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4
README.md
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@@ -12,7 +12,7 @@ Utility functions for dealing with floating point arithmetic, includes functions
### Specific Width Floating Point Types
A set of typedefs similar to those provided by <cstdint> but for floating point types.
A set of typedefs similar to those provided by `<cstdint>` but for floating point types.
### Mathematical Constants
@@ -60,7 +60,7 @@ The integration routines are usable for functions returning complex results - an
### Quaternions and Octonions
Quaternion and Octonians as class templates similar to std::complex.
Quaternion and Octonians are class templates similar to std::complex.
The full documentation is available on [boost.org](http://www.boost.org/doc/libs/release/libs/math).

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<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Chapter&#160;23.&#160;Backgrounders</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Chapter 23. Backgrounders</title>
<link rel="stylesheet" href="math.css" type="text/css">
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<link rel="prev" href="math_toolkit/perf_test_app.html" title="The Performance Test Applications">
<link rel="next" href="math_toolkit/sf_implementation.html" title="Additional Implementation Notes">
</head>
@@ -24,7 +24,7 @@
</div>
<div class="chapter">
<div class="titlepage"><div><div><h1 class="title">
<a name="backgrounders"></a>Chapter&#160;23.&#160;Backgrounders</h1></div></div></div>
<a name="backgrounders"></a>Chapter 23. Backgrounders</h1></div></div></div>
<div class="toc">
<p><b>Table of Contents</b></p>
<dl class="toc">
@@ -52,11 +52,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Chapter&#160;4.&#160;Mathematical Constants</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Chapter 4. Mathematical Constants</title>
<link rel="stylesheet" href="math.css" type="text/css">
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@@ -24,7 +24,7 @@
</div>
<div class="chapter">
<div class="titlepage"><div><div><h1 class="title">
<a name="constants"></a>Chapter&#160;4.&#160;Mathematical Constants</h1></div></div></div>
<a name="constants"></a>Chapter 4. Mathematical Constants</h1></div></div></div>
<div class="toc">
<p><b>Table of Contents</b></p>
<dl class="toc">
@@ -45,11 +45,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

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@@ -1,12 +1,12 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Chapter&#160;3.&#160;Specified-width floating-point typedefs</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Chapter 3. Specified-width floating-point typedefs</title>
<link rel="stylesheet" href="math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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@@ -20,11 +20,11 @@
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="math_toolkit/cond.html"><img src="../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="index.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="math_toolkit/specified_typedefs.html"><img src="../../../../doc/src/images/next.png" alt="Next"></a>
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</div>
<div class="chapter">
<div class="titlepage"><div><div><h1 class="title">
<a name="cstdfloat"></a>Chapter&#160;3.&#160;Specified-width floating-point typedefs</h1></div></div></div>
<a name="cstdfloat"></a>Chapter 3. Specified-width floating-point typedefs</h1></div></div></div>
<div class="toc">
<p><b>Table of Contents</b></p>
<dl class="toc">
@@ -53,11 +53,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>
@@ -65,7 +64,7 @@
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Chapter&#160;5.&#160;Statistical Distributions and Functions</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Chapter 5. Statistical Distributions and Functions</title>
<link rel="stylesheet" href="math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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@@ -24,7 +24,7 @@
</div>
<div class="chapter">
<div class="titlepage"><div><div><h1 class="title">
<a name="dist"></a>Chapter&#160;5.&#160;Statistical Distributions and Functions</h1></div></div></div>
<a name="dist"></a>Chapter 5. Statistical Distributions and Functions</h1></div></div></div>
<div class="toc">
<p><b>Table of Contents</b></p>
<dl class="toc">
@@ -208,11 +208,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Chapter&#160;9.&#160;TR1 and C99 external "C" Functions</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Chapter 9. TR1 and C99 external "C" Functions</title>
<link rel="stylesheet" href="math.css" type="text/css">
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@@ -24,7 +24,7 @@
</div>
<div class="chapter">
<div class="titlepage"><div><div><h1 class="title">
<a name="extern_c"></a>Chapter&#160;9.&#160;TR1 and C99 external "C" Functions</h1></div></div></div>
<a name="extern_c"></a>Chapter 9. TR1 and C99 external "C" Functions</h1></div></div></div>
<div class="toc">
<p><b>Table of Contents</b></p>
<dl class="toc">
@@ -36,11 +36,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

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@@ -0,0 +1,49 @@
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Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -1,13 +1,13 @@
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<title>Chapter&#160;18.&#160;Integer Utilities (Greatest Common Divisor and Least Common Multiple)</title>
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@@ -24,7 +24,7 @@
</div>
<div class="chapter">
<div class="titlepage"><div><div><h1 class="title">
<a name="gcd_lcm"></a>Chapter&#160;18.&#160;Integer Utilities (Greatest Common Divisor and Least Common Multiple)</h1></div></div></div>
<a name="gcd_lcm"></a>Chapter 18. Integer Utilities (Greatest Common Divisor and Least Common Multiple)</h1></div></div></div>
<p>
This code has now been moved to Boost.Integer, please see <a href="http://www.boost.org/doc/libs/release/libs/integer/doc/html/index.html" target="_top">here</a>.
</p>
@@ -37,11 +37,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>)
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26
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<html>
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@@ -22,7 +22,7 @@
<div class="titlepage">
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<div><h1 class="title">
<a name="math_toolkit"></a>Math Toolkit 2.12.0</h1></div>
<a name="math_toolkit"></a>Math Toolkit 2.13.0</h1></div>
<div><div class="authorgroup">
<div class="author"><h3 class="author">
<span class="firstname">Nikhar</span> <span class="surname">Agrawal</span>
@@ -49,13 +49,16 @@
<span class="firstname">John</span> <span class="surname">Maddock</span>
</h3></div>
<div class="author"><h3 class="author">
<span class="firstname">Evan</span> <span class="surname">Miller</span>
</h3></div>
<div class="author"><h3 class="author">
<span class="firstname">Jeremy W.</span> <span class="surname">Murphy</span>
</h3></div>
<div class="author"><h3 class="author">
<span class="firstname">Matthew</span> <span class="surname">Pulver</span>
</h3></div>
<div class="author"><h3 class="author">
<span class="firstname">Johan</span> <span class="surname">R&#229;de</span>
<span class="firstname">Johan</span> <span class="surname">Råde</span>
</h3></div>
<div class="author"><h3 class="author">
<span class="firstname">Benjamin</span> <span class="surname">Sobotta</span>
@@ -80,11 +83,10 @@
This manual is also available in <a href="http://sourceforge.net/projects/boost/files/boost-docs/" target="_top">printer
friendly PDF format</a>, and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0,
Classification 519.2-dc22. </p></div>
<div><p class="copyright">Copyright &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang</p></div>
<div><p class="copyright">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
Thijs van den Berg, Daryle Walker and Xiaogang Zhang</p></div>
<div><div class="legalnotice">
<a name="math_toolkit.legal"></a><p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
@@ -128,7 +130,7 @@ This manual is also available in <a href="http://sourceforge.net/projects/boost/
</div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"><p><small>Last revised: April 04, 2020 at 08:18:24 GMT</small></p></td>
<td align="left"><p><small>Last revised: October 19, 2020 at 17:12:45 GMT</small></p></td>
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19
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@@ -1,11 +1,11 @@
<html>
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<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Chapter&#160;25.&#160;Indexes</title>
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<title>Chapter 25. Indexes</title>
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@@ -24,7 +24,7 @@
</div>
<div class="chapter">
<div class="titlepage"><div><div><h1 class="title">
<a name="indexes"></a>Chapter&#160;25.&#160;Indexes</h1></div></div></div>
<a name="indexes"></a>Chapter 25. Indexes</h1></div></div></div>
<div class="toc">
<p><b>Table of Contents</b></p>
<dl class="toc">
@@ -38,11 +38,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

