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Subject: [Boost-commit] svn:boost r82764 - in trunk: boost/math/special_functions libs/math/doc/sf_and_dist libs/math/doc/sf_and_dist/html/math_toolkit/special libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel libs/math/test
From: john_at_[hidden]
Date: 2013-02-06 20:26:33

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Author: johnmaddock
Date: 2013-02-06 13:53:06 EST (Wed, 06 Feb 2013)
New Revision: 82764
URL: http://svn.boost.org/trac/boost/changeset/82764

Log:
Reordered parameters to Bessel and Airy root finders.
Update to docs with references.
Added:
   trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel0.html (contents, props changed)
Text files modified:
   trunk/boost/math/special_functions/airy.hpp | 8 ++--
   trunk/boost/math/special_functions/bessel.hpp | 8 ++--
   trunk/boost/math/special_functions/math_fwd.hpp | 32 +++++++++++-----------
   trunk/libs/math/doc/sf_and_dist/bessel_jy.qbk | 56 ++++++++++++++++++++++++++++++++-------
   trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel.html | 8 +++--
   trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel.html | 3 ++
   trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/mbessel.html | 3 ++
   trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/sph_bessel.html | 3 ++
   trunk/libs/math/test/test_bessel_airy_zeros.cpp | 2
   9 files changed, 85 insertions(+), 38 deletions(-)

Modified: trunk/boost/math/special_functions/airy.hpp
==============================================================================
--- trunk/boost/math/special_functions/airy.hpp (original)
+++ trunk/boost/math/special_functions/airy.hpp 2013-02-06 13:53:06 EST (Wed, 06 Feb 2013)
@@ -357,8 +357,8 @@

template <class T, class OutputIterator, class Policy>
inline OutputIterator airy_ai_zero(
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it,
 const Policy& pol)
{
@@ -375,8 +375,8 @@

template <class T, class OutputIterator>
inline OutputIterator airy_ai_zero(
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it)
{
 return airy_ai_zero<T>(number_of_zeros, start_index, out_it, policies::policy<>());
@@ -405,8 +405,8 @@

template <class T, class OutputIterator, class Policy>
inline OutputIterator airy_bi_zero(
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it,
 const Policy& pol)
{
@@ -423,8 +423,8 @@

template <class T, class OutputIterator>
inline OutputIterator airy_bi_zero(
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it)
{
 return airy_bi_zero<T>(number_of_zeros, start_index, out_it, policies::policy<>());

Modified: trunk/boost/math/special_functions/bessel.hpp
==============================================================================
--- trunk/boost/math/special_functions/bessel.hpp (original)
+++ trunk/boost/math/special_functions/bessel.hpp 2013-02-06 13:53:06 EST (Wed, 06 Feb 2013)
@@ -557,8 +557,8 @@

template <class T, class OutputIterator, class Policy>
inline OutputIterator cyl_bessel_j_zero(T v,
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it,
 const Policy& pol)
{
@@ -573,8 +573,8 @@

template <class T, class OutputIterator>
inline OutputIterator cyl_bessel_j_zero(T v,
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it)
{
 return cyl_bessel_j_zero(v, number_of_zeros, start_index, out_it, policies::policy<>());
@@ -599,8 +599,8 @@

template <class T, class OutputIterator, class Policy>
inline OutputIterator cyl_neumann_zero(T v,
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it,
 const Policy& pol)
{
@@ -615,8 +615,8 @@

template <class T, class OutputIterator>
inline OutputIterator cyl_neumann_zero(T v,
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it)
{
 return cyl_neumann_zero(v, number_of_zeros, start_index, out_it, policies::policy<>());

Modified: trunk/boost/math/special_functions/math_fwd.hpp
==============================================================================
--- trunk/boost/math/special_functions/math_fwd.hpp (original)
+++ trunk/boost/math/special_functions/math_fwd.hpp 2013-02-06 13:53:06 EST (Wed, 06 Feb 2013)
@@ -622,14 +622,14 @@

 template <class T, class OutputIterator>
 OutputIterator cyl_bessel_j_zero(T v,
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it);

 template <class T, class OutputIterator, class Policy>
 OutputIterator cyl_bessel_j_zero(T v,
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it,
 const Policy&);

