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