307
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@@ -1,12 +1,12 @@
<html>
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<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
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<link rel="prev" href="../indexes.html" title="Chapter&#160;25.&#160;Indexes">
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@@ -24,7 +24,7 @@
</div>
<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="id1942960"></a>Function Index</h2></div></div></div>
<a name="id1955624"></a>Function Index</h2></div></div></div>
<p><a class="link" href="s01.html#idx_id_0">1</a> <a class="link" href="s01.html#idx_id_1">2</a> <a class="link" href="s01.html#idx_id_2">4</a> <a class="link" href="s01.html#idx_id_5">A</a> <a class="link" href="s01.html#idx_id_6">B</a> <a class="link" href="s01.html#idx_id_7">C</a> <a class="link" href="s01.html#idx_id_8">D</a> <a class="link" href="s01.html#idx_id_9">E</a> <a class="link" href="s01.html#idx_id_10">F</a> <a class="link" href="s01.html#idx_id_11">G</a> <a class="link" href="s01.html#idx_id_12">H</a> <a class="link" href="s01.html#idx_id_13">I</a> <a class="link" href="s01.html#idx_id_14">J</a> <a class="link" href="s01.html#idx_id_15">K</a> <a class="link" href="s01.html#idx_id_16">L</a> <a class="link" href="s01.html#idx_id_17">M</a> <a class="link" href="s01.html#idx_id_18">N</a> <a class="link" href="s01.html#idx_id_19">O</a> <a class="link" href="s01.html#idx_id_20">P</a> <a class="link" href="s01.html#idx_id_21">Q</a> <a class="link" href="s01.html#idx_id_22">R</a> <a class="link" href="s01.html#idx_id_23">S</a> <a class="link" href="s01.html#idx_id_24">T</a> <a class="link" href="s01.html#idx_id_25">U</a> <a class="link" href="s01.html#idx_id_26">V</a> <a class="link" href="s01.html#idx_id_27">W</a> <a class="link" href="s01.html#idx_id_28">X</a> <a class="link" href="s01.html#idx_id_29">Y</a> <a class="link" href="s01.html#idx_id_30">Z</a></p>
<div class="variablelist"><dl class="variablelist">
<dt>
@@ -54,17 +54,18 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ellint/ellint_3.html" title="Elliptic Integrals of the Third Kind - Legendre Form"><span class="index-entry-level-1">Elliptic Integrals of the Third Kind - Legendre Form</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_root.html" title="Finding Zeros of Bessel Functions of the First and Second Kinds"><span class="index-entry-level-1">Finding Zeros of Bessel Functions of the First and Second Kinds</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/history1.html" title="History and What's New"><span class="index-entry-level-1">History and What's New</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/kolmogorov_smirnov_dist.html" title="Kolmogorov-Smirnov Distribution"><span class="index-entry-level-1">Kolmogorov-Smirnov Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/legendre_stieltjes.html" title="Legendre-Stieltjes Polynomials"><span class="index-entry-level-1">Legendre-Stieltjes Polynomials</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/constants.html#math_toolkit.constants.mathematical_constants" title="Table&#160;4.1.&#160;Mathematical Constants"><span class="index-entry-level-1">Mathematical Constants</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/constants.html#math_toolkit.constants.mathematical_constants" title="Table 4.1. Mathematical Constants"><span class="index-entry-level-1">Mathematical Constants</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/nc_beta_dist.html" title="Noncentral Beta Distribution"><span class="index-entry-level-1">Noncentral Beta Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Polynomial_Method_Comparison_with_Clang_version_9_0_0_tags_RELEASE_900_final_on_linux" title="Table&#160;22.8.&#160;Polynomial Method Comparison with Clang version 9.0.0 (tags/RELEASE_900/final) on linux"><span class="index-entry-level-1">Polynomial Method Comparison with Clang version 9.0.0 (tags/RELEASE_900/final) on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Polynomial_Method_Comparison_with_GNU_C_version_9_2_1_20191008_on_linux" title="Table&#160;22.6.&#160;Polynomial Method Comparison with GNU C++ version 9.2.1 20191008 on linux"><span class="index-entry-level-1">Polynomial Method Comparison with GNU C++ version 9.2.1 20191008 on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Polynomial_Method_Comparison_with_Intel_C_C_0x_mode_version_1910_on_linux" title="Table&#160;22.10.&#160;Polynomial Method Comparison with Intel C++ C++0x mode version 1910 on linux"><span class="index-entry-level-1">Polynomial Method Comparison with Intel C++ C++0x mode version 1910 on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Polynomial_Method_Comparison_with_Microsoft_Visual_C_version_14_2_on_Windows_x64" title="Table&#160;22.4.&#160;Polynomial Method Comparison with Microsoft Visual C++ version 14.2 on Windows x64"><span class="index-entry-level-1">Polynomial Method Comparison with Microsoft Visual C++ version 14.2 on Windows x64</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Rational_Method_Comparison_with_Clang_version_9_0_0_tags_RELEASE_900_final_on_linux" title="Table&#160;22.9.&#160;Rational Method Comparison with Clang version 9.0.0 (tags/RELEASE_900/final) on linux"><span class="index-entry-level-1">Rational Method Comparison with Clang version 9.0.0 (tags/RELEASE_900/final) on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Rational_Method_Comparison_with_GNU_C_version_9_2_1_20191008_on_linux" title="Table&#160;22.7.&#160;Rational Method Comparison with GNU C++ version 9.2.1 20191008 on linux"><span class="index-entry-level-1">Rational Method Comparison with GNU C++ version 9.2.1 20191008 on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Rational_Method_Comparison_with_Intel_C_C_0x_mode_version_1910_on_linux" title="Table&#160;22.11.&#160;Rational Method Comparison with Intel C++ C++0x mode version 1910 on linux"><span class="index-entry-level-1">Rational Method Comparison with Intel C++ C++0x mode version 1910 on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Rational_Method_Comparison_with_Microsoft_Visual_C_version_14_2_on_Windows_x64" title="Table&#160;22.5.&#160;Rational Method Comparison with Microsoft Visual C++ version 14.2 on Windows x64"><span class="index-entry-level-1">Rational Method Comparison with Microsoft Visual C++ version 14.2 on Windows x64</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Polynomial_Method_Comparison_with_Clang_version_9_0_0_tags_RELEASE_900_final_on_linux" title="Table 22.8. Polynomial Method Comparison with Clang version 9.0.0 (tags/RELEASE_900/final) on linux"><span class="index-entry-level-1">Polynomial Method Comparison with Clang version 9.0.0 (tags/RELEASE_900/final) on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Polynomial_Method_Comparison_with_GNU_C_version_9_2_1_20191008_on_linux" title="Table 22.6. Polynomial Method Comparison with GNU C++ version 9.2.1 20191008 on linux"><span class="index-entry-level-1">Polynomial Method Comparison with GNU C++ version 9.2.1 20191008 on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Polynomial_Method_Comparison_with_Intel_C_C_0x_mode_version_1910_on_linux" title="Table 22.10. Polynomial Method Comparison with Intel C++ C++0x mode version 1910 on linux"><span class="index-entry-level-1">Polynomial Method Comparison with Intel C++ C++0x mode version 1910 on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Polynomial_Method_Comparison_with_Microsoft_Visual_C_version_14_2_on_Windows_x64" title="Table 22.4. Polynomial Method Comparison with Microsoft Visual C++ version 14.2 on Windows x64"><span class="index-entry-level-1">Polynomial Method Comparison with Microsoft Visual C++ version 14.2 on Windows x64</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Rational_Method_Comparison_with_Clang_version_9_0_0_tags_RELEASE_900_final_on_linux" title="Table 22.9. Rational Method Comparison with Clang version 9.0.0 (tags/RELEASE_900/final) on linux"><span class="index-entry-level-1">Rational Method Comparison with Clang version 9.0.0 (tags/RELEASE_900/final) on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Rational_Method_Comparison_with_GNU_C_version_9_2_1_20191008_on_linux" title="Table 22.7. Rational Method Comparison with GNU C++ version 9.2.1 20191008 on linux"><span class="index-entry-level-1">Rational Method Comparison with GNU C++ version 9.2.1 20191008 on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Rational_Method_Comparison_with_Intel_C_C_0x_mode_version_1910_on_linux" title="Table 22.11. Rational Method Comparison with Intel C++ C++0x mode version 1910 on linux"><span class="index-entry-level-1">Rational Method Comparison with Intel C++ C++0x mode version 1910 on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html#math_toolkit.tuning.table_Rational_Method_Comparison_with_Microsoft_Visual_C_version_14_2_on_Windows_x64" title="Table 22.5. Rational Method Comparison with Microsoft Visual C++ version 14.2 on Windows x64"><span class="index-entry-level-1">Rational Method Comparison with Microsoft Visual C++ version 14.2 on Windows x64</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/sph_harm.html" title="Spherical Harmonics"><span class="index-entry-level-1">Spherical Harmonics</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/weibull_dist.html" title="Weibull Distribution"><span class="index-entry-level-1">Weibull Distribution</span></a></p></li>
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@@ -81,6 +82,10 @@
</dt>
<dd><div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">A</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/kolmogorov_smirnov_dist.html" title="Kolmogorov-Smirnov Distribution"><span class="index-entry-level-1">Kolmogorov-Smirnov Distribution</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">abscissa</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/gauss_kronrod.html" title="Gauss-Kronrod Quadrature"><span class="index-entry-level-1">Gauss-Kronrod Quadrature</span></a></p></li>
@@ -118,6 +123,17 @@
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<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">add_fn</span></p>
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</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">agm</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/agm.html" title="Arithmetic-Geometric Mean"><span class="index-entry-level-1">Arithmetic-Geometric Mean</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/constants.html#math_toolkit.constants.mathematical_constants" title="Table 4.1. Mathematical Constants"><span class="index-entry-level-1">Mathematical Constants</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">Ai</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/airy/airy_root.html" title="Finding Zeros of Airy Functions"><span class="index-entry-level-1">Finding Zeros of Airy Functions</span></a></p></li>
@@ -215,7 +231,7 @@
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/comparisons.html#math_toolkit.comparisons.table_Library_Comparison_with_GNU_C_version_9_2_1_20191008_on_linux" title="Table&#160;22.15.&#160;Library Comparison with GNU C++ version 9.2.1 20191008 on linux"><span class="index-entry-level-1">Library Comparison with GNU C++ version 9.2.1 20191008 on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/comparisons.html#math_toolkit.comparisons.table_Library_Comparison_with_GNU_C_version_9_2_1_20191008_on_linux" title="Table 22.15. Library Comparison with GNU C++ version 9.2.1 20191008 on linux"><span class="index-entry-level-1">Library Comparison with GNU C++ version 9.2.1 20191008 on linux</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
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</li>
@@ -294,6 +310,10 @@
</dt>
<dd><div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">background_color</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ulps_plots.html" title="ULPs Plots"><span class="index-entry-level-1">ULPs Plots</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">barycentric_rational</span></p>
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@@ -309,7 +329,7 @@
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_beta/ibeta_function.html#math_toolkit.sf_beta.ibeta_function.table_beta_incomplete_" title="Table&#160;8.20.&#160;Error rates for beta (incomplete)"><span class="index-entry-level-1">Error rates for beta (incomplete)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_beta/ibeta_function.html#math_toolkit.sf_beta.ibeta_function.table_beta_incomplete_" title="Table 8.20. Error rates for beta (incomplete)"><span class="index-entry-level-1">Error rates for beta (incomplete)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_beta/ibeta_function.html" title="Incomplete Beta Functions"><span class="index-entry-level-1">Incomplete Beta Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/nc_beta_dist.html" title="Noncentral Beta Distribution"><span class="index-entry-level-1">Noncentral Beta Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses"><span class="index-entry-level-1">The Incomplete Beta Function Inverses</span></a></p></li>
@@ -440,7 +460,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/c99.html" title="C99 C Functions"><span class="index-entry-level-1">C99 C Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><span class="bold"><strong><a class="link" href="../math_toolkit/powers/cbrt.html" title="cbrt"><span class="index-entry-level-1">cbrt</span></a></strong></span></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)"><span class="index-entry-level-1">Examples of Root-Finding (with and without derivatives)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/comparisons.html#math_toolkit.comparisons.table_Library_Comparison_with_Microsoft_Visual_C_version_14_2_on_Windows_x64" title="Table&#160;22.14.&#160;Library Comparison with Microsoft Visual C++ version 14.2 on Windows x64"><span class="index-entry-level-1">Library Comparison with Microsoft Visual C++ version 14.2 on Windows x64</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/comparisons.html#math_toolkit.comparisons.table_Library_Comparison_with_Microsoft_Visual_C_version_14_2_on_Windows_x64" title="Table 22.14. Library Comparison with Microsoft Visual C++ version 14.2 on Windows x64"><span class="index-entry-level-1">Library Comparison with Microsoft Visual C++ version 14.2 on Windows x64</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_finding_examples/multiprecision_root.html" title="Root-finding using Boost.Multiprecision"><span class="index-entry-level-1">Root-finding using Boost.Multiprecision</span></a></p></li>
</ul></div>
</li>
@@ -510,6 +530,11 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/nmp.html" title="Non-Member Properties"><span class="index-entry-level-1">Non-Member Properties</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">clip</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ulps_plots.html" title="ULPs Plots"><span class="index-entry-level-1">ULPs Plots</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/cohen_acceleration.html" title="Cohen Acceleration"><span class="index-entry-level-0">cohen_acceleration</span></a></p></li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">complement</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/st_eg/tut_mean_intervals.html" title="Calculating confidence intervals on the mean with the Students-t distribution"><span class="index-entry-level-1">Calculating confidence intervals on the mean with the Students-t distribution</span></a></p></li>
@@ -520,7 +545,7 @@
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">complex_newton</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der"><span class="index-entry-level-1">Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schröder"><span class="index-entry-level-1">Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schröder</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">complex_step_derivative</span></p>
@@ -663,7 +688,7 @@
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">convergence</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der"><span class="index-entry-level-1">Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schröder"><span class="index-entry-level-1">Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schröder</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">copysign</span></p>
@@ -700,6 +725,10 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bivariate_statistics.html" title="Bivariate Statistics"><span class="index-entry-level-1">Bivariate Statistics</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">crop_color</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ulps_plots.html" title="ULPs Plots"><span class="index-entry-level-1">ULPs Plots</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">cylindrical</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/oct_create.html" title="Octonion Creation Functions"><span class="index-entry-level-1">Octonion Creation Functions</span></a></p></li>
@@ -720,7 +749,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/logs.html" title="Error Logs For Error Rate Tables"><span class="index-entry-level-1">Error Logs For Error Rate Tables</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/mbessel.html#math_toolkit.bessel.mbessel.table_cyl_bessel_i_integer_orders_" title="Table&#160;8.44.&#160;Error rates for cyl_bessel_i (integer orders)"><span class="index-entry-level-1">Error rates for cyl_bessel_i (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/mbessel.html#math_toolkit.bessel.mbessel.table_cyl_bessel_i_integer_orders_" title="Table 8.44. Error rates for cyl_bessel_i (integer orders)"><span class="index-entry-level-1">Error rates for cyl_bessel_i (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds"><span class="index-entry-level-1">Modified Bessel Functions of the First and Second Kinds</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
</ul></div>
@@ -745,7 +774,7 @@
<p><span class="index-entry-level-0">cyl_bessel_i_prime</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html" title="Derivatives of the Bessel Functions"><span class="index-entry-level-1">Derivatives of the Bessel Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_i_prime_integer_orders_" title="Table&#160;8.50.&#160;Error rates for cyl_bessel_i_prime (integer orders)"><span class="index-entry-level-1">Error rates for cyl_bessel_i_prime (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_i_prime_integer_orders_" title="Table 8.50. Error rates for cyl_bessel_i_prime (integer orders)"><span class="index-entry-level-1">Error rates for cyl_bessel_i_prime (integer orders)</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
@@ -755,7 +784,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/logs.html" title="Error Logs For Error Rate Tables"><span class="index-entry-level-1">Error Logs For Error Rate Tables</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_first.html#math_toolkit.bessel.bessel_first.table_cyl_bessel_j_integer_orders_" title="Table&#160;8.40.&#160;Error rates for cyl_bessel_j (integer orders)"><span class="index-entry-level-1">Error rates for cyl_bessel_j (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_first.html#math_toolkit.bessel.bessel_first.table_cyl_bessel_j_integer_orders_" title="Table 8.40. Error rates for cyl_bessel_j (integer orders)"><span class="index-entry-level-1">Error rates for cyl_bessel_j (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
</ul></div>
</li>
@@ -771,7 +800,7 @@
<p><span class="index-entry-level-0">cyl_bessel_j_prime</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html" title="Derivatives of the Bessel Functions"><span class="index-entry-level-1">Derivatives of the Bessel Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_j_prime_integer_orders_" title="Table&#160;8.52.&#160;Error rates for cyl_bessel_j_prime (integer orders)"><span class="index-entry-level-1">Error rates for cyl_bessel_j_prime (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_j_prime_integer_orders_" title="Table 8.52. Error rates for cyl_bessel_j_prime (integer orders)"><span class="index-entry-level-1">Error rates for cyl_bessel_j_prime (integer orders)</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
@@ -784,7 +813,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/logs.html" title="Error Logs For Error Rate Tables"><span class="index-entry-level-1">Error Logs For Error Rate Tables</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/mbessel.html#math_toolkit.bessel.mbessel.table_cyl_bessel_k_integer_orders_" title="Table&#160;8.46.&#160;Error rates for cyl_bessel_k (integer orders)"><span class="index-entry-level-1">Error rates for cyl_bessel_k (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/mbessel.html#math_toolkit.bessel.mbessel.table_cyl_bessel_k_integer_orders_" title="Table 8.46. Error rates for cyl_bessel_k (integer orders)"><span class="index-entry-level-1">Error rates for cyl_bessel_k (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds"><span class="index-entry-level-1">Modified Bessel Functions of the First and Second Kinds</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
</ul></div>
@@ -809,7 +838,7 @@
<p><span class="index-entry-level-0">cyl_bessel_k_prime</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html" title="Derivatives of the Bessel Functions"><span class="index-entry-level-1">Derivatives of the Bessel Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_k_prime_integer_orders_" title="Table&#160;8.54.&#160;Error rates for cyl_bessel_k_prime (integer orders)"><span class="index-entry-level-1">Error rates for cyl_bessel_k_prime (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_k_prime_integer_orders_" title="Table 8.54. Error rates for cyl_bessel_k_prime (integer orders)"><span class="index-entry-level-1">Error rates for cyl_bessel_k_prime (integer orders)</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
@@ -827,7 +856,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/logs.html" title="Error Logs For Error Rate Tables"><span class="index-entry-level-1">Error Logs For Error Rate Tables</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_first.html#math_toolkit.bessel.bessel_first.table_cyl_neumann_integer_orders_" title="Table&#160;8.42.&#160;Error rates for cyl_neumann (integer orders)"><span class="index-entry-level-1">Error rates for cyl_neumann (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_first.html#math_toolkit.bessel.bessel_first.table_cyl_neumann_integer_orders_" title="Table 8.42. Error rates for cyl_neumann (integer orders)"><span class="index-entry-level-1">Error rates for cyl_neumann (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
</ul></div>
</li>
@@ -843,7 +872,7 @@
<p><span class="index-entry-level-0">cyl_neumann_prime</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_derivatives.html" title="Derivatives of the Bessel Functions"><span class="index-entry-level-1">Derivatives of the Bessel Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/all_table.html#math_toolkit.logs_and_tables.all_table.table_cyl_neumann_prime_integer_orders_" title="Table&#160;23.25.&#160;Error rates for cyl_neumann_prime (integer orders)"><span class="index-entry-level-1">Error rates for cyl_neumann_prime (integer orders)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/all_table.html#math_toolkit.logs_and_tables.all_table.table_cyl_neumann_prime_integer_orders_" title="Table 23.25. Error rates for cyl_neumann_prime (integer orders)"><span class="index-entry-level-1">Error rates for cyl_neumann_prime (integer orders)</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
@@ -897,11 +926,19 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/autodiff.html" title="Automatic Differentiation"><span class="index-entry-level-1">Automatic Differentiation</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_finding_examples/cbrt_eg.html" title="Finding the Cubed Root With and Without Derivatives"><span class="index-entry-level-1">Finding the Cubed Root With and Without Derivatives</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/history1.html" title="History and What's New"><span class="index-entry-level-1">History and What's New</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der"><span class="index-entry-level-1">Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schröder"><span class="index-entry-level-1">Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schröder</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/digamma.html" title="Digamma"><span class="index-entry-level-0">digamma</span></a></p></li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">dis</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/empirical_cdf.html" title="Empirical Cumulative Distribution Function"><span class="index-entry-level-1">Empirical Cumulative Distribution Function</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/anderson_darling.html" title="The Anderson-Darling Test"><span class="index-entry-level-1">The Anderson-Darling Test</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ljung_box.html" title="The Ljung-Box Test"><span class="index-entry-level-1">The Ljung-Box Test</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">distribution</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/bernoulli_dist.html" title="Bernoulli Distribution"><span class="index-entry-level-1">Bernoulli Distribution</span></a></p></li>
@@ -916,12 +953,27 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/overview/generic.html" title="Generic operations common to all distributions are non-member functions"><span class="index-entry-level-1">Generic operations common to all distributions are non-member functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/history1.html" title="History and What's New"><span class="index-entry-level-1">History and What's New</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/hyperexponential_dist.html" title="Hyperexponential Distribution"><span class="index-entry-level-1">Hyperexponential Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/kolmogorov_smirnov_dist.html" title="Kolmogorov-Smirnov Distribution"><span class="index-entry-level-1">Kolmogorov-Smirnov Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/normal_example/normal_misc.html" title="Some Miscellaneous Examples of the Normal (Gaussian) Distribution"><span class="index-entry-level-1">Some Miscellaneous Examples of the Normal (Gaussian) Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/nccs_eg/nccs_power_eg.html" title="Tables of the power function of the chi2 test."><span class="index-entry-level-1">Tables of the power function of the chi 2 test.</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/triangular_dist.html" title="Triangular Distribution"><span class="index-entry-level-1">Triangular Distribution</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">domain</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/cubic_hermite.html" title="Cubic Hermite interpolation"><span class="index-entry-level-1">Cubic Hermite interpolation</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta1.html" title="Jacobi Theta Function θ1"><span class="index-entry-level-1">Jacobi Theta Function θ 1</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta2.html" title="Jacobi Theta Function θ2"><span class="index-entry-level-1">Jacobi Theta Function θ 2</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta3.html" title="Jacobi Theta Function θ3"><span class="index-entry-level-1">Jacobi Theta Function θ 3</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta4.html" title="Jacobi Theta Function θ4"><span class="index-entry-level-1">Jacobi Theta Function θ 4</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta_overview.html" title="Overview of the Jacobi Theta Functions"><span class="index-entry-level-1">Overview of the Jacobi Theta Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/owens_t.html" title="Owen's T function"><span class="index-entry-level-1">Owen's T function</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/polynomials.html" title="Polynomials"><span class="index-entry-level-1">Polynomials</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/quintic_hermite.html" title="Quintic Hermite interpolation"><span class="index-entry-level-1">Quintic Hermite interpolation</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">double</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/cbrt_comparison.html" title="Comparison of Cube Root Finding Algorithms"><span class="index-entry-level-1">Comparison of Cube Root Finding Algorithms</span></a></p></li>
@@ -942,7 +994,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ellint/ellint_1.html" title="Elliptic Integrals of the First Kind - Legendre Form"><span class="index-entry-level-1">Elliptic Integrals of the First Kind - Legendre Form</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/all_table.html#math_toolkit.logs_and_tables.all_table.table_ellint_1_complete_" title="Table&#160;23.28.&#160;Error rates for ellint_1 (complete)"><span class="index-entry-level-1">Error rates for ellint_1 (complete)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/all_table.html#math_toolkit.logs_and_tables.all_table.table_ellint_1_complete_" title="Table 23.28. Error rates for ellint_1 (complete)"><span class="index-entry-level-1">Error rates for ellint_1 (complete)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
</ul></div>
</li>
@@ -969,7 +1021,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ellint/ellint_2.html" title="Elliptic Integrals of the Second Kind - Legendre Form"><span class="index-entry-level-1">Elliptic Integrals of the Second Kind - Legendre Form</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/logs.html" title="Error Logs For Error Rate Tables"><span class="index-entry-level-1">Error Logs For Error Rate Tables</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/all_table.html#math_toolkit.logs_and_tables.all_table.table_ellint_2_complete_" title="Table&#160;23.30.&#160;Error rates for ellint_2 (complete)"><span class="index-entry-level-1">Error rates for ellint_2 (complete)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/all_table.html#math_toolkit.logs_and_tables.all_table.table_ellint_2_complete_" title="Table 23.30. Error rates for ellint_2 (complete)"><span class="index-entry-level-1">Error rates for ellint_2 (complete)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
</ul></div>
</li>
@@ -997,7 +1049,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ellint/ellint_d.html" title="Elliptic Integral D - Legendre Form"><span class="index-entry-level-1">Elliptic Integral D - Legendre Form</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ellint/ellint_3.html" title="Elliptic Integrals of the Third Kind - Legendre Form"><span class="index-entry-level-1">Elliptic Integrals of the Third Kind - Legendre Form</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/logs.html" title="Error Logs For Error Rate Tables"><span class="index-entry-level-1">Error Logs For Error Rate Tables</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/all_table.html#math_toolkit.logs_and_tables.all_table.table_ellint_3_complete_" title="Table&#160;23.32.&#160;Error rates for ellint_3 (complete)"><span class="index-entry-level-1">Error rates for ellint_3 (complete)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/all_table.html#math_toolkit.logs_and_tables.all_table.table_ellint_3_complete_" title="Table 23.32. Error rates for ellint_3 (complete)"><span class="index-entry-level-1">Error rates for ellint_3 (complete)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
</ul></div>
</li>
@@ -1021,7 +1073,7 @@
<p><span class="index-entry-level-0">ellint_d</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ellint/ellint_d.html" title="Elliptic Integral D - Legendre Form"><span class="index-entry-level-1">Elliptic Integral D - Legendre Form</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ellint/ellint_d.html#math_toolkit.ellint.ellint_d.table_ellint_d_complete_" title="Table&#160;8.66.&#160;Error rates for ellint_d (complete)"><span class="index-entry-level-1">Error rates for ellint_d (complete)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ellint/ellint_d.html#math_toolkit.ellint.ellint_d.table_ellint_d_complete_" title="Table 8.66. Error rates for ellint_d (complete)"><span class="index-entry-level-1">Error rates for ellint_d (complete)</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
@@ -1049,6 +1101,10 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/nmp.html" title="Non-Member Properties"><span class="index-entry-level-1">Non-Member Properties</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">envelope_color</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ulps_plots.html" title="ULPs Plots"><span class="index-entry-level-1">ULPs Plots</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">epsilon</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/brent_minima.html" title="Locating Function Minima using Brent's algorithm"><span class="index-entry-level-1">Locating Function Minima using Brent's algorithm</span></a></p></li></ul></div>
</li>
@@ -1175,8 +1231,8 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/logs.html" title="Error Logs For Error Rate Tables"><span class="index-entry-level-1">Error Logs For Error Rate Tables</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/expint/expint_i.html#math_toolkit.expint.expint_i.table_expint_Ei_" title="Table&#160;8.78.&#160;Error rates for expint (Ei)"><span class="index-entry-level-1">Error rates for expint (Ei)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/expint/expint_n.html#math_toolkit.expint.expint_n.table_expint_En_" title="Table&#160;8.77.&#160;Error rates for expint (En)"><span class="index-entry-level-1">Error rates for expint (En)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/expint/expint_i.html#math_toolkit.expint.expint_i.table_expint_Ei_" title="Table 8.78. Error rates for expint (Ei)"><span class="index-entry-level-1">Error rates for expint (Ei)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/expint/expint_n.html#math_toolkit.expint.expint_n.table_expint_En_" title="Table 8.77. Error rates for expint (En)"><span class="index-entry-level-1">Error rates for expint (En)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/expint/expint_i.html" title="Exponential Integral Ei"><span class="index-entry-level-1">Exponential Integral Ei</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/expint/expint_n.html" title="Exponential Integral En"><span class="index-entry-level-1">Exponential Integral En</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
@@ -1405,6 +1461,10 @@
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">font_color</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ulps_plots.html" title="ULPs Plots"><span class="index-entry-level-1">ULPs Plots</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">forward_cardinal_b_spline</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/cardinal_b_splines.html" title="Cardinal B-splines"><span class="index-entry-level-1">Cardinal B-splines</span></a></p></li></ul></div>
</li>
@@ -1438,6 +1498,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution"><span class="index-entry-level-1">Inverse Gaussian (or Inverse Normal) Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ellint/jacobi_zeta.html" title="Jacobi Zeta Function"><span class="index-entry-level-1">Jacobi Zeta Function</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/nmp.html" title="Non-Member Properties"><span class="index-entry-level-1">Non-Member Properties</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta_overview.html" title="Overview of the Jacobi Theta Functions"><span class="index-entry-level-1">Overview of the Jacobi Theta Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/pareto.html" title="Pareto Distribution"><span class="index-entry-level-1">Pareto Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/rational.html" title="Polynomial and Rational Function Evaluation"><span class="index-entry-level-1">Polynomial and Rational Function Evaluation</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/variates.html" title="Random Variates and Distribution Parameters"><span class="index-entry-level-1">Random Variates and Distribution Parameters</span></a></p></li>
@@ -1453,6 +1514,8 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/history1.html" title="History and What's New"><span class="index-entry-level-1">History and What's New</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta3.html" title="Jacobi Theta Function θ3"><span class="index-entry-level-1">Jacobi Theta Function θ 3</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta4.html" title="Jacobi Theta Function θ4"><span class="index-entry-level-1">Jacobi Theta Function θ 4</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/nc_t_dist.html" title="Noncentral T Distribution"><span class="index-entry-level-1">Noncentral T Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sinc/sinc_overview.html" title="Sinus Cardinal and Hyperbolic Sinus Cardinal Functions Overview"><span class="index-entry-level-1">Sinus Cardinal and Hyperbolic Sinus Cardinal Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
@@ -1469,7 +1532,10 @@
<dd><div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">G</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/owens_t.html" title="Owen's T function"><span class="index-entry-level-1">Owen's T function</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/kolmogorov_smirnov_dist.html" title="Kolmogorov-Smirnov Distribution"><span class="index-entry-level-1">Kolmogorov-Smirnov Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/owens_t.html" title="Owen's T function"><span class="index-entry-level-1">Owen's T function</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">gamma_p</span></p>
@@ -1546,7 +1612,7 @@
<dd><div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">halley_iterate</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der"><span class="index-entry-level-1">Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schröder"><span class="index-entry-level-1">Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schröder</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">hamming_distance</span></p>
@@ -1590,6 +1656,10 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ellint/heuman_lambda.html" title="Heuman Lambda Function"><span class="index-entry-level-1">Heuman Lambda Function</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">horizontal_lines</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ulps_plots.html" title="ULPs Plots"><span class="index-entry-level-1">ULPs Plots</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">hoyer_sparsity</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/signal_statistics.html" title="Signal Statistics"><span class="index-entry-level-1">Signal Statistics</span></a></p></li></ul></div>
</li>
@@ -1813,8 +1883,8 @@
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">inverse</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.table_gamma_p_inv" title="Table&#160;8.13.&#160;Error rates for gamma_p_inv"><span class="index-entry-level-1">Error rates for gamma_p_inv</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.table_gamma_q_inv" title="Table&#160;8.14.&#160;Error rates for gamma_q_inv"><span class="index-entry-level-1">Error rates for gamma_q_inv</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.table_gamma_p_inv" title="Table 8.13. Error rates for gamma_p_inv"><span class="index-entry-level-1">Error rates for gamma_p_inv</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.table_gamma_q_inv" title="Table 8.14. Error rates for gamma_q_inv"><span class="index-entry-level-1">Error rates for gamma_q_inv</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
@@ -1929,6 +1999,54 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi/jacobi_sn.html" title="Jacobi Elliptic Function sn"><span class="index-entry-level-1">Jacobi Elliptic Function sn</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">jacobi_theta1</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta1.html" title="Jacobi Theta Function θ1"><span class="index-entry-level-1">Jacobi Theta Function θ 1</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">jacobi_theta1tau</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta1.html" title="Jacobi Theta Function θ1"><span class="index-entry-level-1">Jacobi Theta Function θ 1</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">jacobi_theta2</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta2.html" title="Jacobi Theta Function θ2"><span class="index-entry-level-1">Jacobi Theta Function θ 2</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">jacobi_theta2tau</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta2.html" title="Jacobi Theta Function θ2"><span class="index-entry-level-1">Jacobi Theta Function θ 2</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">jacobi_theta3</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta3.html" title="Jacobi Theta Function θ3"><span class="index-entry-level-1">Jacobi Theta Function θ 3</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">jacobi_theta3m1</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta3.html" title="Jacobi Theta Function θ3"><span class="index-entry-level-1">Jacobi Theta Function θ 3</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">jacobi_theta3m1tau</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta3.html" title="Jacobi Theta Function θ3"><span class="index-entry-level-1">Jacobi Theta Function θ 3</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">jacobi_theta3tau</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta3.html" title="Jacobi Theta Function θ3"><span class="index-entry-level-1">Jacobi Theta Function θ 3</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">jacobi_theta4</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta4.html" title="Jacobi Theta Function θ4"><span class="index-entry-level-1">Jacobi Theta Function θ 4</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">jacobi_theta4m1</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta4.html" title="Jacobi Theta Function θ4"><span class="index-entry-level-1">Jacobi Theta Function θ 4</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">jacobi_theta4m1tau</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta4.html" title="Jacobi Theta Function θ4"><span class="index-entry-level-1">Jacobi Theta Function θ 4</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">jacobi_theta4tau</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta4.html" title="Jacobi Theta Function θ4"><span class="index-entry-level-1">Jacobi Theta Function θ 4</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">jacobi_zeta</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ellint/jacobi_zeta.html" title="Jacobi Zeta Function"><span class="index-entry-level-1">Jacobi Zeta Function</span></a></p></li></ul></div>
</li>
@@ -1941,6 +2059,7 @@
<p><span class="index-entry-level-0">k</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_finding_examples/elliptic_eg.html" title="A More complex example - Inverting the Elliptic Integrals"><span class="index-entry-level-1">A More complex example - Inverting the Elliptic Integrals</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/agm.html" title="Arithmetic-Geometric Mean"><span class="index-entry-level-1">Arithmetic-Geometric Mean</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/cardinal_b_splines.html" title="Cardinal B-splines"><span class="index-entry-level-1">Cardinal B-splines</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/catmull_rom.html" title="Catmull-Rom Splines"><span class="index-entry-level-1">Catmull-Rom Splines</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ellint/ellint_intro.html" title="Elliptic Integral Overview"><span class="index-entry-level-1">Elliptic Integral Overview</span></a></p></li>
@@ -2015,8 +2134,8 @@
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_tr1.html" title="C99 and TR1 C Functions Overview"><span class="index-entry-level-1">C99 and TR1 C Functions Overview</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/laguerre.html#math_toolkit.sf_poly.laguerre.table_laguerre_n_m_x_" title="Table&#160;8.36.&#160;Error rates for laguerre(n, m, x)"><span class="index-entry-level-1">Error rates for laguerre(n, m, x)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/laguerre.html#math_toolkit.sf_poly.laguerre.table_laguerre_n_x_" title="Table&#160;8.35.&#160;Error rates for laguerre(n, x)"><span class="index-entry-level-1">Error rates for laguerre(n, x)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/laguerre.html#math_toolkit.sf_poly.laguerre.table_laguerre_n_m_x_" title="Table 8.36. Error rates for laguerre(n, m, x)"><span class="index-entry-level-1">Error rates for laguerre(n, m, x)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/laguerre.html#math_toolkit.sf_poly.laguerre.table_laguerre_n_x_" title="Table 8.35. Error rates for laguerre(n, x)"><span class="index-entry-level-1">Error rates for laguerre(n, x)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/laguerre.html" title="Laguerre (and Associated) Polynomials"><span class="index-entry-level-1">Laguerre (and Associated) Polynomials</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
</ul></div>
@@ -2093,7 +2212,7 @@
<p><span class="index-entry-level-0">legendre_p</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/logs.html" title="Error Logs For Error Rate Tables"><span class="index-entry-level-1">Error Logs For Error Rate Tables</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/legendre.html#math_toolkit.sf_poly.legendre.table_legendre_p_associated_" title="Table&#160;8.34.&#160;Error rates for legendre_p (associated)"><span class="index-entry-level-1">Error rates for legendre_p (associated)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/legendre.html#math_toolkit.sf_poly.legendre.table_legendre_p_associated_" title="Table 8.34. Error rates for legendre_p (associated)"><span class="index-entry-level-1">Error rates for legendre_p (associated)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/legendre.html" title="Legendre (and Associated) Polynomials"><span class="index-entry-level-1">Legendre (and Associated) Polynomials</span></a></p></li>
</ul></div>
</li>
@@ -2418,6 +2537,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/history1.html" title="History and What's New"><span class="index-entry-level-1">History and What's New</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/nmp.html" title="Non-Member Properties"><span class="index-entry-level-1">Non-Member Properties</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/triangular_dist.html" title="Triangular Distribution"><span class="index-entry-level-1">Triangular Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/univariate_statistics.html" title="Univariate Statistics"><span class="index-entry-level-1">Univariate Statistics</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
@@ -2445,6 +2565,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_finding_examples/cbrt_eg.html" title="Finding the Cubed Root With and Without Derivatives"><span class="index-entry-level-1">Finding the Cubed Root With and Without Derivatives</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/overview/generic.html" title="Generic operations common to all distributions are non-member functions"><span class="index-entry-level-1">Generic operations common to all distributions are non-member functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/history1.html" title="History and What's New"><span class="index-entry-level-1">History and What's New</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/kolmogorov_smirnov_dist.html" title="Kolmogorov-Smirnov Distribution"><span class="index-entry-level-1">Kolmogorov-Smirnov Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/legendre_stieltjes.html" title="Legendre-Stieltjes Polynomials"><span class="index-entry-level-1">Legendre-Stieltjes Polynomials</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tr1_ref.html" title="TR1 C Functions Quick Reference"><span class="index-entry-level-1">TR1 C Functions Quick Reference</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/trapezoidal.html" title="Trapezoidal Quadrature"><span class="index-entry-level-1">Trapezoidal Quadrature</span></a></p></li>
@@ -2474,6 +2595,10 @@
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">nan_color</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ulps_plots.html" title="ULPs Plots"><span class="index-entry-level-1">ULPs Plots</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">nearbyint</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/overview_tr1.html" title="C99 and C++ TR1 C-style Functions"><span class="index-entry-level-1">C99 and C++ TR1 C-style Functions</span></a></p></li>
@@ -2496,7 +2621,7 @@
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<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">newton_raphson_iterate</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der"><span class="index-entry-level-1">Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schröder"><span class="index-entry-level-1">Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schröder</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">nextafter</span></p>
@@ -2623,6 +2748,14 @@
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">one</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/main_faq.html" title="Boost.Math Frequently Asked Questions (FAQs)"><span class="index-entry-level-1">Boost.Math Frequently Asked Questions (FAQs)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/cs_eg/chi_sq_size.html" title="Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation"><span class="index-entry-level-1">Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/remez.html" title="The Remez Method"><span class="index-entry-level-1">The Remez Method</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">one_sample_t_test</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/t_test.html" title="t-tests"><span class="index-entry-level-1">t -tests</span></a></p></li></ul></div>
</li>
@@ -2637,6 +2770,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/cardinal_trigonometric.html" title="Cardinal Trigonometric interpolation"><span class="index-entry-level-1">Cardinal Trigonometric interpolation</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/catmull_rom.html" title="Catmull-Rom Splines"><span class="index-entry-level-1">Catmull-Rom Splines</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/chebyshev.html" title="Chebyshev Polynomials"><span class="index-entry-level-1">Chebyshev Polynomials</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/cohen_acceleration.html" title="Cohen Acceleration"><span class="index-entry-level-1">Cohen Acceleration</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_finding_examples/5th_root_eg.html" title="Computing the Fifth Root"><span class="index-entry-level-1">Computing the Fifth Root</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/cond.html" title="Condition Numbers"><span class="index-entry-level-1">Condition Numbers</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/cf.html" title="Continued Fraction Evaluation"><span class="index-entry-level-1">Continued Fraction Evaluation</span></a></p></li>
@@ -2726,7 +2860,7 @@
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">powm1</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/all_table.html#math_toolkit.logs_and_tables.all_table.table_boost_math_powm1" title="Table&#160;23.7.&#160;Error rates for boost::math::powm1"><span class="index-entry-level-1">Error rates for boost::math::powm1</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/logs_and_tables/all_table.html#math_toolkit.logs_and_tables.all_table.table_boost_math_powm1" title="Table 23.7. Error rates for boost::math::powm1"><span class="index-entry-level-1">Error rates for boost::math::powm1</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><span class="bold"><strong><a class="link" href="../math_toolkit/powers/powm1.html" title="powm1"><span class="index-entry-level-1">powm1</span></a></strong></span></p></li>
</ul></div>
</li>
@@ -2767,6 +2901,10 @@
</dt>
<dd><div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">q</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/jacobi_theta/jacobi_theta_overview.html" title="Overview of the Jacobi Theta Functions"><span class="index-entry-level-1">Overview of the Jacobi Theta Functions</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">quad</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/chebyshev.html" title="Chebyshev Polynomials"><span class="index-entry-level-1">Chebyshev Polynomials</span></a></p></li></ul></div>
</li>
@@ -2826,6 +2964,13 @@
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">RealType</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/real_concepts.html" title="Conceptual Requirements for Real Number Types"><span class="index-entry-level-1">Conceptual Requirements for Real Number Types</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/directories.html" title="Directory and File Structure"><span class="index-entry-level-1">Directory and File Structure</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">refinement</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/lambert_w.html" title="Lambert W function"><span class="index-entry-level-1">Lambert W function</span></a></p></li></ul></div>
</li>
@@ -2940,16 +3085,16 @@
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">root</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_11_0" title="Table&#160;10.8.&#160;11th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_AVX"><span class="index-entry-level-1">11th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_AVX</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_11_1" title="Table&#160;10.11.&#160;11th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_SSE2"><span class="index-entry-level-1">11th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_SSE2</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_11" title="Table&#160;10.5.&#160;11th root(28) for float, double, long double and cpp_bin_float_50 types, using _X86_SSE2"><span class="index-entry-level-1">11th root(28) for float, double, long double and cpp_bin_float_50 types, using _X86_SSE2</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_5_0" title="Table&#160;10.6.&#160;5th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_AVX"><span class="index-entry-level-1">5th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_AVX</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_5_1" title="Table&#160;10.9.&#160;5th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_SSE2"><span class="index-entry-level-1">5th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_SSE2</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_5" title="Table&#160;10.3.&#160;5th root(28) for float, double, long double and cpp_bin_float_50 types, using _X86_SSE2"><span class="index-entry-level-1">5th root(28) for float, double, long double and cpp_bin_float_50 types, using _X86_SSE2</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_7_0" title="Table&#160;10.7.&#160;7th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_AVX"><span class="index-entry-level-1">7th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_AVX</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_7_1" title="Table&#160;10.10.&#160;7th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_SSE2"><span class="index-entry-level-1">7th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_SSE2</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_7" title="Table&#160;10.4.&#160;7th root(28) for float, double, long double and cpp_bin_float_50 types, using _X86_SSE2"><span class="index-entry-level-1">7th root(28) for float, double, long double and cpp_bin_float_50 types, using _X86_SSE2</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/cbrt_comparison.html#math_toolkit.root_comparison.cbrt_comparison.cbrt_4" title="Table&#160;10.1.&#160;Cube root(28) for float, double, long double and cpp_bin_float_50"><span class="index-entry-level-1">Cube root(28) for float, double, long double and cpp_bin_float_50</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_11_0" title="Table 10.8. 11th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_AVX"><span class="index-entry-level-1">11th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_AVX</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_11_1" title="Table 10.11. 11th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_SSE2"><span class="index-entry-level-1">11th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_SSE2</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_11" title="Table 10.5. 11th root(28) for float, double, long double and cpp_bin_float_50 types, using _X86_SSE2"><span class="index-entry-level-1">11th root(28) for float, double, long double and cpp_bin_float_50 types, using _X86_SSE2</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_5_0" title="Table 10.6. 5th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_AVX"><span class="index-entry-level-1">5th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_AVX</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_5_1" title="Table 10.9. 5th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_SSE2"><span class="index-entry-level-1">5th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_SSE2</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_5" title="Table 10.3. 5th root(28) for float, double, long double and cpp_bin_float_50 types, using _X86_SSE2"><span class="index-entry-level-1">5th root(28) for float, double, long double and cpp_bin_float_50 types, using _X86_SSE2</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_7_0" title="Table 10.7. 7th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_AVX"><span class="index-entry-level-1">7th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_AVX</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_7_1" title="Table 10.10. 7th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_SSE2"><span class="index-entry-level-1">7th root(28) for float, double, long double and cpp_bin_float_50 types, using _X64_SSE2</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.root_7" title="Table 10.4. 7th root(28) for float, double, long double and cpp_bin_float_50 types, using _X86_SSE2"><span class="index-entry-level-1">7th root(28) for float, double, long double and cpp_bin_float_50 types, using _X86_SSE2</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_comparison/cbrt_comparison.html#math_toolkit.root_comparison.cbrt_comparison.cbrt_4" title="Table 10.1. Cube root(28) for float, double, long double and cpp_bin_float_50"><span class="index-entry-level-1">Cube root(28) for float, double, long double and cpp_bin_float_50</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)"><span class="index-entry-level-1">Examples of Root-Finding (with and without derivatives)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_finding_examples/cbrt_eg.html" title="Finding the Cubed Root With and Without Derivatives"><span class="index-entry-level-1">Finding the Cubed Root With and Without Derivatives</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_root.html" title="Finding Zeros of Bessel Functions of the First and Second Kinds"><span class="index-entry-level-1">Finding Zeros of Bessel Functions of the First and Second Kinds</span></a></p></li>
@@ -2982,6 +3127,10 @@
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">rsqrt</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/powers/rsqrt.html" title="Reciprocal square root"><span class="index-entry-level-1">Reciprocal square root</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">runs_above_and_below_median</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/runs_test.html" title="Runs tests"><span class="index-entry-level-1">Runs tests</span></a></p></li></ul></div>
</li>
@@ -3072,7 +3221,7 @@
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">schroder_iterate</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der"><span class="index-entry-level-1">Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schröder"><span class="index-entry-level-1">Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schröder</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">semipolar</span></p>
@@ -3138,6 +3287,7 @@
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/bernoulli_dist.html" title="Bernoulli Distribution"><span class="index-entry-level-1">Bernoulli Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/geometric_dist.html" title="Geometric Distribution"><span class="index-entry-level-1">Geometric Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/kolmogorov_smirnov_dist.html" title="Kolmogorov-Smirnov Distribution"><span class="index-entry-level-1">Kolmogorov-Smirnov Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/nmp.html" title="Non-Member Properties"><span class="index-entry-level-1">Non-Member Properties</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/triangular_dist.html" title="Triangular Distribution"><span class="index-entry-level-1">Triangular Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/univariate_statistics.html" title="Univariate Statistics"><span class="index-entry-level-1">Univariate Statistics</span></a></p></li>
@@ -3155,6 +3305,10 @@
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">sorted_mode</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/univariate_statistics.html" title="Univariate Statistics"><span class="index-entry-level-1">Univariate Statistics</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">spherical</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/oct_create.html" title="Octonion Creation Functions"><span class="index-entry-level-1">Octonion Creation Functions</span></a></p></li>
@@ -3270,6 +3424,10 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/digamma.html" title="Digamma"><span class="index-entry-level-1">Digamma</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">sum</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/cond.html" title="Condition Numbers"><span class="index-entry-level-1">Condition Numbers</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">sum_series</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/series_evaluation.html" title="Series Evaluation"><span class="index-entry-level-1">Series Evaluation</span></a></p></li></ul></div>
</li>
@@ -3306,9 +3464,9 @@
<p><span class="index-entry-level-0">t</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/st_eg/tut_mean_intervals.html" title="Calculating confidence intervals on the mean with the Students-t distribution"><span class="index-entry-level-1">Calculating confidence intervals on the mean with the Students-t distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.table_non_central_t_CDF" title="Table&#160;5.8.&#160;Error rates for non central t CDF"><span class="index-entry-level-1">Error rates for non central t CDF</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.table_non_central_t_CDF_complement" title="Table&#160;5.9.&#160;Error rates for non central t CDF complement"><span class="index-entry-level-1">Error rates for non central t CDF complement</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/owens_t.html#math_toolkit.owens_t.table_owens_t" title="Table&#160;8.86.&#160;Error rates for owens_t"><span class="index-entry-level-1">Error rates for owens_t</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.table_non_central_t_CDF" title="Table 5.8. Error rates for non central t CDF"><span class="index-entry-level-1">Error rates for non central t CDF</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.table_non_central_t_CDF_complement" title="Table 5.9. Error rates for non central t CDF complement"><span class="index-entry-level-1">Error rates for non central t CDF complement</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/owens_t.html#math_toolkit.owens_t.table_owens_t" title="Table 8.86. Error rates for owens_t"><span class="index-entry-level-1">Error rates for owens_t</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/special_tut/special_tut_impl.html" title="Implementation"><span class="index-entry-level-1">Implementation</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/issues.html" title="Known Issues, and TODO List"><span class="index-entry-level-1">Known Issues, and TODO List</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/rational.html" title="Polynomial and Rational Function Evaluation"><span class="index-entry-level-1">Polynomial and Rational Function Evaluation</span></a></p></li>
@@ -3344,10 +3502,10 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/c99.html" title="C99 C Functions"><span class="index-entry-level-1">C99 C Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/pol_tutorial/user_def_err_pol.html" title="Calling User Defined Error Handlers"><span class="index-entry-level-1">Calling User Defined Error Handlers</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/pol_tutorial/ad_hoc_sf_policies.html" title="Changing the Policy on an Ad Hoc Basis for the Special Functions"><span class="index-entry-level-1">Changing the Policy on an Ad Hoc Basis for the Special Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma.html#math_toolkit.sf_gamma.igamma.table_gamma_p" title="Table&#160;8.9.&#160;Error rates for gamma_p"><span class="index-entry-level-1">Error rates for gamma_p</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma.html#math_toolkit.sf_gamma.igamma.table_gamma_q" title="Table&#160;8.10.&#160;Error rates for gamma_q"><span class="index-entry-level-1">Error rates for gamma_q</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma.html#math_toolkit.sf_gamma.igamma.table_tgamma_incomplete_" title="Table&#160;8.12.&#160;Error rates for tgamma (incomplete)"><span class="index-entry-level-1">Error rates for tgamma (incomplete)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma.html#math_toolkit.sf_gamma.igamma.table_tgamma_lower" title="Table&#160;8.11.&#160;Error rates for tgamma_lower"><span class="index-entry-level-1">Error rates for tgamma_lower</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma.html#math_toolkit.sf_gamma.igamma.table_gamma_p" title="Table 8.9. Error rates for gamma_p"><span class="index-entry-level-1">Error rates for gamma_p</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma.html#math_toolkit.sf_gamma.igamma.table_gamma_q" title="Table 8.10. Error rates for gamma_q"><span class="index-entry-level-1">Error rates for gamma_q</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma.html#math_toolkit.sf_gamma.igamma.table_tgamma_incomplete_" title="Table 8.12. Error rates for tgamma (incomplete)"><span class="index-entry-level-1">Error rates for tgamma (incomplete)</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma.html#math_toolkit.sf_gamma.igamma.table_tgamma_lower" title="Table 8.11. Error rates for tgamma_lower"><span class="index-entry-level-1">Error rates for tgamma_lower</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/tgamma.html" title="Gamma"><span class="index-entry-level-1">Gamma</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/igamma.html" title="Incomplete Gamma Functions"><span class="index-entry-level-1">Incomplete Gamma Functions</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/lgamma.html" title="Log Gamma"><span class="index-entry-level-1">Log Gamma</span></a></p></li>
@@ -3358,7 +3516,7 @@
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">tgamma1pm1</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/tgamma.html#math_toolkit.sf_gamma.tgamma.table_tgamma1pm1" title="Table&#160;8.2.&#160;Error rates for tgamma1pm1"><span class="index-entry-level-1">Error rates for tgamma1pm1</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/tgamma.html#math_toolkit.sf_gamma.tgamma.table_tgamma1pm1" title="Table 8.2. Error rates for tgamma1pm1"><span class="index-entry-level-1">Error rates for tgamma1pm1</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/tgamma.html" title="Gamma"><span class="index-entry-level-1">Gamma</span></a></p></li>
</ul></div>
</li>
@@ -3394,6 +3552,10 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/gamma_ratios.html" title="Ratios of Gamma Functions"><span class="index-entry-level-1">Ratios of Gamma Functions</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">title</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ulps_plots.html" title="ULPs Plots"><span class="index-entry-level-1">ULPs Plots</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">tol</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/root_finding_examples/elliptic_eg.html" title="A More complex example - Inverting the Elliptic Integrals"><span class="index-entry-level-1">A More complex example - Inverting the Elliptic Integrals</span></a></p></li>
@@ -3457,6 +3619,10 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/next_float/ulp.html" title="Obtaining the Size of a Unit In the Last Place - ULP"><span class="index-entry-level-1">Obtaining the Size of a Unit In the Last Place - ULP</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">ulp_envelope</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ulps_plots.html" title="ULPs Plots"><span class="index-entry-level-1">ULPs Plots</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">unchecked_bernoulli_b2n</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/number_series/bernoulli_numbers.html" title="Bernoulli Numbers"><span class="index-entry-level-1">Bernoulli Numbers</span></a></p></li></ul></div>
</li>
@@ -3605,6 +3771,10 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/univariate_statistics.html" title="Univariate Statistics"><span class="index-entry-level-1">Univariate Statistics</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">vertical_lines</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ulps_plots.html" title="ULPs Plots"><span class="index-entry-level-1">ULPs Plots</span></a></p></li></ul></div>
</li>
</ul></div></dd>
<dt>
<a name="idx_id_27"></a><span class="term">W</span>
@@ -3621,6 +3791,10 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/gauss.html" title="Gauss-Legendre quadrature"><span class="index-entry-level-1">Gauss-Legendre quadrature</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">width</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ulps_plots.html" title="ULPs Plots"><span class="index-entry-level-1">ULPs Plots</span></a></p></li></ul></div>
</li>
</ul></div></dd>
<dt>
<a name="idx_id_28"></a><span class="term">X</span>
@@ -3681,11 +3855,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