@@ -641,14 +641,14 @@

 template <class T, class OutputIterator>
 OutputIterator cyl_neumann_zero(T v,
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it);

 template <class T, class OutputIterator, class Policy>
 OutputIterator cyl_neumann_zero(T v,
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it,
 const Policy&);

@@ -707,13 +707,13 @@

 template <class OutputIterator>
 OutputIterator airy_ai_zero(
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it);
 template <class OutputIterator, class Policy>
 OutputIterator airy_ai_zero(
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it,
 const Policy&);

@@ -724,13 +724,13 @@

 template <class OutputIterator>
 OutputIterator airy_bi_zero(
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it);
 template <class OutputIterator, class Policy>
 OutputIterator airy_bi_zero(
- unsigned number_of_zeros,
 unsigned start_index,
+ unsigned number_of_zeros,
 OutputIterator out_it,
 const Policy&);

@@ -1220,10 +1220,10 @@
\
template <class OutputIterator, class T>\
 inline void cyl_bessel_j_zero(T v,\
- unsigned number_of_zeros,\
 unsigned start_index,\
+ unsigned number_of_zeros,\
 OutputIterator out_it)\
- { boost::math::cyl_bessel_j_zero(v, number_of_zeros, start_index, out_it, Policy()); }\
+ { boost::math::cyl_bessel_j_zero(v, start_index, number_of_zeros, out_it, Policy()); }\
\
 template <class T>\
 inline typename boost::math::detail::bessel_traits<T, T, Policy >::result_type cyl_neumann_zero(T v, unsigned m)\
@@ -1231,10 +1231,10 @@
\
template <class OutputIterator, class T>\
 inline void cyl_neumann_zero(T v,\
- unsigned number_of_zeros,\
 unsigned start_index,\
+ unsigned number_of_zeros,\
 OutputIterator out_it)\
- { boost::math::cyl_neumann_zero(v, number_of_zeros, start_index, out_it, Policy()); }\
+ { boost::math::cyl_neumann_zero(v, start_index, number_of_zeros, out_it, Policy()); }\
\
 template <class T>\
 inline typename boost::math::tools::promote_args<T>::type sin_pi(T x){ return boost::math::sin_pi(x); }\
@@ -1388,15 +1388,15 @@
 inline T airy_ai_zero(unsigned m)\
 { return boost::math::airy_ai_zero<T>(m, Policy()); }\
 template <class T, class OutputIterator>\
- OutputIterator airy_ai_zero(unsigned number_of_zeros, unsigned start_index, OutputIterator out_it)\
- { return boost::math::airy_ai_zero<T>(number_of_zeros, start_index, out_it, Policy()); }\
+ OutputIterator airy_ai_zero(unsigned start_index, unsigned number_of_zeros, OutputIterator out_it)\
+ { return boost::math::airy_ai_zero<T>(start_index, number_of_zeros, out_it, Policy()); }\
 \
 template <class T>\
 inline T airy_bi_zero(unsigned m)\
 { return boost::math::airy_bi_zero<T>(m, Policy()); }\
 template <class T, class OutputIterator>\
- OutputIterator airy_bi_zero(unsigned number_of_zeros, unsigned start_index, OutputIterator out_it)\
- { return boost::math::airy_bi_zero<T>(number_of_zeros, start_index, out_it, Policy()); }\
+ OutputIterator airy_bi_zero(unsigned start_index, unsigned number_of_zeros, OutputIterator out_it)\
+ { return boost::math::airy_bi_zero<T>(start_index, number_of_zeros, out_it, Policy()); }\
 \

Modified: trunk/libs/math/doc/sf_and_dist/bessel_jy.qbk
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/bessel_jy.qbk (original)
+++ trunk/libs/math/doc/sf_and_dist/bessel_jy.qbk 2013-02-06 13:53:06 EST (Wed, 06 Feb 2013)
@@ -281,20 +281,29 @@

[h4 Synopsis]

+`#include <boost/math/special_functions/bessel.hpp>`
+
Functions for obtaining both a single zero or root of the Bessel function,
and placing multiple zeros into a container like `std::vector` by providing an output iterator.

