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-</div>
-<div class="section">
-<div class="titlepage"><div><div><h3 class="title">
-<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
- of the First and Second Kinds</a>
-</h3></div></div></div>
-<h5>
-<a name="math_toolkit.bessel.sph_bessel.h0"></a>
- <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>
- </h5>
-<p>
- <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>
- </p>
-<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>
-<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">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>
-<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>
-
-<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>
-<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">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>
-<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>
-</pre>
-<h5>
-<a name="math_toolkit.bessel.sph_bessel.h1"></a>
- <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>
- </h5>
-<p>
- The functions <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_bessel</a>
- and <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_neumann</a> return
- the result of the Spherical Bessel functions of the first and second kinds
- respectively:
- </p>
-<p>
- sph_bessel(v, x) = j<sub>v</sub>(x)
- </p>
-<p>
- sph_neumann(v, x) = y<sub>v</sub>(x) = n<sub>v</sub>(x)
- </p>
-<p>
- where:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span>
- </p>
-<p>
- 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
- type calculation rules</em></span></a> for the single argument type T.
- </p>
-<p>
- The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
- be used to control the behaviour of the function: how it handles errors,
- 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
- documentation for more details</a>.
- </p>
-<p>
- The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
- 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>.
- </p>
-<p>
- The j<sub>v</sub>   function is cyclic like J<sub>v</sub>   but differs in its behaviour at the origin:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/sph_bessel.svg" align="middle"></span>
- </p>
-<p>
- Likewise y<sub>v</sub>   is also cyclic for large x, but tends to -∞  
-for small <span class="emphasis"><em>x</em></span>:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/sph_neumann.svg" align="middle"></span>
- </p>
-<h5>
-<a name="math_toolkit.bessel.sph_bessel.h2"></a>
- <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>
- </h5>
-<p>
- There are two sets of test values: spot values calculated using <a href="http://functions.wolfram.com/" target="_top">functions.wolfram.com</a>,
- and a much larger set of tests computed using a simplified version of this
- implementation (with all the special case handling removed).
- </p>
-<h5>
-<a name="math_toolkit.bessel.sph_bessel.h3"></a>
- <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>
- </h5>
-<div class="table">
-<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>
-<div class="table-contents"><table class="table" summary="Error rates for sph_bessel">
-<colgroup>
-<col>
-<col>
-<col>
-<col>
-<col>
-</colgroup>
-<thead><tr>
-<th>
- </th>
-<th>
- <p>
- Microsoft Visual C++ version 12.0<br> Win32<br> double
- </p>
- </th>
-<th>
- <p>
- GNU C++ version 5.1.0<br> linux<br> long double
- </p>
- </th>
-<th>
- <p>
- GNU C++ version 5.1.0<br> linux<br> double
- </p>
- </th>
-<th>
- <p>
- Sun compiler version 0x5130<br> Sun Solaris<br> long double
- </p>
- </th>
-</tr></thead>
-<tbody><tr>
-<td>
- <p>
- Bessel j: Random Data
- </p>
- </td>
-<td>
- <p>
- <span class="blue">Max = 245ε (Mean = 16.3ε)</span>
- </p>
- </td>
-<td>
- <p>
- <span class="blue">Max = 243ε (Mean = 13.3ε)</span><br> <br>
- (<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 1.91e+06ε (Mean =
- 1.09e+05ε))
- </p>
- </td>
-<td>
- <p>
- <span class="blue">Max = 0.978ε (Mean = 0.0539ε)</span><br>
- <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.79e+03ε (Mean = 107ε))
- </p>
- </td>
-<td>
- <p>
- <span class="blue">Max = 243ε (Mean = 33.7ε)</span>
- </p>
- </td>
-</tr></tbody>
-</table></div>
-</div>
-<br class="table-break"><div class="table">
-<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>
-<div class="table-contents"><table class="table" summary="Error rates for sph_neumann">
-<colgroup>
-<col>
-<col>
-<col>
-<col>
-<col>
-</colgroup>
-<thead><tr>
-<th>
- </th>
-<th>
- <p>
- Microsoft Visual C++ version 12.0<br> Win32<br> double
- </p>
- </th>
-<th>
- <p>
- GNU C++ version 5.1.0<br> linux<br> double
- </p>
- </th>
-<th>
- <p>
- GNU C++ version 5.1.0<br> linux<br> long double
- </p>
- </th>
-<th>
- <p>
- Sun compiler version 0x5130<br> Sun Solaris<br> long double
- </p>
- </th>
-</tr></thead>
-<tbody><tr>
-<td>
- <p>
- y: Random Data
- </p>
- </td>
-<td>
- <p>
- <span class="blue">Max = 281ε (Mean = 31.1ε)</span>
- </p>
- </td>
-<td>
- <p>
- <span class="blue">Max = 0.995ε (Mean = 0.0665ε)</span><br>
- <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 8.5e+04ε (Mean = 5.33e+03ε))
- </p>
- </td>
-<td>
- <p>
- <span class="blue">Max = 234ε (Mean = 19.5ε)</span><br> <br>
- (<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 1.6e+06ε (Mean = 1.4e+05ε))
- </p>
- </td>
-<td>
- <p>
- <span class="blue">Max = 234ε (Mean = 19.8ε)</span>
- </p>
- </td>
-</tr></tbody>
-</table></div>
-</div>
-<br class="table-break"><h5>
-<a name="math_toolkit.bessel.sph_bessel.h4"></a>
- <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>
- </h5>
-<p>
- Other than error handling and a couple of special cases these functions are
- implemented directly in terms of their definitions:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span>
- </p>
-<p>
- The special cases occur for:
- </p>
-<p>
- j<sub>0</sub>  = <a class="link" href="../sinc/sinc_pi.html" title="sinc_pi">sinc_pi</a>(x) = sin(x)
- / x
- </p>
-<p>
- and for small <span class="emphasis"><em>x < 1</em></span>, we can use the series:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/sbessel5.svg"></span>
- </p>
-<p>
- which neatly avoids the problem of calculating 0/0 that can occur with the
- main definition as x → 0.
- </p>
-</div>
-<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
-<td align="left"></td>
-<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
- Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
- Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
- Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
- Distributed under the Boost Software License, Version 1.0. (See accompanying
- 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>)
- </p>
-</div></td>
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