40
doc/html/indexes/s02.html
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@@ -1,11 +1,11 @@
<html>
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<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
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@@ -24,8 +24,8 @@
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<div class="section">
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<a name="id1986067"></a>Class Index</h2></div></div></div>
<p><a class="link" href="s02.html#idx_id_36">A</a> <a class="link" href="s02.html#idx_id_37">B</a> <a class="link" href="s02.html#idx_id_38">C</a> <a class="link" href="s02.html#idx_id_39">D</a> <a class="link" href="s02.html#idx_id_40">E</a> <a class="link" href="s02.html#idx_id_41">F</a> <a class="link" href="s02.html#idx_id_42">G</a> <a class="link" href="s02.html#idx_id_43">H</a> <a class="link" href="s02.html#idx_id_44">I</a> <a class="link" href="s02.html#idx_id_46">K</a> <a class="link" href="s02.html#idx_id_47">L</a> <a class="link" href="s02.html#idx_id_48">M</a> <a class="link" href="s02.html#idx_id_49">N</a> <a class="link" href="s02.html#idx_id_50">O</a> <a class="link" href="s02.html#idx_id_51">P</a> <a class="link" href="s02.html#idx_id_52">Q</a> <a class="link" href="s02.html#idx_id_53">R</a> <a class="link" href="s02.html#idx_id_54">S</a> <a class="link" href="s02.html#idx_id_55">T</a> <a class="link" href="s02.html#idx_id_56">U</a> <a class="link" href="s02.html#idx_id_57">V</a> <a class="link" href="s02.html#idx_id_58">W</a></p>
<div class="variablelist"><dl class="variablelist">
<dt>
<a name="idx_id_36"></a><span class="term">A</span>
@@ -92,6 +92,10 @@
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/cauchy_dist.html" title="Cauchy-Lorentz Distribution"><span class="index-entry-level-1">Cauchy-Lorentz Distribution</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">centered_continued_fraction</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/centered_continued_fraction.html" title="Centered Continued Fractions"><span class="index-entry-level-1">Centered Continued Fractions</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">chebyshev_transform</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_poly/chebyshev.html" title="Chebyshev Polynomials"><span class="index-entry-level-1">Chebyshev Polynomials</span></a></p></li></ul></div>
</li>
@@ -142,6 +146,7 @@
</dt>
<dd><div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/empirical_cdf.html" title="Empirical Cumulative Distribution Function"><span class="index-entry-level-0">empirical_cumulative_distribution_function</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/engel_expansion.html" title="Engel Expansion"><span class="index-entry-level-0">engel_expansion</span></a></p></li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">eps_tolerance</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/roots_noderiv/root_termination.html" title="Termination Condition Functors"><span class="index-entry-level-1">Termination Condition Functors</span></a></p></li></ul></div>
@@ -226,6 +231,10 @@
</li>
</ul></div></dd>
<dt>
<a name="idx_id_46"></a><span class="term">K</span>
</dt>
<dd><div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/kolmogorov_smirnov_dist.html" title="Kolmogorov-Smirnov Distribution"><span class="index-entry-level-0">kolmogorov_smirnov_distribution</span></a></p></li></ul></div></dd>
<dt>
<a name="idx_id_47"></a><span class="term">L</span>
</dt>
<dd><div class="index"><ul class="index" style="list-style-type: none; ">
@@ -243,6 +252,10 @@
<p><span class="index-entry-level-0">lognormal_distribution</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/lognormal_dist.html" title="Log Normal Distribution"><span class="index-entry-level-1">Log Normal Distribution</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">luroth_expansion</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/luroth_expansion.html" title="Luroth Expansions"><span class="index-entry-level-1">Luroth Expansions</span></a></p></li></ul></div>
</li>
</ul></div></dd>
<dt>
<a name="idx_id_48"></a><span class="term">M</span>
@@ -365,6 +378,10 @@
</dt>
<dd><div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">simple_continued_fraction</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/internals/simple_continued_fraction.html" title="Simple Continued Fractions"><span class="index-entry-level-1">Simple Continued Fractions</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">sinh_sinh</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/double_exponential/de_overview.html" title="Overview"><span class="index-entry-level-1">Overview</span></a></p></li>
@@ -401,6 +418,10 @@
<a name="idx_id_56"></a><span class="term">U</span>
</dt>
<dd><div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">ulps_plot</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/ulps_plots.html" title="ULPs Plots"><span class="index-entry-level-1">ULPs Plots</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/uniform_dist.html" title="Uniform Distribution"><span class="index-entry-level-0">uniform_distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">upper_incomplete_gamma_fract</span></p>
@@ -431,11 +452,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