-The signature of the single value function is:
+The signature of the single value functions are:

 template <class T>
- inline typename detail::bessel_traits<T, T, policies::policy<> >::result_type
- cyl_bessel_j_zero(T v, // Floating-point value for Jv.
- unsigned m); // start index.
+ T cyl_bessel_j_zero(T v, // Floating-point value for Jv.
+ unsigned m); // index of zero
+ template <class T>
+ T cyl_neumann_zero(T v, // Floating-point value for Jv.
+ unsigned m); // index of zero

and for multiple zeros:

 template <class T, class OutputIterator>
- inline OutputIterator cyl_bessel_j_zero(T v, // Floating-point value for Jv.
+ OutputIterator cyl_bessel_j_zero(T v, // Floating-point value for Jv.
+ unsigned start_index,
+ unsigned number_of_zeros,
+ OutputIterator out_it); //
+ template <class T, class OutputIterator>
+ OutputIterator cyl_neumann_zero(T v, // Floating-point value for Jv.
 unsigned start_index,
 unsigned number_of_zeros,
 OutputIterator out_it); //
@@ -302,18 +311,44 @@
There are also versions which allows control of the
__policy_section for error handling and precision.

+ template <class T>
+ T cyl_bessel_j_zero(T v, // Floating-point value for Jv.
+ unsigned m, // start index.
+ const Policy& pol); // Policy
+ template <class T>
+ T cyl_neumann_zero(T v, // Floating-point value for Jv.
+ unsigned m, // start index.
+ const Policy& pol); // Policy
+
+ template <class T, class OutputIterator, class Policy>
+ OutputIterator cyl_bessel_j_zero(T v, // Floating-point value for Jv.
+ unsigned start_index,
+ unsigned number_of_zeros,
+ OutputIterator out_it,
+ const Policy& pol);
 template <class T, class OutputIterator, class Policy>
- inline OutputIterator cyl_bessel_j_zero(T v, // Floating-point value for Jv.
+ OutputIterator cyl_neumann_zero(T v, // Floating-point value for Jv.
 unsigned start_index,
 unsigned number_of_zeros,
 OutputIterator out_it,
 const Policy& pol);

[h4 Description]
+
The zeros or roots (values of x where the function crosses the y = 0 axis)
-of the Bessel and Neumann functions are computed by two functions.
+of the Bessel and Neumann functions are computed by two functions,
+`cyl_bessel_j_zero` and `cyl_neumann_zero`.

+In each case the index of the zero
+returned is 1-based, which is to say:

+ cyl_bessel_j_zero(v, 1);
+
+returns the first zero of Bessel J, and
+
+ cyl_bessel_j_zero(v, 0);
+
+results in a __domain_error being raised.

[h4 Examples of finding Bessel and Neumann zeros]

@@ -324,17 +359,18 @@
[bessel_zero_example_2]

[bessel_zero_example_iterator_1]
+
[bessel_zero_example_iterator_2]

The full code for this example for Bessel `cyl_bessel_j_zeros` is at
-[../../example/bessel_zeros_example.cpp Bessel, Neumann and Airy zeros].
+[@../../../example/bessel_zeros_example.cpp Bessel, Neumann and Airy zeros].

[h4 Implementation and testing]

The method uses Newton-Raphson method starting with McMahon's approximation.
-TODO needa good reference here
-
+See "Handbook of Mathematical Functions", M. Abramowitz and I. A. Stegan, section 9.5.
+Also: NIST Digital Library of Mathematical Functions, [@http://dlmf.nist.gov/10.21#vi Bessel Function Zeros].

