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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">&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;15.&#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;15.&#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;15.&#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;15.&#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 <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&#160;15.&#160;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&#160;15.&#160;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">&lt;</span> <span class="number">0</span></code>.
  84       </p>
  85 <p>
  86         The j<sub>v</sub> &#160; function is cyclic like J<sub>v</sub> &#160; 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> &#160; is also cyclic for large x, but tends to -&#8734; &#160;
  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&#160;6.48.&#160;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 = 245&#949; (Mean = 16.3&#949;)</span>
 154                 </p>
 155               </td>
 156 <td>
 157                 <p>
 158                   <span class="blue">Max = 243&#949; (Mean = 13.3&#949;)</span><br> <br>
 159                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.91e+06&#949; (Mean =
 160                   1.09e+05&#949;))
 161                 </p>
 162               </td>
 163 <td>
 164                 <p>
 165                   <span class="blue">Max = 0.978&#949; (Mean = 0.0539&#949;)</span><br>
 166                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.79e+03&#949; (Mean = 107&#949;))
 167                 </p>
 168               </td>
 169 <td>
 170                 <p>
 171                   <span class="blue">Max = 243&#949; (Mean = 33.7&#949;)</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&#160;6.49.&#160;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 = 281&#949; (Mean = 31.1&#949;)</span>
 220                 </p>
 221               </td>
 222 <td>
 223                 <p>
 224                   <span class="blue">Max = 0.995&#949; (Mean = 0.0665&#949;)</span><br>
 225                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 8.5e+04&#949; (Mean = 5.33e+03&#949;))
 226                 </p>
 227               </td>
 228 <td>
 229                 <p>
 230                   <span class="blue">Max = 234&#949; (Mean = 19.5&#949;)</span><br> <br>
 231                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.6e+06&#949; (Mean = 1.4e+05&#949;))
 232                 </p>
 233               </td>
 234 <td>
 235                 <p>
 236                   <span class="blue">Max = 234&#949; (Mean = 19.8&#949;)</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> &#160;= <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 &lt; 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 &#8594; 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 &#169; 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 R&#229;de, 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>
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