28
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Typedef Index</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../indexes.html" title="Chapter&#160;25.&#160;Indexes">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../indexes.html" title="Chapter 25. Indexes">
<link rel="prev" href="s02.html" title="Class Index">
<link rel="next" href="s04.html" title="Macro Index">
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@@ -24,8 +24,8 @@
</div>
<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="id1976122"></a>Typedef Index</h2></div></div></div>
<p><a class="link" href="s03.html#idx_id_67">A</a> <a class="link" href="s03.html#idx_id_68">B</a> <a class="link" href="s03.html#idx_id_69">C</a> <a class="link" href="s03.html#idx_id_70">D</a> <a class="link" href="s03.html#idx_id_71">E</a> <a class="link" href="s03.html#idx_id_72">F</a> <a class="link" href="s03.html#idx_id_73">G</a> <a class="link" href="s03.html#idx_id_74">H</a> <a class="link" href="s03.html#idx_id_75">I</a> <a class="link" href="s03.html#idx_id_78">L</a> <a class="link" href="s03.html#idx_id_80">N</a> <a class="link" href="s03.html#idx_id_81">O</a> <a class="link" href="s03.html#idx_id_82">P</a> <a class="link" href="s03.html#idx_id_84">R</a> <a class="link" href="s03.html#idx_id_85">S</a> <a class="link" href="s03.html#idx_id_86">T</a> <a class="link" href="s03.html#idx_id_87">U</a> <a class="link" href="s03.html#idx_id_88">V</a> <a class="link" href="s03.html#idx_id_89">W</a></p>
<a name="id1988647"></a>Typedef Index</h2></div></div></div>
<p><a class="link" href="s03.html#idx_id_67">A</a> <a class="link" href="s03.html#idx_id_68">B</a> <a class="link" href="s03.html#idx_id_69">C</a> <a class="link" href="s03.html#idx_id_70">D</a> <a class="link" href="s03.html#idx_id_71">E</a> <a class="link" href="s03.html#idx_id_72">F</a> <a class="link" href="s03.html#idx_id_73">G</a> <a class="link" href="s03.html#idx_id_74">H</a> <a class="link" href="s03.html#idx_id_75">I</a> <a class="link" href="s03.html#idx_id_77">K</a> <a class="link" href="s03.html#idx_id_78">L</a> <a class="link" href="s03.html#idx_id_80">N</a> <a class="link" href="s03.html#idx_id_81">O</a> <a class="link" href="s03.html#idx_id_82">P</a> <a class="link" href="s03.html#idx_id_84">R</a> <a class="link" href="s03.html#idx_id_85">S</a> <a class="link" href="s03.html#idx_id_86">T</a> <a class="link" href="s03.html#idx_id_87">U</a> <a class="link" href="s03.html#idx_id_88">V</a> <a class="link" href="s03.html#idx_id_89">W</a></p>
<div class="variablelist"><dl class="variablelist">
<dt>
<a name="idx_id_67"></a><span class="term">A</span>
@@ -197,6 +197,13 @@
</li>
</ul></div></dd>
<dt>
<a name="idx_id_77"></a><span class="term">K</span>
</dt>
<dd><div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">kolmogorov_smirnov</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/kolmogorov_smirnov_dist.html" title="Kolmogorov-Smirnov Distribution"><span class="index-entry-level-1">Kolmogorov-Smirnov Distribution</span></a></p></li></ul></div>
</li></ul></div></dd>
<dt>
<a name="idx_id_78"></a><span class="term">L</span>
</dt>
<dd><div class="index"><ul class="index" style="list-style-type: none; ">
@@ -294,6 +301,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution"><span class="index-entry-level-1">Inverse Chi Squared Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/inverse_gamma_dist.html" title="Inverse Gamma Distribution"><span class="index-entry-level-1">Inverse Gamma Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution"><span class="index-entry-level-1">Inverse Gaussian (or Inverse Normal) Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/kolmogorov_smirnov_dist.html" title="Kolmogorov-Smirnov Distribution"><span class="index-entry-level-1">Kolmogorov-Smirnov Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/laplace_dist.html" title="Laplace Distribution"><span class="index-entry-level-1">Laplace Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/lognormal_dist.html" title="Log Normal Distribution"><span class="index-entry-level-1">Log Normal Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/logistic_dist.html" title="Logistic Distribution"><span class="index-entry-level-1">Logistic Distribution</span></a></p></li>
@@ -399,6 +407,7 @@
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution"><span class="index-entry-level-1">Inverse Chi Squared Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/inverse_gamma_dist.html" title="Inverse Gamma Distribution"><span class="index-entry-level-1">Inverse Gamma Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution"><span class="index-entry-level-1">Inverse Gaussian (or Inverse Normal) Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/kolmogorov_smirnov_dist.html" title="Kolmogorov-Smirnov Distribution"><span class="index-entry-level-1">Kolmogorov-Smirnov Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/laplace_dist.html" title="Laplace Distribution"><span class="index-entry-level-1">Laplace Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/lognormal_dist.html" title="Log Normal Distribution"><span class="index-entry-level-1">Log Normal Distribution</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/logistic_dist.html" title="Logistic Distribution"><span class="index-entry-level-1">Logistic Distribution</span></a></p></li>
@@ -440,11 +449,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