The precision of evaluation of zeros was tested at 50 decimal digits using `cpp_dec_float_50`
and found identical with spot values computed by __WolframAlpha.

Modified: trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel.html
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel.html (original)
+++ trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel.html 2013-02-06 13:53:06 EST (Wed, 06 Feb 2013)
@@ -31,6 +31,8 @@
 Overview</a></dt>
<dt><a href="bessel/bessel.html">Bessel Functions
 of the First and Second Kinds</a></dt>
+<dt><a href="bessel/bessel0.html">Finding Zeros
+ of Bessel Functions of the First and Second Kinds</a></dt>
<dt><a href="bessel/mbessel.html">Modified Bessel
 Functions of the First and Second Kinds</a></dt>
<dt><a href="bessel/sph_bessel.html">Spherical Bessel
@@ -39,9 +41,9 @@
</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 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang
- Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs van den Berg and Benjamin
- Sobotta
+<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow,
+ Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs
+ van den Berg, Benjamin Sobotta and Christopher Kormanyos
 Distributed under the Boost Software License, Version 1.0. (See accompanying
 file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)

Modified: trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel.html
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel.html (original)
+++ trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel.html 2013-02-06 13:53:06 EST (Wed, 06 Feb 2013)
@@ -31,6 +31,9 @@
<a name="math_toolkit.special.bessel.bessel.h0"></a>
<a name="math_toolkit.special.bessel.bessel.synopsis"></a><a class="link" href="bessel.html#math_toolkit.special.bessel.bessel.synopsis">Synopsis</a>
</h5>
+
+ <code class="computeroutput">#include <boost/math/special_functions/bessel.hpp></code>
+ 
<pre class="programlisting">template <class T1, class T2>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result">calculated-result-type</a> cyl_bessel_j(T1 v, T2 x);