57
doc/html/indexes/s04.html
View File

@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Macro Index</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../indexes.html" title="Chapter&#160;25.&#160;Indexes">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../indexes.html" title="Chapter 25. Indexes">
<link rel="prev" href="s03.html" title="Typedef Index">
<link rel="next" href="s05.html" title="Index">
</head>
@@ -24,7 +24,7 @@
</div>
<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="id1980158"></a>Macro Index</h2></div></div></div>
<a name="id1992066"></a>Macro Index</h2></div></div></div>
<p><a class="link" href="s04.html#idx_id_99">B</a> <a class="link" href="s04.html#idx_id_103">F</a></p>
<div class="variablelist"><dl class="variablelist">
<dt>
@@ -91,7 +91,7 @@
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_FPU_EXCEPTION_GUARD</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/special_tut/special_tut_impl.html" title="Implementation"><span class="index-entry-level-1">Implementation</span></a></p></li>
</ul></div>
</li>
@@ -109,11 +109,11 @@
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_CONTROL_FP</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_DECLARE_DISTRIBUTIONS</span></p>
@@ -142,7 +142,7 @@
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_DISABLE_FLOAT128</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_DISCRETE_QUANTILE_POLICY</span></p>
@@ -176,31 +176,31 @@
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_INSTRUMENT_CODE</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_INSTRUMENT_FPU</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_INSTRUMENT_VARIABLE</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_INT_TABLE_TYPE</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_tuning" title="Table&#160;1.12.&#160;Boost.Math Tuning"><span class="index-entry-level-1">Boost.Math Tuning</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_tuning" title="Table 1.12. Boost.Math Tuning"><span class="index-entry-level-1">Boost.Math Tuning</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html" title="Performance Tuning Macros"><span class="index-entry-level-1">Performance Tuning Macros</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_INT_VALUE_SUFFIX</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_tuning" title="Table&#160;1.12.&#160;Boost.Math Tuning"><span class="index-entry-level-1">Boost.Math Tuning</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_tuning" title="Table 1.12. Boost.Math Tuning"><span class="index-entry-level-1">Boost.Math Tuning</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_MAX_POLY_ORDER</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_tuning" title="Table&#160;1.12.&#160;Boost.Math Tuning"><span class="index-entry-level-1">Boost.Math Tuning</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_tuning" title="Table 1.12. Boost.Math Tuning"><span class="index-entry-level-1">Boost.Math Tuning</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html" title="Performance Tuning Macros"><span class="index-entry-level-1">Performance Tuning Macros</span></a></p></li>
</ul></div>
</li>
@@ -225,22 +225,22 @@
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/special_tut/special_tut_test.html" title="Testing"><span class="index-entry-level-1">Testing</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/compilers_overview.html#math_toolkit.compilers_overview.supported_tested_compilers" title="Table&#160;1.9.&#160;Supported/Tested Compilers"><span class="index-entry-level-1">Supported/Tested Compilers</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/compilers_overview.html#math_toolkit.compilers_overview.supported_tested_compilers" title="Table 1.9. Supported/Tested Compilers"><span class="index-entry-level-1">Supported/Tested Compilers</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/special_tut/special_tut_test.html" title="Testing"><span class="index-entry-level-1">Testing</span></a></p></li>
</ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_NO_REAL_CONCEPT_TESTS</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/special_tut/special_tut_test.html" title="Testing"><span class="index-entry-level-1">Testing</span></a></p></li>
</ul></div>
</li>
@@ -262,7 +262,7 @@
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_POLY_METHOD</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_tuning" title="Table&#160;1.12.&#160;Boost.Math Tuning"><span class="index-entry-level-1">Boost.Math Tuning</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_tuning" title="Table 1.12. Boost.Math Tuning"><span class="index-entry-level-1">Boost.Math Tuning</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html" title="Performance Tuning Macros"><span class="index-entry-level-1">Performance Tuning Macros</span></a></p></li>
</ul></div>
</li>
@@ -277,7 +277,7 @@
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_RATIONAL_METHOD</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_tuning" title="Table&#160;1.12.&#160;Boost.Math Tuning"><span class="index-entry-level-1">Boost.Math Tuning</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_tuning" title="Table 1.12. Boost.Math Tuning"><span class="index-entry-level-1">Boost.Math Tuning</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/tuning.html" title="Performance Tuning Macros"><span class="index-entry-level-1">Performance Tuning Macros</span></a></p></li>
</ul></div>
</li>
@@ -287,12 +287,12 @@
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_SMALL_CONSTANT</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_STD_USING</span></p>
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/new_const.html" title="Defining New Constants"><span class="index-entry-level-1">Defining New Constants</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/special_tut/special_tut_impl.html" title="Implementation"><span class="index-entry-level-1">Implementation</span></a></p></li>
</ul></div>
@@ -307,11 +307,11 @@
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_USE_C99</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
</li>
<li class="listitem" style="list-style-type: none">
<p><span class="index-entry-level-0">BOOST_MATH_USE_FLOAT128</span></p>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table&#160;1.11.&#160;Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
<div class="index"><ul class="index" style="list-style-type: none; "><li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/config_macros.html#math_toolkit.config_macros.boost_math_macros" title="Table 1.11. Boost.Math Macros"><span class="index-entry-level-1">Boost.Math Macros</span></a></p></li></ul></div>
</li>
</ul></div></dd>
<dt>
@@ -343,11 +343,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

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@@ -1,12 +1,12 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Chapter&#160;19.&#160;Internal Details: Series, Rationals and Continued Fractions, Testing, and Development Tools</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Chapter 19. Internal Details: Series, Rationals and Continued Fractions, Testing, and Development Tools</title>
<link rel="stylesheet" href="math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="index.html" title="Math Toolkit 2.12.0">
<link rel="prev" href="gcd_lcm.html" title="Chapter&#160;18.&#160;Integer Utilities (Greatest Common Divisor and Least Common Multiple)">
<link rel="home" href="index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="index.html" title="Math Toolkit 2.13.0">
<link rel="prev" href="gcd_lcm.html" title="Chapter 18. Integer Utilities (Greatest Common Divisor and Least Common Multiple)">
<link rel="next" href="math_toolkit/internals_overview.html" title="Overview">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
@@ -24,7 +24,7 @@
</div>
<div class="chapter">
<div class="titlepage"><div><div><h1 class="title">
<a name="internals"></a>Chapter&#160;19.&#160;Internal Details: Series, Rationals and Continued Fractions, Testing,
<a name="internals"></a>Chapter 19. Internal Details: Series, Rationals and Continued Fractions, Testing,
and Development Tools</h1></div></div></div>
<div class="toc">
<p><b>Table of Contents</b></p>
@@ -33,9 +33,17 @@
<dt><span class="section"><a href="math_toolkit/internals.html">Internal tools</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/internals/series_evaluation.html">Series Evaluation</a></span></dt>
<dt><span class="section"><a href="math_toolkit/internals/agm.html">Arithmetic-Geometric Mean</a></span></dt>
<dt><span class="section"><a href="math_toolkit/internals/cf.html">Continued Fraction Evaluation</a></span></dt>
<dt><span class="section"><a href="math_toolkit/internals/simple_continued_fraction.html">Simple
Continued Fractions</a></span></dt>
<dt><span class="section"><a href="math_toolkit/internals/centered_continued_fraction.html">Centered
Continued Fractions</a></span></dt>
<dt><span class="section"><a href="math_toolkit/internals/luroth_expansion.html">Luroth Expansions</a></span></dt>
<dt><span class="section"><a href="math_toolkit/internals/engel_expansion.html">Engel Expansion</a></span></dt>
<dt><span class="section"><a href="math_toolkit/internals/recurrence.html">Tools For 3-Term Recurrence
Relations</a></span></dt>
<dt><span class="section"><a href="math_toolkit/internals/cohen_acceleration.html">Cohen Acceleration</a></span></dt>
<dt><span class="section"><a href="math_toolkit/internals/tuples.html">Tuples</a></span></dt>
<dt><span class="section"><a href="math_toolkit/internals/minimax.html">Minimax Approximations
and the Remez Algorithm</a></span></dt>
@@ -49,11 +57,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

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<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Chapter&#160;12.&#160;Interpolation</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Chapter 12. Interpolation</title>
<link rel="stylesheet" href="math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="index.html" title="Math Toolkit 2.12.0">
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<link rel="home" href="index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="index.html" title="Math Toolkit 2.13.0">
<link rel="prev" href="math_toolkit/rational.html" title="Polynomial and Rational Function Evaluation">
<link rel="next" href="math_toolkit/cardinal_cubic_b.html" title="Cardinal Cubic B-spline interpolation">
</head>
@@ -24,7 +24,7 @@
</div>
<div class="chapter">
<div class="titlepage"><div><div><h1 class="title">
<a name="interpolation"></a>Chapter&#160;12.&#160;Interpolation</h1></div></div></div>
<a name="interpolation"></a>Chapter 12. Interpolation</h1></div></div></div>
<div class="toc">
<p><b>Table of Contents</b></p>
<dl class="toc">
@@ -50,11 +50,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Chapter&#160;15.&#160;Complex Number Functions</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Chapter 15. Complex Number Functions</title>
<link rel="stylesheet" href="math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="index.html" title="Math Toolkit 2.12.0">
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<link rel="home" href="index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="index.html" title="Math Toolkit 2.13.0">
<link rel="prev" href="math_toolkit/daubechies_filters.html" title="Daubechies Filters">
<link rel="next" href="math_toolkit/complex_implementation.html" title="Implementation and Accuracy">
</head>
@@ -24,7 +24,7 @@
</div>
<div class="chapter">
<div class="titlepage"><div><div><h1 class="title">
<a name="inverse_complex"></a>Chapter&#160;15.&#160;Complex Number Functions</h1></div></div></div>
<a name="inverse_complex"></a>Chapter 15. Complex Number Functions</h1></div></div></div>
<div class="toc">
<p><b>Table of Contents</b></p>
<dl class="toc">
@@ -48,11 +48,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

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@@ -126,20 +126,24 @@ span.alignjustify
Names at http://www.w3.org/TR/2002/WD-css3-color-20020219/ 4.3. X11 color keywords.
Quickbook Usage: [role red Some red text]
inline-block - Flows a element inline with the text, but allows width and height to be specified.
https://stackoverflow.com/questions/3043021/is-there-any-guide-on-when-to-use-displayblock-when-inline-and-when-inline-b#:~:text=The%20use%20cases%20for%20block,it%27s%20used%20naturally%20for%20images
*/
span.red { inline-block; color: red; }
span.green { color: green; }
span.lime { color: #00FF00; }
span.blue { color: blue; }
span.navy { color: navy; }
span.yellow { color: yellow; }
span.magenta { color: magenta; }
span.indigo { color: #4B0082; }
span.cyan { color: cyan; }
span.purple { color: purple; }
span.gold { color: gold; }
span.silver { color: silver; } /* lighter gray */
span.gray { color: #808080; } /* light gray */
span.red { display:inline-block; color: red; }
span.green { display:inline-block; color: green; }
span.lime { display:inline-block; color: #00FF00; }
span.blue { display:inline-block; color: blue; }
span.navy { display:inline-block; color: navy; }
span.yellow { display:inline-block; color: yellow; }
span.magenta { display:inline-block; color: magenta; }
span.indigo { display:inline-block; color: #4B0082; }
span.cyan { display:inline-block; color: cyan; }
span.purple { display:inline-block; color: purple; }
span.gold { display:inline-block; color: gold; }
span.silver { display:inline-block; color: silver; } /* lighter gray */
span.gray { display:inline-block; color: #808080; } /* light gray */
/* role for inline Unicode mathematical equations,
making font an italic (as is conventional for equations)
@@ -147,7 +151,7 @@ span.gray { color: #808080; } /* light gray */
and a little bigger (* 125%) because the serif font appears smaller than the default sans serif fonts.
Used, for example: [role serif_italic This is in serif font and italic].
Used in turn by template for inline expressions to match equations as SVG or PNG images.
Can be combined with colors and bold.
*/
span.serif_italic {
font-family: serif;

15
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Acknowledgements</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../quaternions.html" title="Chapter&#160;16.&#160;Quaternions">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../quaternions.html" title="Chapter 16. Quaternions">
<link rel="prev" href="exp.html" title="The Quaternionic Exponential">
<link rel="next" href="quat_history.html" title="History">
</head>
@@ -35,11 +35,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Acknowledgements</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../octonions.html" title="Chapter&#160;17.&#160;Octonions">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../octonions.html" title="Chapter 17. Octonions">
<link rel="prev" href="oct_tests.html" title="Test Program">
<link rel="next" href="oct_history.html" title="History">
</head>
@@ -35,11 +35,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>acos</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../inverse_complex.html" title="Chapter&#160;15.&#160;Complex Number Functions">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../inverse_complex.html" title="Chapter 15. Complex Number Functions">
<link rel="prev" href="asin.html" title="asin">
<link rel="next" href="atan.html" title="atan">
</head>
@@ -49,11 +49,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>acosh</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../inverse_complex.html" title="Chapter&#160;15.&#160;Complex Number Functions">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../inverse_complex.html" title="Chapter 15. Complex Number Functions">
<link rel="prev" href="asinh.html" title="asinh">
<link rel="next" href="atanh.html" title="atanh">
</head>
@@ -49,11 +49,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

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<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Airy Functions</title>
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../special.html" title="Chapter&#160;8.&#160;Special Functions">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../special.html" title="Chapter 8. Special Functions">
<link rel="prev" href="hankel/sph_hankel.html" title="Spherical Hankel Functions">
<link rel="next" href="airy/ai.html" title="Airy Ai Function">
</head>
@@ -36,11 +36,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

19
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Airy Ai Function</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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<link rel="up" href="../airy.html" title="Airy Functions">
<link rel="prev" href="../airy.html" title="Airy Functions">
<link rel="next" href="bi.html" title="Airy Bi Function">
@@ -63,9 +63,9 @@
function</a>.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. 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&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">policy
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
documentation for more details</a>.
</p>
<p>
@@ -86,7 +86,7 @@
functions for detailed accuracy information.
</p>
<p>
In general though, the relative error is low (less than 100 &#949;) for <span class="emphasis"><em>x
In general though, the relative error is low (less than 100 ε) for <span class="emphasis"><em>x
&gt; 0</em></span> while only the absolute error is low for <span class="emphasis"><em>x &lt;
0</em></span> as the following error plot illustrates:
</p>
@@ -117,11 +117,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

19
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Airy Ai' Function</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../airy.html" title="Airy Functions">
<link rel="prev" href="bi.html" title="Airy Bi Function">
<link rel="next" href="bip.html" title="Airy Bi' Function">
@@ -56,9 +56,9 @@
</p></blockquote></div>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. 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&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">policy
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
documentation for more details</a>.
</p>
<p>
@@ -79,7 +79,7 @@
functions for detailed accuracy information.
</p>
<p>
In general though, the relative error is low (less than 100 &#949;) for <span class="emphasis"><em>x
In general though, the relative error is low (less than 100 ε) for <span class="emphasis"><em>x
&gt; 0</em></span> while only the absolute error is low for <span class="emphasis"><em>x &lt;
0</em></span> as the following error plot illustrates:
</p>
@@ -109,11 +109,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

17
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Finding Zeros of Airy Functions</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../airy.html" title="Airy Functions">
<link rel="prev" href="bip.html" title="Airy Bi' Function">
<link rel="next" href="../ellint.html" title="Elliptic Integrals">
@@ -65,7 +65,7 @@
<span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span> <span class="comment">// Destination for zeros.</span>
</pre>
<p>
There are also versions which allow control of the <a class="link" href="../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policies</a>
There are also versions which allow control of the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policies</a>
for error handling and precision.
</p>
<pre class="programlisting"> <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
@@ -313,7 +313,7 @@
<span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span> <span class="comment">// Destination for zeros.</span>
</pre>
<p>
There are also versions which allows control of the <a class="link" href="../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policies</a>
There are also versions which allows control of the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policies</a>
for error handling and precision.
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
@@ -462,11 +462,10 @@ airy_ai_zeros:
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

19
doc/html/math_toolkit/airy/bi.html
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Airy Bi Function</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../airy.html" title="Airy Functions">
<link rel="prev" href="ai.html" title="Airy Ai Function">
<link rel="next" href="aip.html" title="Airy Ai' Function">
@@ -55,9 +55,9 @@
</p></blockquote></div>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. 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&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">policy
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
documentation for more details</a>.
</p>
<p>
@@ -78,7 +78,7 @@
- refer to those functions for detailed accuracy information.
</p>
<p>
In general though, the relative error is low (less than 100 &#949;) for <span class="emphasis"><em>x
In general though, the relative error is low (less than 100 ε) for <span class="emphasis"><em>x
&gt; 0</em></span> while only the absolute error is low for <span class="emphasis"><em>x &lt;
0</em></span> as the following error plot illustrate:
</p>
@@ -108,11 +108,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

19
doc/html/math_toolkit/airy/bip.html
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Airy Bi' Function</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../airy.html" title="Airy Functions">
<link rel="prev" href="aip.html" title="Airy Ai' Function">
<link rel="next" href="airy_root.html" title="Finding Zeros of Airy Functions">
@@ -56,9 +56,9 @@
</p></blockquote></div>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. 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&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">policy
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
documentation for more details</a>.
</p>
<p>
@@ -79,7 +79,7 @@
- refer to those functions for detailed accuracy information.
</p>
<p>
In general though, the relative error is low (less than 100 &#949;) for <span class="emphasis"><em>x
In general though, the relative error is low (less than 100 ε) for <span class="emphasis"><em>x
&gt; 0</em></span> while only the absolute error is low for <span class="emphasis"><em>x &lt;
0</em></span> as the following error plot illustrates:
</p>
@@ -109,11 +109,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>The Anderson-Darling Test</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../statistics.html" title="Chapter&#160;6.&#160;Statistics">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../statistics.html" title="Chapter 6. Statistics">
<link rel="prev" href="signal_statistics.html" title="Signal Statistics">
<link rel="next" href="t_test.html" title="t-tests">
</head>
@@ -192,11 +192,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

17
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@@ -1,13 +1,13 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Conceptual Archetypes for Reals and Distributions</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../using_udt.html" title="Chapter&#160;20.&#160;Use with User-Defined Floating-Point Types - Boost.Multiprecision and others">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../using_udt.html" title="Chapter 20. Use with User-Defined Floating-Point Types - Boost.Multiprecision and others">
<link rel="prev" href="dist_concept.html" title="Conceptual Requirements for Distribution Types">
<link rel="next" href="../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">
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</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
@@ -179,11 +179,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>asin</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../inverse_complex.html" title="Chapter&#160;15.&#160;Complex Number Functions">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../inverse_complex.html" title="Chapter 15. Complex Number Functions">
<link rel="prev" href="complex_implementation.html" title="Implementation and Accuracy">
<link rel="next" href="acos.html" title="acos">
</head>
@@ -49,11 +49,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>asinh</title>
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<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../inverse_complex.html" title="Chapter 15. Complex Number Functions">
<link rel="prev" href="atan.html" title="atan">
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@@ -49,11 +49,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>atan</title>
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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<link rel="up" href="../inverse_complex.html" title="Chapter 15. Complex Number Functions">
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@@ -49,11 +49,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>atanh</title>
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</head>
@@ -49,11 +49,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

39
doc/html/math_toolkit/autodiff.html
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Automatic Differentiation</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
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<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../quadrature.html" title="Chapter 13. Quadrature and Differentiation">
<link rel="prev" href="diff.html" title="Numerical Differentiation">
<link rel="next" href="diff0.html" title="Lanczos Smoothing Derivatives">
</head>
@@ -102,9 +102,9 @@
<p>
The essential idea of autodiff is the substitution of numbers with polynomials
in the evaluation of <span class="emphasis"><em>f(x<sub>0</sub>)</em></span>. By substituting the number
<span class="emphasis"><em>x<sub>0</sub></em></span> with the first-order polynomial <span class="emphasis"><em>x<sub>0</sub>+&#949;</em></span>,
and using the same algorithm to compute <span class="emphasis"><em>f(x<sub>0</sub>+&#949;)</em></span>,
the resulting polynomial in <span class="emphasis"><em>&#949;</em></span> contains the function's
<span class="emphasis"><em>x<sub>0</sub></em></span> with the first-order polynomial <span class="emphasis"><em>x<sub>0</sub>+ε</em></span>,
and using the same algorithm to compute <span class="emphasis"><em>f(x<sub>0</sub>+ε)</em></span>,
the resulting polynomial in <span class="emphasis"><em>ε</em></span> contains the function's
derivatives <span class="emphasis"><em>f'(x<sub>0</sub>)</em></span>, <span class="emphasis"><em>f''(x<sub>0</sub>)</em></span>, <span class="emphasis"><em>f'''(x<sub>0</sub>)</em></span>,
... within the coefficients. Each coefficient is equal to the derivative of
its respective order, divided by the factorial of the order.
@@ -112,8 +112,8 @@
<p>
In greater detail, assume one is interested in calculating the first <span class="emphasis"><em>N</em></span>
derivatives of <span class="emphasis"><em>f</em></span> at <span class="emphasis"><em>x<sub>0</sub></em></span>. Without loss
of precision to the calculation of the derivatives, all terms <span class="emphasis"><em>O(&#949;<sup>N+1</sup>)</em></span>
that include powers of <span class="emphasis"><em>&#949;</em></span> greater than <span class="emphasis"><em>N</em></span>
of precision to the calculation of the derivatives, all terms <span class="emphasis"><em>O(ε<sup>N+1</sup>)</em></span>
that include powers of <span class="emphasis"><em>ε</em></span> greater than <span class="emphasis"><em>N</em></span>
can be discarded. (This is due to the fact that each term in a polynomial depends
only upon equal and lower-order terms under arithmetic operations.) Under these
truncation rules, <span class="emphasis"><em>f</em></span> provides a polynomial-to-polynomial
@@ -126,10 +126,10 @@
C++'s ability to overload operators and functions allows for the creation of
a class <code class="computeroutput"><span class="identifier">fvar</span></code> (<span class="underline">f</span>orward-mode
autodiff <span class="underline">var</span>iable) that represents polynomials
in <span class="emphasis"><em>&#949;</em></span>. Thus the same algorithm <span class="emphasis"><em>f</em></span>
in <span class="emphasis"><em>ε</em></span>. Thus the same algorithm <span class="emphasis"><em>f</em></span>
that calculates the numeric value of <span class="emphasis"><em>y<sub>0</sub>=f(x<sub>0</sub>)</em></span>, when written
to accept and return variables of a generic (template) type, is also used to
calculate the polynomial <span class="emphasis"><em>&#931;<sub>n</sub>y<sub>n</sub>&#949;<sup>n</sup>=f(x<sub>0</sub>+&#949;)</em></span>.
calculate the polynomial <span class="emphasis"><em>Σ<sub>n</sub>y<sub>n</sub>ε<sup>n</sup>=f(x<sub>0</sub>+ε)</em></span>.
The derivatives <span class="emphasis"><em>f<sup>(n)</sup>(x<sub>0</sub>)</em></span> are then found from the product
of the respective factorial <span class="emphasis"><em>n!</em></span> and coefficient <span class="emphasis"><em>y<sub>n</sub></em></span>:
</p>
@@ -153,7 +153,7 @@
<p>
In this example, <code class="computeroutput"><span class="identifier">make_fvar</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span>
<span class="identifier">Order</span><span class="special">&gt;(</span><span class="number">2.0</span><span class="special">)</span></code> instantiates
the polynomial 2+<span class="emphasis"><em>&#949;</em></span>. The <code class="computeroutput"><span class="identifier">Order</span><span class="special">=</span><span class="number">5</span></code> means that
the polynomial 2+<span class="emphasis"><em>ε</em></span>. The <code class="computeroutput"><span class="identifier">Order</span><span class="special">=</span><span class="number">5</span></code> means that
enough space is allocated (on the stack) to hold a polynomial of up to degree
5 during the proceeding computation.
</p>
@@ -382,12 +382,12 @@ put gamma = 0.00199852
</li>
<li class="listitem">
Accuracy of derivatives over finite difference methods. Single-iteration
finite difference methods always include a <span class="emphasis"><em>&#916;x</em></span>
free variable that must be carefully chosen for each application. If <span class="emphasis"><em>&#916;x</em></span>
is too small, then numerical errors become large. If <span class="emphasis"><em>&#916;x</em></span>
finite difference methods always include a <span class="emphasis"><em>Δx</em></span>
free variable that must be carefully chosen for each application. If <span class="emphasis"><em>Δx</em></span>
is too small, then numerical errors become large. If <span class="emphasis"><em>Δx</em></span>
is too large, then mathematical errors become large. With autodiff, there
are no free variables to set and the accuracy of the answer is generally
superior to finite difference methods even with the best choice of <span class="emphasis"><em>&#916;x</em></span>.
superior to finite difference methods even with the best choice of <span class="emphasis"><em>Δx</em></span>.
</li>
</ul></div>
<h2>
@@ -401,11 +401,10 @@ put gamma = 0.00199852
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

63
doc/html/math_toolkit/bad_guess.html
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>The Effect of a Poor Initial Guess</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../root_finding.html" title="Chapter&#160;10.&#160;Root Finding &amp; Minimization Algorithms">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../root_finding.html" title="Chapter 10. Root Finding &amp; Minimization Algorithms">
<link rel="prev" href="root_finding_examples/elliptic_eg.html" title="A More complex example - Inverting the Elliptic Integrals">
<link rel="next" href="bad_roots.html" title="Examples Where Root Finding Goes Wrong">
</head>
@@ -55,62 +55,62 @@
</th>
<th>
<p>
-500% (&#8776;1.323)
-500% (≈1.323)
</p>
</th>
<th>
<p>
-100% (&#8776;3.97)
-100% (≈3.97)
</p>
</th>
<th>
<p>
-50% (&#8776;3.96)
-50% (≈3.96)
</p>
</th>
<th>
<p>
-20% (&#8776;6.35)
-20% (≈6.35)
</p>
</th>
<th>
<p>
-10% (&#8776;7.14)
-10% (≈7.14)
</p>
</th>
<th>
<p>
-5% (&#8776;7.54)
-5% (≈7.54)
</p>
</th>
<th>
<p>
5% (&#8776;8.33)
5% (≈8.33)
</p>
</th>
<th>
<p>
10% (&#8776;8.73)
10% (≈8.73)
</p>
</th>
<th>
<p>
20% (&#8776;9.52)
20% (≈9.52)
</p>
</th>
<th>
<p>
50% (&#8776;11.91)
50% (≈11.91)
</p>
</th>
<th>
<p>
100% (&#8776;15.87)
100% (≈15.87)
</p>
</th>
<th>
<p>
500 (&#8776;47.6)
500 (≈47.6)
</p>
</th>
</tr></thead>
@@ -422,62 +422,62 @@
</th>
<th>
<p>
-500% (&#8776;20.6)
-500% (≈20.6)
</p>
</th>
<th>
<p>
-100% (&#8776;61.81)
-100% (≈61.81)
</p>
</th>
<th>
<p>
-50% (&#8776;61.81)
-50% (≈61.81)
</p>
</th>
<th>
<p>
-20% (&#8776;98.9)
-20% (≈98.9)
</p>
</th>
<th>
<p>
-10% (&#8776;111.3)
-10% (≈111.3)
</p>
</th>
<th>
<p>
-5% (&#8776;117.4)
-5% (≈117.4)
</p>
</th>
<th>
<p>
5% (&#8776;129.8)
5% (≈129.8)
</p>
</th>
<th>
<p>
10% (&#8776;136)
10% (≈136)
</p>
</th>
<th>
<p>
20% (&#8776;148.3)
20% (≈148.3)
</p>
</th>
<th>
<p>
50% (&#8776;185.4)
50% (≈185.4)
</p>
</th>
<th>
<p>
100% (&#8776;247.2)
100% (≈247.2)
</p>
</th>
<th>
<p>
500 (&#8776;741.7)
500 (≈741.7)
</p>
</th>
</tr></thead>
@@ -759,11 +759,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
doc/html/math_toolkit/bad_roots.html
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Examples Where Root Finding Goes Wrong</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../root_finding.html" title="Chapter&#160;10.&#160;Root Finding &amp; Minimization Algorithms">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../root_finding.html" title="Chapter 10. Root Finding &amp; Minimization Algorithms">
<link rel="prev" href="bad_guess.html" title="The Effect of a Poor Initial Guess">
<link rel="next" href="brent_minima.html" title="Locating Function Minima using Brent's algorithm">
</head>
@@ -110,11 +110,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

29
doc/html/math_toolkit/barycentric.html
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Barycentric Rational Interpolation</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../interpolation.html" title="Chapter&#160;12.&#160;Interpolation">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../interpolation.html" title="Chapter 12. Interpolation">
<link rel="prev" href="whittaker_shannon.html" title="Whittaker-Shannon interpolation">
<link rel="next" href="vector_barycentric.html" title="Vector-valued Barycentric Rational Interpolation">
</head>
@@ -61,8 +61,8 @@
</h4>
<p>
Barycentric rational interpolation is a high-accuracy interpolation method
for non-uniformly spaced samples. It requires &#119926;(<span class="emphasis"><em>N</em></span>) time
for construction, and &#119926;(<span class="emphasis"><em>N</em></span>) time for each evaluation. Linear
for non-uniformly spaced samples. It requires 𝑶(<span class="emphasis"><em>N</em></span>) time
for construction, and 𝑶(<span class="emphasis"><em>N</em></span>) time for each evaluation. Linear
time evaluation is not optimal; for instance the cubic B-spline can be evaluated
in constant time. However, using the cubic B-spline requires uniformly-spaced
samples, which are not always available.
@@ -82,8 +82,8 @@
</pre>
<p>
This implicitly calls the constructor with approximation order 3, and hence
the accuracy is &#119926;(<span class="emphasis"><em>h</em></span><sup>4</sup>). In general, if you require an approximation
order <span class="emphasis"><em>d</em></span>, then the error is &#119926;(<span class="emphasis"><em>h</em></span><sup><span class="emphasis"><em>d</em></span>+1</sup>).
the accuracy is 𝑶(<span class="emphasis"><em>h</em></span><sup>4</sup>). In general, if you require an approximation
order <span class="emphasis"><em>d</em></span>, then the error is 𝑶(<span class="emphasis"><em>h</em></span><sup><span class="emphasis"><em>d</em></span>+1</sup>).
A call to the constructor with an explicit approximation order is demonstrated
below
</p>
@@ -122,9 +122,9 @@
</p>
<p>
A desirable property of any interpolator <span class="emphasis"><em>f</em></span> is that for
all <span class="emphasis"><em>x</em></span><sub>min</sub> &#8804; <span class="emphasis"><em>x</em></span> &#8804; <span class="emphasis"><em>x</em></span><sub>max</sub>,
also <span class="emphasis"><em>y</em></span><sub>min</sub> &#8804; <span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>)
&#8804; <span class="emphasis"><em>y</em></span><sub>max</sub>.
all <span class="emphasis"><em>x</em></span><sub>min</sub> ≤ <span class="emphasis"><em>x</em></span> ≤ <span class="emphasis"><em>x</em></span><sub>max</sub>,
also <span class="emphasis"><em>y</em></span><sub>min</sub> ≤ <span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>)
≤ <span class="emphasis"><em>y</em></span><sub>max</sub>.
</p>
<p>
<span class="emphasis"><em>This property does not hold for the barycentric rational interpolator.</em></span>
@@ -270,11 +270,10 @@ Root was found in 10 iterations.
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

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@@ -1,11 +1,11 @@
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<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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@@ -42,11 +42,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