Added: trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel0.html
==============================================================================
--- (empty file)
+++ trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel0.html 2013-02-06 13:53:06 EST (Wed, 06 Feb 2013)
@@ -0,0 +1,168 @@
+<html>
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
+<title>Finding Zeros of Bessel Functions of the First and Second Kinds</title>
+<link rel="stylesheet" href="../../../../../../../../doc/src/boostbook.css" type="text/css">
+<meta name="generator" content="DocBook XSL Stylesheets V1.77.1">
+<link rel="home" href="../../../index.html" title="Math Toolkit">
+<link rel="up" href="../bessel.html" title="Bessel Functions">
+<link rel="prev" href="bessel.html" title="Bessel Functions of the First and Second Kinds">
+<link rel="next" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">
+</head>
+<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
+<table cellpadding="2" width="100%"><tr>
+<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../boost.png"></td>
+<td align="center">Home</td>
+<td align="center">Libraries</td>
+<td align="center">People</td>
+<td align="center">FAQ</td>
+<td align="center">More</td>
+</tr></table>
+<hr>
+<div class="spirit-nav">
+<a accesskey="p" href="bessel.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="mbessel.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a>
+</div>
+<div class="section math_toolkit_special_bessel_bessel0">
+<div class="titlepage"><div><div><h4 class="title">
+<a name="math_toolkit.special.bessel.bessel0"></a><a class="link" href="bessel0.html" title="Finding Zeros of Bessel Functions of the First and Second Kinds">Finding Zeros
+ of Bessel Functions of the First and Second Kinds</a>
+</h4></div></div></div>
+<h5>
+<a name="math_toolkit.special.bessel.bessel.h5"></a>
+ <a name="math_toolkit.special.bessel.bessel.synopsis0"></a><a class="link" href="bessel0.html#math_toolkit.special.bessel.bessel.synopsis0">Synopsis</a>
+ </h5>
+
+ <code class="computeroutput">#include <boost/math/special_functions/bessel.hpp></code>
+ 
+
+ Functions for obtaining both a single zero or root of the Bessel function,
+ and placing multiple zeros into a container like <code class="computeroutput">std::vector</code>
+ by providing an output iterator.
+ 
+
+ The signature of the single value functions are:
+ 
+<pre class="programlisting">template <class T>
+T cyl_bessel_j_zero(T v, // Floating-point value for Jv.
+ unsigned m); // index of zero
+template <class T>
+T cyl_neumann_zero(T v, // Floating-point value for Jv.
+ unsigned m); // index of zero
+</pre>
+
+ and for multiple zeros:
+ 
+<pre class="programlisting">template <class T, class OutputIterator>
+OutputIterator cyl_bessel_j_zero(T v, // Floating-point value for Jv.
+ unsigned start_index,
+ unsigned number_of_zeros,
+ OutputIterator out_it); //
+template <class T, class OutputIterator>
+OutputIterator cyl_neumann_zero(T v, // Floating-point value for Jv.
+ unsigned start_index,
+ unsigned number_of_zeros,
+ OutputIterator out_it); //
+</pre>
+
+ There are also versions which allows control of the <a class="link" href="../../policy.html" title="Policies">Policies</a>
+ for error handling and precision.
+ 
+<pre class="programlisting">template <class T>
+T cyl_bessel_j_zero(T v, // Floating-point value for Jv.
+ unsigned m, // start index.
+ const Policy& pol); // Policy
+template <class T>
+T cyl_neumann_zero(T v, // Floating-point value for Jv.
+ unsigned m, // start index.
+ const Policy& pol); // Policy
+
+template <class T, class OutputIterator, class Policy>
+OutputIterator cyl_bessel_j_zero(T v, // Floating-point value for Jv.
+ unsigned start_index,
+ unsigned number_of_zeros,
+ OutputIterator out_it,
+ const Policy& pol);
+template <class T, class OutputIterator, class Policy>
+OutputIterator cyl_neumann_zero(T v, // Floating-point value for Jv.
+ unsigned start_index,
+ unsigned number_of_zeros,
+ OutputIterator out_it,
+ const Policy& pol);
+</pre>
+<h5>
+<a name="math_toolkit.special.bessel.bessel.h6"></a>
+ <a name="math_toolkit.special.bessel.bessel.description0"></a><a class="link" href="bessel0.html#math_toolkit.special.bessel.bessel.description0">Description</a>
+ </h5>
+
+ The zeros or roots (values of x where the function crosses the y = 0 axis)
+ of the Bessel and Neumann functions are computed by two functions, <code class="computeroutput">cyl_bessel_j_zero</code> and <code class="computeroutput">cyl_neumann_zero</code>.
+ 
+
+ In each case the index of the zero returned is 1-based, which is to say:
+ 
+<pre class="programlisting">cyl_bessel_j_zero(v, 1);
+</pre>
+
+ returns the first zero of Bessel J, and
+ 
+<pre class="programlisting">cyl_bessel_j_zero(v, 0);
+</pre>
+
+ results in a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> being raised.
+ 
+<h5>
+<a name="math_toolkit.special.bessel.bessel.h7"></a>
+ <a name="math_toolkit.special.bessel.bessel.examples_of_finding_bessel_and_neumann_zeros"></a><a class="link" href="bessel0.html#math_toolkit.special.bessel.bessel.examples_of_finding_bessel_and_neumann_zeros">Examples
+ of finding Bessel and Neumann zeros</a>
+ </h5>
+
+ [bessel_zero_example_1]
+ 
+
+ [bessel_zero_example_2]
+ 
+
+ [bessel_zero_example_iterator_1]
+ 
+
+ [bessel_zero_example_iterator_2]
+ 
+
+ The full code for this example for Bessel <code class="computeroutput">cyl_bessel_j_zeros</code>
+ is at <a href="../../../../../../example/bessel_zeros_example.cpp" target="_top">Bessel, Neumann
+ and Airy zeros</a>.
+ 
+<h5>
+<a name="math_toolkit.special.bessel.bessel.h8"></a>
+ <a name="math_toolkit.special.bessel.bessel.implementation_and_testing"></a><a class="link" href="bessel0.html#math_toolkit.special.bessel.bessel.implementation_and_testing">Implementation
+ and testing</a>
+ </h5>
+
+ The method uses Newton-Raphson method starting with McMahon's approximation.
+ See "Handbook of Mathematical Functions", M. Abramowitz and I.
+ A. Stegan, section 9.5. Also: NIST Digital Library of Mathematical Functions,
+ Bessel Function Zeros.
+ 
+
+ The precision of evaluation of zeros was tested at 50 decimal digits using
+ <code class="computeroutput">cpp_dec_float_50</code> and found
+ identical with spot values computed by <a href="http://www.wolframalpha.com/" target="_top">Wolfram
+ Alpha</a>.
+ 
+</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, 2013 John Maddock, Paul A. Bristow,
+ Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs
+ van den Berg, Benjamin Sobotta and Christopher Kormanyos
+ Distributed under the Boost Software License, Version 1.0. (See accompanying
+ file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+ 
+</div></td>
+</tr></table>
+<hr>
+<div class="spirit-nav">
+<a accesskey="p" href="bessel.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="mbessel.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a>
+</div>
+</body>
+</html>