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<html>
<head>
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@@ -39,7 +39,7 @@
</p></blockquote></div>
<p>
where &#957; is the <span class="emphasis"><em>order</em></span> of the equation, and may be an arbitrary
where ν is the <span class="emphasis"><em>order</em></span> of the equation, and may be an arbitrary
real or complex number, although integer orders are the most common occurrence.
</p>
<p>
@@ -217,11 +217,10 @@ is also known as the spherical
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Finding Zeros of Bessel Functions of the First and Second Kinds</title>
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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@@ -70,7 +70,7 @@
<span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span> <span class="comment">// Destination for zeros.</span>
</pre>
<p>
There are also versions which allow control of the <a class="link" href="../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policies</a>
There are also versions which allow control of the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policies</a>
for error handling and precision.
</p>
<pre class="programlisting"> <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
@@ -107,21 +107,21 @@
<span class="phrase"><a name="math_toolkit.bessel.bessel_root.description"></a></span><a class="link" href="bessel_root.html#math_toolkit.bessel.bessel_root.description">Description</a>
</h5>
<p>
Every real order &#957; cylindrical Bessel and Neumann functions have an infinite
Every real order ν cylindrical Bessel and Neumann functions have an infinite
number of zeros on the positive real axis. The real zeros on the positive
real axis can be found by solving for the roots of
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>J<sub>&#957;</sub>(j<sub>&#957;, m</sub>) = 0</em></span>
<span class="emphasis"><em>J<sub>ν</sub>(j<sub>ν, m</sub>) = 0</em></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>Y<sub>&#957;</sub>(y<sub>&#957;, m</sub>) = 0</em></span>
<span class="emphasis"><em>Y<sub>ν</sub>(y<sub>ν, m</sub>) = 0</em></span>
</p></blockquote></div>
<p>
Here, <span class="emphasis"><em>j<sub>&#957;, m</sub></em></span> represents the <span class="emphasis"><em>m<sup>th</sup></em></span> root
of the cylindrical Bessel function of order <span class="emphasis"><em>&#957;</em></span>, and <span class="emphasis"><em>y<sub>&#957;,
Here, <span class="emphasis"><em>j<sub>ν, m</sub></em></span> represents the <span class="emphasis"><em>m<sup>th</sup></em></span> root
of the cylindrical Bessel function of order <span class="emphasis"><em>ν</em></span>, and <span class="emphasis"><em>y<sub>ν,
m</sub></em></span> represents the <span class="emphasis"><em>m<sup>th</sup></em></span> root of the cylindrical
Neumann function of order <span class="emphasis"><em>&#957;</em></span>.
Neumann function of order <span class="emphasis"><em>ν</em></span>.
</p>
<p>
The zeros or roots (values of <code class="computeroutput"><span class="identifier">x</span></code>
@@ -546,47 +546,47 @@
<span class="phrase"><a name="math_toolkit.bessel.bessel_root.implementation"></a></span><a class="link" href="bessel_root.html#math_toolkit.bessel.bessel_root.implementation">Implementation</a>
</h4>
<p>
Various methods are used to compute initial estimates for <span class="emphasis"><em>j<sub>&#957;, m</sub></em></span>
and <span class="emphasis"><em>y<sub>&#957;, m</sub></em></span> ; these are described in detail below.
Various methods are used to compute initial estimates for <span class="emphasis"><em>j<sub>ν, m</sub></em></span>
and <span class="emphasis"><em>y<sub>ν, m</sub></em></span> ; these are described in detail below.
</p>
<p>
After finding the initial estimate of a given root, its precision is subsequently
refined to the desired level using Newton-Raphson iteration from Boost.Math's
<a class="link" href="../roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der">root-finding with derivatives</a>
<a class="link" href="../roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schröder">root-finding with derivatives</a>
utilities combined with the functions <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
and <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>.
</p>
<p>
Newton iteration requires both <span class="emphasis"><em>J<sub>&#957;</sub>(x)</em></span> or <span class="emphasis"><em>Y<sub>&#957;</sub>(x)</em></span>
as well as its derivative. The derivatives of <span class="emphasis"><em>J<sub>&#957;</sub>(x)</em></span> and
<span class="emphasis"><em>Y<sub>&#957;</sub>(x)</em></span> with respect to <span class="emphasis"><em>x</em></span> are given
Newton iteration requires both <span class="emphasis"><em>J<sub>ν</sub>(x)</em></span> or <span class="emphasis"><em>Y<sub>ν</sub>(x)</em></span>
as well as its derivative. The derivatives of <span class="emphasis"><em>J<sub>ν</sub>(x)</em></span> and
<span class="emphasis"><em>Y<sub>ν</sub>(x)</em></span> with respect to <span class="emphasis"><em>x</em></span> are given
by M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, NBS
(1964). In particular,
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">d/<sub>dx</sub> <span class="emphasis"><em>J<sub>&#957;</sub>(x)</em></span> = <span class="emphasis"><em>J<sub>&#957;-1</sub>(x)</em></span>
- &#957; J<sub>&#957;</sub>(x) / x</span>
<span class="serif_italic">d/<sub>dx</sub> <span class="emphasis"><em>J<sub>ν</sub>(x)</em></span> = <span class="emphasis"><em>J<sub>ν-1</sub>(x)</em></span>
- ν J<sub>ν</sub>(x) / x</span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">d/<sub>dx</sub> <span class="emphasis"><em>Y<sub>&#957;</sub>(x)</em></span> = <span class="emphasis"><em>Y<sub>&#957;-1</sub>(x)</em></span>
- &#957; Y<sub>&#957;</sub>(x) / x</span>
<span class="serif_italic">d/<sub>dx</sub> <span class="emphasis"><em>Y<sub>ν</sub>(x)</em></span> = <span class="emphasis"><em>Y<sub>ν-1</sub>(x)</em></span>
- ν Y<sub>ν</sub>(x) / x</span>
</p></blockquote></div>
<p>
Enumeration of the rank of a root (in other words the index of a root) begins
with one and counts up, in other words <span class="emphasis"><em>m,=1,2,3,&#8230;</em></span> The
with one and counts up, in other words <span class="emphasis"><em>m,=1,2,3,…</em></span> The
value of the first root is always greater than zero.
</p>
<p>
For certain special parameters, cylindrical Bessel functions and cylindrical
Neumann functions have a root at the origin. For example, <span class="emphasis"><em>J<sub>&#957;</sub>(x)</em></span>
has a root at the origin for every positive order <span class="emphasis"><em>&#957; &gt; 0</em></span>,
and for every negative integer order <span class="emphasis"><em>&#957; = -n</em></span> with <span class="emphasis"><em>n
&#8712; &#8469; <sup>+</sup></em></span> and <span class="emphasis"><em>n &#8800; 0</em></span>.
Neumann functions have a root at the origin. For example, <span class="emphasis"><em>J<sub>ν</sub>(x)</em></span>
has a root at the origin for every positive order <span class="emphasis"><em>ν &gt; 0</em></span>,
and for every negative integer order <span class="emphasis"><em>ν = -n</em></span> with <span class="emphasis"><em>n
∈ ℕ <sup>+</sup></em></span> and <span class="emphasis"><em>n ≠ 0</em></span>.
</p>
<p>
In addition, <span class="emphasis"><em>Y<sub>&#957;</sub>(x)</em></span> has a root at the origin for every
negative half-integer order <span class="emphasis"><em>&#957; = -n/2</em></span>, with <span class="emphasis"><em>n
&#8712; &#8469; <sup>+</sup></em></span> and and <span class="emphasis"><em>n &#8800; 0</em></span>.
In addition, <span class="emphasis"><em>Y<sub>ν</sub>(x)</em></span> has a root at the origin for every
negative half-integer order <span class="emphasis"><em>ν = -n/2</em></span>, with <span class="emphasis"><em>n
∈ ℕ <sup>+</sup></em></span> and and <span class="emphasis"><em>n ≠ 0</em></span>.
</p>
<p>
For these special parameter values, the origin with a value of <span class="emphasis"><em>x
@@ -599,89 +599,89 @@
is made between positive order and negative order, and different methods
are used for these. In addition, different algorithms are used for the first
root <span class="emphasis"><em>m = 1</em></span> and for subsequent roots with higher rank
<span class="emphasis"><em>m &#8805; 2</em></span>. Furthermore, estimates of the roots for Bessel
functions with order above and below a cutoff at <span class="emphasis"><em>&#957; = 2.2</em></span>
<span class="emphasis"><em>m ≥ 2</em></span>. Furthermore, estimates of the roots for Bessel
functions with order above and below a cutoff at <span class="emphasis"><em>ν = 2.2</em></span>
are calculated with different methods.
</p>
<p>
Calculations of the estimates of <span class="emphasis"><em>j<sub>&#957;,1</sub></em></span> and <span class="emphasis"><em>y<sub>&#957;,1</sub></em></span>
with <span class="emphasis"><em>0 &#8804; &#957; &lt; 2.2</em></span> use empirically tabulated values. The
Calculations of the estimates of <span class="emphasis"><em>j<sub>ν,1</sub></em></span> and <span class="emphasis"><em>y<sub>ν,1</sub></em></span>
with <span class="emphasis"><em>0 ≤ ν &lt; 2.2</em></span> use empirically tabulated values. The
coefficients for these have been generated by a computer algebra system.
</p>
<p>
Calculations of the estimates of <span class="emphasis"><em>j<sub>&#957;,1</sub></em></span> and <span class="emphasis"><em>y<sub>&#957;,1</sub></em></span>
with <span class="emphasis"><em>&#957;&#8805; 2.2</em></span> use Eqs.9.5.14 and 9.5.15 in M. Abramowitz
Calculations of the estimates of <span class="emphasis"><em>j<sub>ν,1</sub></em></span> and <span class="emphasis"><em>y<sub>ν,1</sub></em></span>
with <span class="emphasis"><em>ν≥ 2.2</em></span> use Eqs.9.5.14 and 9.5.15 in M. Abramowitz
and I. A. Stegun, Handbook of Mathematical Functions, NBS (1964).
</p>
<p>
In particular,
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">j<sub>&#957;,1</sub> &#8773; &#957; + 1.85575 &#957;<sup>&#8531;</sup> + 1.033150 &#957;<sup>-&#8531;</sup> - 0.00397 &#957;<sup>-1</sup> - 0.0908
&#957;<sup>-5/3</sup> + 0.043 &#957;<sup>-7/3</sup> + &#8230;</span>
<span class="serif_italic">j<sub>ν,1</sub> ≅ ν + 1.85575 ν<sup>⅓</sup> + 1.033150 ν<sup>-⅓</sup> - 0.00397 ν<sup>-1</sup> - 0.0908
ν<sup>-5/3</sup> + 0.043 ν<sup>-7/3</sup> + …</span>
</p></blockquote></div>
<p>
and
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">y<sub>&#957;,1</sub> &#8773; &#957; + 0.93157 &#957;<sup>&#8531;</sup> + 0.26035 &#957;<sup>-&#8531;</sup> + 0.01198 &#957;<sup>-1</sup> - 0.0060
&#957;<sup>-5/3</sup> - 0.001 &#957;<sup>-7/3</sup> + &#8230;</span>
<span class="serif_italic">y<sub>ν,1</sub> ≅ ν + 0.93157 ν<sup>⅓</sup> + 0.26035 ν<sup>-⅓</sup> + 0.01198 ν<sup>-1</sup> - 0.0060
ν<sup>-5/3</sup> - 0.001 ν<sup>-7/3</sup> + …</span>
</p></blockquote></div>
<p>
Calculations of the estimates of <span class="emphasis"><em>j<sub>&#957;, m</sub></em></span> and <span class="emphasis"><em>y<sub>&#957;,
m</sub></em></span> with rank <span class="emphasis"><em>m &gt; 2</em></span> and <span class="emphasis"><em>0 &#8804; &#957; &lt;
Calculations of the estimates of <span class="emphasis"><em>j<sub>ν, m</sub></em></span> and <span class="emphasis"><em>y<sub>ν,
m</sub></em></span> with rank <span class="emphasis"><em>m &gt; 2</em></span> and <span class="emphasis"><em>0 ≤ ν &lt;
2.2</em></span> use McMahon's approximation, as described in M. Abramowitz
and I. A. Stegan, Section 9.5 and 9.5.12. In particular,
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>j<sub>&#957;,m</sub>, y<sub>&#957;,m</sub> &#8773;</em></span>
<span class="emphasis"><em>j<sub>ν,m</sub>, y<sub>ν,m</sub> ≅</em></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
&#946; - (&#956;-1) / 8&#946;
β - (μ-1) / 8β
</p></blockquote></div></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>- 4(&#956;-1)(7&#956; - 31) / 3(8&#946;)<sup>3</sup></em></span>
<span class="emphasis"><em>- 4(μ-1)(7μ - 31) / 3(8β)<sup>3</sup></em></span>
</p></blockquote></div></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>-32(&#956;-1)(83&#956;&#178; - 982&#956; + 3779) / 15(8&#946;)<sup>5</sup></em></span>
<span class="emphasis"><em>-32(μ-1)(83μ² - 982μ + 3779) / 15(8β)<sup>5</sup></em></span>
</p></blockquote></div></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>-64(&#956;-1)(6949&#956;<sup>3</sup> - 153855&#956;&#178; + 1585743&#956;- 6277237) / 105(8a)<sup>7</sup></em></span>
<span class="emphasis"><em>-64(μ-1)(6949μ<sup>3</sup> - 153855μ² + 1585743μ- 6277237) / 105(8a)<sup>7</sup></em></span>
</p></blockquote></div></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>- &#8230;</em></span> &#8193; (5)
<span class="emphasis"><em>- …</em></span>   (5)
</p></blockquote></div></blockquote></div>
<p>
where <span class="emphasis"><em>&#956; = 4&#957;<sup>2</sup></em></span> and <span class="emphasis"><em>&#946; = (m + &#189;&#957; - &#188;)&#960;</em></span> for
<span class="emphasis"><em>j<sub>&#957;,m</sub></em></span> and <span class="emphasis"><em>&#946; = (m + &#189;&#957; -&#190;)&#960; for <span class="emphasis"><em>y<sub>&#957;,m</sub></em></span></em></span>.
where <span class="emphasis"><em>μ = 4ν<sup>2</sup></em></span> and <span class="emphasis"><em>β = (m + ½ν - ¼)π</em></span> for
<span class="emphasis"><em>j<sub>ν,m</sub></em></span> and <span class="emphasis"><em>β = (m + ½ν -¾)π for <span class="emphasis"><em>y<sub>ν,m</sub></em></span></em></span>.
</p>
<p>
Calculations of the estimates of <span class="emphasis"><em>j<sub>&#957;, m</sub></em></span> and <span class="emphasis"><em>y<sub>&#957;,
m</sub></em></span> with <span class="emphasis"><em>&#957; &#8805; 2.2</em></span> use one term in the asymptotic
Calculations of the estimates of <span class="emphasis"><em>j<sub>ν, m</sub></em></span> and <span class="emphasis"><em>y<sub>ν,
m</sub></em></span> with <span class="emphasis"><em>ν ≥ 2.2</em></span> use one term in the asymptotic
expansion given in Eq.9.5.22 and top line of Eq.9.5.26 combined with Eq.
9.3.39, all in M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,
NBS (1964) explicit and easy-to-understand treatment for asymptotic expansion
of zeros. The latter two equations are expressed for argument <span class="emphasis"><em>(x)</em></span>
greater than one. (Olver also gives the series form of the equations in
<a href="http://dlmf.nist.gov/10.21#vi" target="_top">&#167;10.21(vi) McMahon's Asymptotic
<a href="http://dlmf.nist.gov/10.21#vi" target="_top">§10.21(vi) McMahon's Asymptotic
Expansions for Large Zeros</a> - using slightly different variable names).
</p>
<p>
In summary,
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">j<sub>&#957;, m</sub> &#8764; &#957;x(-&#950;) + f<sub>1</sub>(-&#950;/&#957;)</span>
<span class="serif_italic">j<sub>ν, m</sub> ∼ νx(-ζ) + f<sub>1</sub>(-ζ/ν)</span>
</p></blockquote></div>
<p>
where <span class="emphasis"><em>-&#950; = &#957;<sup>-2/3</sup>a<sub>m</sub></em></span> and <span class="emphasis"><em>a<sub>m</sub></em></span> is the absolute
where <span class="emphasis"><em>-ζ = ν<sup>-2/3</sup>a<sub>m</sub></em></span> and <span class="emphasis"><em>a<sub>m</sub></em></span> is the absolute
value of the <span class="emphasis"><em>m<sup>th</sup></em></span> root of <span class="emphasis"><em>Ai(x)</em></span>
on the negative real axis.
</p>
<p>
Here <span class="emphasis"><em>x = x(-&#950;)</em></span> is the inverse of the function
Here <span class="emphasis"><em>x = x(-ζ)</em></span> is the inverse of the function
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">&#8532;(-&#950;)<sup>3/2</sup> = &#8730;(x&#178; - 1) - cos&#8315;&#185;(1/x)</span>
<span class="serif_italic">⅔(-ζ)<sup>3/2</sup> = √(x² - 1) - cos⁻¹(1/x)</span>
</p></blockquote></div>
<p>
(7)
@@ -690,48 +690,48 @@
Furthermore,
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">f<sub>1</sub>(-&#950;) = &#189;x(-&#950;) {h(-&#950;)}&#178; &#8901; b<sub>0</sub>(-&#950;)</span>
<span class="serif_italic">f<sub>1</sub>(-ζ) = ½x(-ζ) {h(-ζ)}² ⋅ b<sub>0</sub>(-ζ)</span>
</p></blockquote></div>
<p>
where
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">h(-&#950;) = {4(-&#950;) / (x&#178; - 1)}<sup>4</sup></span>
<span class="serif_italic">h(-ζ) = {4(-ζ) / (x² - 1)}<sup>4</sup></span>
</p></blockquote></div>
<p>
and
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">b<sub>0</sub>(-&#950;) = -5/(48&#950;&#178;) + 1/(-&#950;)<sup>&#189;</sup> &#8901; { 5/(24(x<sup>2</sup>-1)<sup>3/2</sup>) +
1/(8(x<sup>2</sup>-1)<sup>&#189;)</sup>}</span>
<span class="serif_italic">b<sub>0</sub>(-ζ) = -5/(48ζ²) + 1/(-ζ)<sup>½</sup> ⋅ { 5/(24(x<sup>2</sup>-1)<sup>3/2</sup>) +
1/(8(x<sup>2</sup>-1)<sup>½)</sup>}</span>
</p></blockquote></div>
<p>
When solving for <span class="emphasis"><em>x(-&#950;)</em></span> in Eq. 7 above, the right-hand-side
When solving for <span class="emphasis"><em>x(-ζ)</em></span> in Eq. 7 above, the right-hand-side
is expanded to order 2 in a Taylor series for large <span class="emphasis"><em>x</em></span>.
This results in
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">&#8532;(-&#950;)<sup>3/2</sup> &#8776; x + 1/2x - &#960;/2</span>
<span class="serif_italic">⅔(-ζ)<sup>3/2</sup> ≈ x + 1/2x - π/2</span>
</p></blockquote></div>
<p>
The positive root of the resulting quadratic equation is used to find an
initial estimate <span class="emphasis"><em>x(-&#950;)</em></span>. This initial estimate is subsequently
initial estimate <span class="emphasis"><em>x(-ζ)</em></span>. This initial estimate is subsequently
refined with several steps of Newton-Raphson iteration in Eq. 7.
</p>
<p>
Estimates of the roots of cylindrical Bessel functions of negative order
on the positive real axis are found using interlacing relations. For example,
the <span class="emphasis"><em>m<sup>th</sup></em></span> root of the cylindrical Bessel function <span class="emphasis"><em>j<sub>-&#957;,m</sub></em></span>
the <span class="emphasis"><em>m<sup>th</sup></em></span> root of the cylindrical Bessel function <span class="emphasis"><em>j<sub>-ν,m</sub></em></span>
is bracketed by the <span class="emphasis"><em>m<sup>th</sup></em></span> root and the <span class="emphasis"><em>(m+1)<sup>th</sup></em></span>
root of the Bessel function of corresponding positive integer order. In other
words,
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">j<sub>n&#957;, m</sub> &lt; j<sub>-&#957;, m</sub> &lt; j<sub>n&#957;, m+1</sub></span>
<span class="serif_italic">j<sub>nν, m</sub> &lt; j<sub>-ν, m</sub> &lt; j<sub>nν, m+1</sub></span>
</p></blockquote></div>
<p>
where <span class="emphasis"><em>m &gt; 1</em></span> and <span class="emphasis"><em>n<sub>&#957;</sub></em></span> represents
the integral floor of the absolute value of <span class="emphasis"><em>|-&#957;|</em></span>.
where <span class="emphasis"><em>m &gt; 1</em></span> and <span class="emphasis"><em>n<sub>ν</sub></em></span> represents
the integral floor of the absolute value of <span class="emphasis"><em>|-ν|</em></span>.
</p>
<p>
Similar bracketing relations are used to find estimates of the roots of Neumann
@@ -742,7 +742,7 @@
Bracketing relations do not hold for the first root of cylindrical Bessel
functions and cylindrical Neumann functions with negative order. Therefore,
iterative algorithms combined with root-finding via bisection are used to
localize <span class="emphasis"><em>j<sub>-&#957;,1</sub></em></span> and <span class="emphasis"><em>y<sub>-&#957;,1</sub></em></span>.
localize <span class="emphasis"><em>j<sub>-ν,1</sub></em></span> and <span class="emphasis"><em>y<sub>-ν,1</sub></em></span>.
</p>
<h4>
<a name="math_toolkit.bessel.bessel_root.h7"></a>
@@ -757,11 +757,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