Modified: trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/mbessel.html
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/mbessel.html (original)
+++ trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/mbessel.html 2013-02-06 13:53:06 EST (Wed, 06 Feb 2013)
@@ -31,6 +31,9 @@
<a name="math_toolkit.special.bessel.mbessel.h0"></a>
<a name="math_toolkit.special.bessel.mbessel.synopsis"></a><a class="link" href="mbessel.html#math_toolkit.special.bessel.mbessel.synopsis">Synopsis</a>
</h5>
+
+ <code class="computeroutput">#include <boost/math/special_functions/bessel.hpp></code>
+ 
<pre class="programlisting">template <class T1, class T2>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result">calculated-result-type</a> cyl_bessel_i(T1 v, T2 x);

Modified: trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/sph_bessel.html
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/sph_bessel.html (original)
+++ trunk/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/sph_bessel.html 2013-02-06 13:53:06 EST (Wed, 06 Feb 2013)
@@ -31,6 +31,9 @@
<a name="math_toolkit.special.bessel.sph_bessel.h0"></a>
<a name="math_toolkit.special.bessel.sph_bessel.synopsis"></a><a class="link" href="sph_bessel.html#math_toolkit.special.bessel.sph_bessel.synopsis">Synopsis</a>
</h5>
+
+ <code class="computeroutput">#include <boost/math/special_functions/bessel.hpp></code>
+ 
<pre class="programlisting">template <class T1, class T2>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result">calculated-result-type</a> sph_bessel(unsigned v, T2 x);

Modified: trunk/libs/math/test/test_bessel_airy_zeros.cpp
==============================================================================
--- trunk/libs/math/test/test_bessel_airy_zeros.cpp (original)
+++ trunk/libs/math/test/test_bessel_airy_zeros.cpp 2013-02-06 13:53:06 EST (Wed, 06 Feb 2013)
@@ -130,7 +130,7 @@

 { // Same test using the multiple zeros version.
 std::vector<RealType> zeros;
- cyl_bessel_j_zero(static_cast<RealType>(0.0), 3, 1U, std::back_inserter(zeros) );
+ cyl_bessel_j_zero(static_cast<RealType>(0.0), 1U, 3, std::back_inserter(zeros) );
 BOOST_CHECK_CLOSE_FRACTION(zeros[0], static_cast<RealType>(2.4048255576957727686216318793264546431242449091460L), tolerance);
 BOOST_CHECK_CLOSE_FRACTION(zeros[1], static_cast<RealType>(5.5200781102863106495966041128130274252218654787829L), tolerance);
 BOOST_CHECK_CLOSE_FRACTION(zeros[2], static_cast<RealType>(8.6537279129110122169541987126609466855657952312754L), tolerance);

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