337
doc/html/math_toolkit/bessel/mbessel.html
View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Modified Bessel Functions of the First and Second Kinds</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../bessel.html" title="Bessel Functions">
<link rel="prev" href="bessel_root.html" title="Finding Zeros of Bessel Functions of the First and Second Kinds">
<link rel="next" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">
@@ -37,14 +37,14 @@
<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>
<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">cyl_bessel_i</span><span class="special">(</span><span class="identifier">T1</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">&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;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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">cyl_bessel_i</span><span class="special">(</span><span class="identifier">T1</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;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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">cyl_bessel_i</span><span class="special">(</span><span class="identifier">T1</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">&amp;);</span>
<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>
<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">cyl_bessel_k</span><span class="special">(</span><span class="identifier">T1</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">&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;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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">cyl_bessel_k</span><span class="special">(</span><span class="identifier">T1</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;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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">cyl_bessel_k</span><span class="special">(</span><span class="identifier">T1</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">&amp;);</span>
</pre>
<h5>
<a name="math_toolkit.bessel.mbessel.h1"></a>
@@ -80,9 +80,9 @@
an integer.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. 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&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">policy
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
documentation for more details</a>.
</p>
<p>
@@ -134,7 +134,7 @@
of the worst problem cases in other libraries:
</p>
<div class="table">
<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_i_integer_orders_"></a><p class="title"><b>Table&#160;8.44.&#160;Error rates for cyl_bessel_i (integer orders)</b></p>
<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_i_integer_orders_"></a><p class="title"><b>Table 8.44. Error rates for cyl_bessel_i (integer orders)</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_i (integer orders)">
<colgroup>
<col>
@@ -176,27 +176,27 @@
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.79&#949; (Mean = 0.482&#949;))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.52&#949; (Mean = 0.622&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_integer_orders__Rmath_3_2_3_Bessel_I0_Mathworld_Data_Integer_Version_">And
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.79ε (Mean = 0.482ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.52ε (Mean = 0.622ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_integer_orders__Rmath_3_2_3_Bessel_I0_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.95&#949; (Mean = 0.738&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 8.49&#949; (Mean = 3.46&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i_integer_orders___cmath__Bessel_I0_Mathworld_Data_Integer_Version_">And
<span class="blue">Max = 1.95ε (Mean = 0.738ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 8.49ε (Mean = 3.46ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i_integer_orders___cmath__Bessel_I0_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.95&#949; (Mean = 0.661&#949;)</span>
<span class="blue">Max = 1.95ε (Mean = 0.661ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.762&#949; (Mean = 0.329&#949;)</span>
<span class="blue">Max = 0.762ε (Mean = 0.329ε)</span>
</p>
</td>
</tr>
@@ -208,27 +208,27 @@
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.82&#949; (Mean = 0.456&#949;))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.53&#949; (Mean = 0.483&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_integer_orders__Rmath_3_2_3_Bessel_I1_Mathworld_Data_Integer_Version_">And
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.82ε (Mean = 0.456ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.53ε (Mean = 0.483ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_integer_orders__Rmath_3_2_3_Bessel_I1_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.64&#949; (Mean = 0.202&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 5&#949; (Mean = 2.15&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i_integer_orders___cmath__Bessel_I1_Mathworld_Data_Integer_Version_">And
<span class="blue">Max = 0.64ε (Mean = 0.202ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 5ε (Mean = 2.15ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i_integer_orders___cmath__Bessel_I1_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.64&#949; (Mean = 0.202&#949;)</span>
<span class="blue">Max = 0.64ε (Mean = 0.202ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.767&#949; (Mean = 0.398&#949;)</span>
<span class="blue">Max = 0.767ε (Mean = 0.398ε)</span>
</p>
</td>
</tr>
@@ -240,28 +240,28 @@
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 5.15&#949; (Mean = 2.13&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_integer_orders__GSL_2_1_Bessel_In_Mathworld_Data_Integer_Version_">And
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 5.15ε (Mean = 2.13ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_integer_orders__GSL_2_1_Bessel_In_Mathworld_Data_Integer_Version_">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 1.73&#949; (Mean = 0.601&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_integer_orders__Rmath_3_2_3_Bessel_In_Mathworld_Data_Integer_Version_">And
Max = 1.73ε (Mean = 0.601ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_integer_orders__Rmath_3_2_3_Bessel_In_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.8&#949; (Mean = 1.33&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 430&#949; (Mean = 163&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i_integer_orders___cmath__Bessel_In_Mathworld_Data_Integer_Version_">And
<span class="blue">Max = 1.8ε (Mean = 1.33ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 430ε (Mean = 163ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i_integer_orders___cmath__Bessel_In_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 463&#949; (Mean = 140&#949;)</span>
<span class="blue">Max = 463ε (Mean = 140ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.46&#949; (Mean = 1.32&#949;)</span>
<span class="blue">Max = 3.46ε (Mean = 1.32ε)</span>
</p>
</td>
</tr>
@@ -269,7 +269,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_i"></a><p class="title"><b>Table&#160;8.45.&#160;Error rates for cyl_bessel_i</b></p>
<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_i"></a><p class="title"><b>Table 8.45. Error rates for cyl_bessel_i</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_i">
<colgroup>
<col>
@@ -311,28 +311,28 @@
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 270&#949; (Mean = 91.6&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_I0_Mathworld_Data">And
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 270ε (Mean = 91.6ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_I0_Mathworld_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 1.52&#949; (Mean = 0.622&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_Rmath_3_2_3_Bessel_I0_Mathworld_Data">And
Max = 1.52ε (Mean = 0.622ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_Rmath_3_2_3_Bessel_I0_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.95&#949; (Mean = 0.738&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 8.49&#949; (Mean = 3.46&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_I0_Mathworld_Data">And
<span class="blue">Max = 1.95ε (Mean = 0.738ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 8.49ε (Mean = 3.46ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_I0_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.95&#949; (Mean = 0.661&#949;)</span>
<span class="blue">Max = 1.95ε (Mean = 0.661ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.762&#949; (Mean = 0.329&#949;)</span>
<span class="blue">Max = 0.762ε (Mean = 0.329ε)</span>
</p>
</td>
</tr>
@@ -344,28 +344,28 @@
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 128&#949; (Mean = 41&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_I1_Mathworld_Data">And
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 128ε (Mean = 41ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_I1_Mathworld_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 1.53&#949; (Mean = 0.483&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_Rmath_3_2_3_Bessel_I1_Mathworld_Data">And
Max = 1.53ε (Mean = 0.483ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_Rmath_3_2_3_Bessel_I1_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.64&#949; (Mean = 0.202&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 5&#949; (Mean = 2.15&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_I1_Mathworld_Data">And
<span class="blue">Max = 0.64ε (Mean = 0.202ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 5ε (Mean = 2.15ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_I1_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.64&#949; (Mean = 0.202&#949;)</span>
<span class="blue">Max = 0.64ε (Mean = 0.202ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.767&#949; (Mean = 0.398&#949;)</span>
<span class="blue">Max = 0.767ε (Mean = 0.398ε)</span>
</p>
</td>
</tr>
@@ -377,28 +377,28 @@
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 2.31&#949; (Mean = 0.838&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_In_Mathworld_Data">And
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 2.31ε (Mean = 0.838ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_In_Mathworld_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 1.73&#949; (Mean = 0.601&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_Rmath_3_2_3_Bessel_In_Mathworld_Data">And
Max = 1.73ε (Mean = 0.601ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_Rmath_3_2_3_Bessel_In_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.8&#949; (Mean = 1.33&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 430&#949; (Mean = 163&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_In_Mathworld_Data">And
<span class="blue">Max = 1.8ε (Mean = 1.33ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 430ε (Mean = 163ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_In_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 463&#949; (Mean = 140&#949;)</span>
<span class="blue">Max = 463ε (Mean = 140ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.46&#949; (Mean = 1.32&#949;)</span>
<span class="blue">Max = 3.46ε (Mean = 1.32ε)</span>
</p>
</td>
</tr>
@@ -410,27 +410,27 @@
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 5.95&#949; (Mean = 2.08&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_Iv_Mathworld_Data">And
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 5.95ε (Mean = 2.08ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_Iv_Mathworld_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 3.53&#949; (Mean = 1.39&#949;))
Max = 3.53ε (Mean = 1.39ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 4.12&#949; (Mean = 1.85&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 616&#949; (Mean = 221&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_Iv_Mathworld_Data">And
<span class="blue">Max = 4.12ε (Mean = 1.85ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 616ε (Mean = 221ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_Iv_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 4.12&#949; (Mean = 1.95&#949;)</span>
<span class="blue">Max = 4.12ε (Mean = 1.95ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.97&#949; (Mean = 1.24&#949;)</span>
<span class="blue">Max = 2.97ε (Mean = 1.24ε)</span>
</p>
</td>
</tr>
@@ -442,26 +442,26 @@
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 261&#949; (Mean = 53.2&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_In_Random_Data">And
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 261ε (Mean = 53.2ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_In_Random_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 7.37&#949; (Mean = 2.4&#949;))
Max = 7.37ε (Mean = 2.4ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 4.62&#949; (Mean = 1.06&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 645&#949; (Mean = 132&#949;))
<span class="blue">Max = 4.62ε (Mean = 1.06ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 645ε (Mean = 132ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 176&#949; (Mean = 39.1&#949;)</span>
<span class="blue">Max = 176ε (Mean = 39.1ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 9.67&#949; (Mean = 1.88&#949;)</span>
<span class="blue">Max = 9.67ε (Mean = 1.88ε)</span>
</p>
</td>
</tr>
@@ -473,29 +473,29 @@
</td>
<td>
<p>
<span class="blue">Max = 0.661&#949; (Mean = 0.0441&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 6.18e+03&#949; (Mean = 1.55e+03&#949;)
<span class="blue">Max = 0.661ε (Mean = 0.0441ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 6.18e+03ε (Mean = 1.55e+03ε)
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_Iv_Random_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
<span class="red">Max = 4.28e+08&#949; (Mean = 2.85e+07&#949;))</span>
<span class="red">Max = 4.28e+08ε (Mean = 2.85e+07ε))</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 8.35&#949; (Mean = 1.62&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.05e+03&#949; (Mean = 224&#949;)
<span class="blue">Max = 8.35ε (Mean = 1.62ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.05e+03ε (Mean = 224ε)
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_Iv_Random_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 283&#949; (Mean = 88.4&#949;)</span>
<span class="blue">Max = 283ε (Mean = 88.4ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 7.46&#949; (Mean = 1.71&#949;)</span>
<span class="blue">Max = 7.46ε (Mean = 1.71ε)</span>
</p>
</td>
</tr>
@@ -507,28 +507,28 @@
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 37&#949; (Mean = 18&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_Iv_Mathworld_Data_large_values_">And
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 37ε (Mean = 18ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_Iv_Mathworld_Data_large_values_">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
<span class="red">Max = 3.77e+168&#949; (Mean = 2.39e+168&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_Rmath_3_2_3_Bessel_Iv_Mathworld_Data_large_values_">And
<span class="red">Max = 3.77e+168ε (Mean = 2.39e+168ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_Rmath_3_2_3_Bessel_Iv_Mathworld_Data_large_values_">And
other failures.</a>)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 14.7&#949; (Mean = 6.66&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 118&#949; (Mean = 57.2&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_Iv_Mathworld_Data_large_values_">And
<span class="blue">Max = 14.7ε (Mean = 6.66ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 118ε (Mean = 57.2ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_Iv_Mathworld_Data_large_values_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 14.7&#949; (Mean = 6.59&#949;)</span>
<span class="blue">Max = 14.7ε (Mean = 6.59ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.67&#949; (Mean = 1.64&#949;)</span>
<span class="blue">Max = 3.67ε (Mean = 1.64ε)</span>
</p>
</td>
</tr>
@@ -536,7 +536,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_k_integer_orders_"></a><p class="title"><b>Table&#160;8.46.&#160;Error rates for cyl_bessel_k (integer orders)</b></p>
<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_k_integer_orders_"></a><p class="title"><b>Table 8.46. Error rates for cyl_bessel_k (integer orders)</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_k (integer orders)">
<colgroup>
<col>
@@ -578,25 +578,25 @@
</td>
<td>
<p>
<span class="blue">Max = 0.833&#949; (Mean = 0.436&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 9.33&#949; (Mean = 3.25&#949;))
<span class="blue">Max = 0.833ε (Mean = 0.436ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 9.33ε (Mean = 3.25ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.2&#949; (Mean = 0.733&#949;))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0.833&#949; (Mean = 0.601&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.2ε (Mean = 0.733ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0.833ε (Mean = 0.601ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.833&#949; (Mean = 0.436&#949;)</span>
<span class="blue">Max = 0.833ε (Mean = 0.436ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.833&#949; (Mean = 0.552&#949;)</span>
<span class="blue">Max = 0.833ε (Mean = 0.552ε)</span>
</p>
</td>
</tr>
@@ -608,25 +608,25 @@
</td>
<td>
<p>
<span class="blue">Max = 0.786&#949; (Mean = 0.329&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 8.94&#949; (Mean = 3.19&#949;))
<span class="blue">Max = 0.786ε (Mean = 0.329ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 8.94ε (Mean = 3.19ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.626&#949; (Mean = 0.333&#949;))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0.894&#949; (Mean = 0.516&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.626ε (Mean = 0.333ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0.894ε (Mean = 0.516ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.786&#949; (Mean = 0.329&#949;)</span>
<span class="blue">Max = 0.786ε (Mean = 0.329ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.786&#949; (Mean = 0.39&#949;)</span>
<span class="blue">Max = 0.786ε (Mean = 0.39ε)</span>
</p>
</td>
</tr>
@@ -638,26 +638,26 @@
</td>
<td>
<p>
<span class="blue">Max = 2.6&#949; (Mean = 1.21&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 12.9&#949; (Mean = 4.91&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k_integer_orders___cmath__Bessel_Kn_Mathworld_Data_Integer_Version_">And
<span class="blue">Max = 2.6ε (Mean = 1.21ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 12.9ε (Mean = 4.91ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k_integer_orders___cmath__Bessel_Kn_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 168&#949; (Mean = 59.5&#949;))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 8.48&#949; (Mean = 2.98&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 168ε (Mean = 59.5ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 8.48ε (Mean = 2.98ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.6&#949; (Mean = 1.21&#949;)</span>
<span class="blue">Max = 2.6ε (Mean = 1.21ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.63&#949; (Mean = 1.46&#949;)</span>
<span class="blue">Max = 3.63ε (Mean = 1.46ε)</span>
</p>
</td>
</tr>
@@ -665,7 +665,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_k"></a><p class="title"><b>Table&#160;8.47.&#160;Error rates for cyl_bessel_k</b></p>
<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_k"></a><p class="title"><b>Table 8.47. Error rates for cyl_bessel_k</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_k">
<colgroup>
<col>
@@ -707,25 +707,25 @@
</td>
<td>
<p>
<span class="blue">Max = 0.833&#949; (Mean = 0.436&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 9.33&#949; (Mean = 3.25&#949;))
<span class="blue">Max = 0.833ε (Mean = 0.436ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 9.33ε (Mean = 3.25ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 6.04&#949; (Mean = 2.16&#949;))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0.833&#949; (Mean = 0.601&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 6.04ε (Mean = 2.16ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0.833ε (Mean = 0.601ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.833&#949; (Mean = 0.436&#949;)</span>
<span class="blue">Max = 0.833ε (Mean = 0.436ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.833&#949; (Mean = 0.552&#949;)</span>
<span class="blue">Max = 0.833ε (Mean = 0.552ε)</span>
</p>
</td>
</tr>
@@ -737,25 +737,25 @@
</td>
<td>
<p>
<span class="blue">Max = 0.786&#949; (Mean = 0.329&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 8.94&#949; (Mean = 3.19&#949;))
<span class="blue">Max = 0.786ε (Mean = 0.329ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 8.94ε (Mean = 3.19ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 6.26&#949; (Mean = 2.21&#949;))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0.894&#949; (Mean = 0.516&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 6.26ε (Mean = 2.21ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0.894ε (Mean = 0.516ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.786&#949; (Mean = 0.329&#949;)</span>
<span class="blue">Max = 0.786ε (Mean = 0.329ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.786&#949; (Mean = 0.39&#949;)</span>
<span class="blue">Max = 0.786ε (Mean = 0.39ε)</span>
</p>
</td>
</tr>
@@ -767,27 +767,27 @@
</td>
<td>
<p>
<span class="blue">Max = 2.6&#949; (Mean = 1.21&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 12.9&#949; (Mean = 4.91&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k__cmath__Bessel_Kn_Mathworld_Data">And
<span class="blue">Max = 2.6ε (Mean = 1.21ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 12.9ε (Mean = 4.91ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k__cmath__Bessel_Kn_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 3.36&#949; (Mean = 1.43&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_k_GSL_2_1_Bessel_Kn_Mathworld_Data">And
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 3.36ε (Mean = 1.43ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_k_GSL_2_1_Bessel_Kn_Mathworld_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 8.48&#949; (Mean = 2.98&#949;))
Max = 8.48ε (Mean = 2.98ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.6&#949; (Mean = 1.21&#949;)</span>
<span class="blue">Max = 2.6ε (Mean = 1.21ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.63&#949; (Mean = 1.46&#949;)</span>
<span class="blue">Max = 3.63ε (Mean = 1.46ε)</span>
</p>
</td>
</tr>
@@ -799,27 +799,27 @@
</td>
<td>
<p>
<span class="blue">Max = 3.58&#949; (Mean = 2.39&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 13&#949; (Mean = 4.81&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k__cmath__Bessel_Kv_Mathworld_Data">And
<span class="blue">Max = 3.58ε (Mean = 2.39ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 13ε (Mean = 4.81ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k__cmath__Bessel_Kv_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 5.47&#949; (Mean = 2.04&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_k_GSL_2_1_Bessel_Kv_Mathworld_Data">And
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 5.47ε (Mean = 2.04ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_k_GSL_2_1_Bessel_Kv_Mathworld_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 3.15&#949; (Mean = 1.35&#949;))
Max = 3.15ε (Mean = 1.35ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 5.21&#949; (Mean = 2.53&#949;)</span>
<span class="blue">Max = 5.21ε (Mean = 2.53ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 4.78&#949; (Mean = 2.19&#949;)</span>
<span class="blue">Max = 4.78ε (Mean = 2.19ε)</span>
</p>
</td>
</tr>
@@ -831,27 +831,27 @@
</td>
<td>
<p>
<span class="blue">Max = 42.3&#949; (Mean = 21&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 42.3&#949; (Mean = 19.8&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k__cmath__Bessel_Kv_Mathworld_Data_large_values_">And
<span class="blue">Max = 42.3ε (Mean = 21ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 42.3ε (Mean = 19.8ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k__cmath__Bessel_Kv_Mathworld_Data_large_values_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 308&#949; (Mean = 142&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_k_GSL_2_1_Bessel_Kv_Mathworld_Data_large_values_">And
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 308ε (Mean = 142ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_k_GSL_2_1_Bessel_Kv_Mathworld_Data_large_values_">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 84.6&#949; (Mean = 37.8&#949;))
Max = 84.6ε (Mean = 37.8ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 42.3&#949; (Mean = 21&#949;)</span>
<span class="blue">Max = 42.3ε (Mean = 21ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 59.8&#949; (Mean = 26.9&#949;)</span>
<span class="blue">Max = 59.8ε (Mean = 26.9ε)</span>
</p>
</td>
</tr>
@@ -863,27 +863,27 @@
</td>
<td>
<p>
<span class="blue">Max = 4.55&#949; (Mean = 1.12&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 13.9&#949; (Mean = 2.91&#949;))
<span class="blue">Max = 4.55ε (Mean = 1.12ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 13.9ε (Mean = 2.91ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.764&#949; (Mean = 0.0348&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 8.71&#949; (Mean = 1.76&#949;)
<span class="blue">Max = 0.764ε (Mean = 0.0348ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 8.71ε (Mean = 1.76ε)
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_k_GSL_2_1_Bessel_Kn_Random_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 7.47&#949; (Mean = 1.34&#949;))
Max = 7.47ε (Mean = 1.34ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 4.55&#949; (Mean = 1.12&#949;)</span>
<span class="blue">Max = 4.55ε (Mean = 1.12ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 9.34&#949; (Mean = 1.7&#949;)</span>
<span class="blue">Max = 9.34ε (Mean = 1.7ε)</span>
</p>
</td>
</tr>
@@ -895,28 +895,28 @@
</td>
<td>
<p>
<span class="blue">Max = 7.88&#949; (Mean = 1.48&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 13.6&#949; (Mean = 2.68&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k__cmath__Bessel_Kv_Random_Data">And
<span class="blue">Max = 7.88ε (Mean = 1.48ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 13.6ε (Mean = 2.68ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k__cmath__Bessel_Kv_Random_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.507&#949; (Mean = 0.0313&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 9.71&#949; (Mean = 1.47&#949;)
<span class="blue">Max = 0.507ε (Mean = 0.0313ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 9.71ε (Mean = 1.47ε)
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_k_GSL_2_1_Bessel_Kv_Random_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 7.37&#949; (Mean = 1.49&#949;))
Max = 7.37ε (Mean = 1.49ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 7.88&#949; (Mean = 1.47&#949;)</span>
<span class="blue">Max = 7.88ε (Mean = 1.47ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 8.33&#949; (Mean = 1.62&#949;)</span>
<span class="blue">Max = 8.33ε (Mean = 1.62ε)</span>
</p>
</td>
</tr>
@@ -985,9 +985,9 @@
The following are handled as special cases first:
</p>
<p>
When computing I<sub>v</sub> for <span class="emphasis"><em>x &lt; 0</em></span>, then &#957; must be an integer
or a domain error occurs. If &#957; is an integer, then the function is odd if &#957; is
odd and even if &#957; is even, and we can reflect to <span class="emphasis"><em>x &gt; 0</em></span>.
When computing I<sub>v</sub> for <span class="emphasis"><em>x &lt; 0</em></span>, then ν must be an integer
or a domain error occurs. If ν is an integer, then the function is odd if ν is
odd and even if ν is even, and we can reflect to <span class="emphasis"><em>x &gt; 0</em></span>.
</p>
<p>
For I<sub>v</sub> with v equal to 0, 1 or 0.5 are handled as special cases.
@@ -1039,7 +1039,7 @@
The 0.5 case is a simple trigonometric function:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
I<sub>0.5</sub>(x) = sqrt(2 / &#960;x) * sinh(x)
I<sub>0.5</sub>(x) = sqrt(2 / πx) * sinh(x)
</p></blockquote></div>
<p>
For K<sub>v</sub> with <span class="emphasis"><em>v</em></span> an integer, the result is calculated using
@@ -1063,7 +1063,7 @@
</p></blockquote></div>
<p>
In the general case, we first normalize &#957; to [<code class="literal">0, [inf]</code>)
In the general case, we first normalize ν to [<code class="literal">0, [inf]</code>)
with the help of the reflection formulae:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
@@ -1075,9 +1075,9 @@
</p></blockquote></div>
<p>
Let &#956; = &#957; - floor(&#957; + 1/2), then &#956; is the fractional part of &#957; such that |&#956;| &lt;= 1/2
(we need this for convergence later). The idea is to calculate K<sub>&#956;</sub>(x) and K<sub>&#956;+1</sub>(x),
and use them to obtain I<sub>&#957;</sub>(x) and K<sub>&#957;</sub>(x).
Let μ = ν - floor(ν + 1/2), then μ is the fractional part of ν such that |μ| &lt;= 1/2
(we need this for convergence later). The idea is to calculate K<sub>μ</sub>(x) and K<sub>μ+1</sub>(x),
and use them to obtain I<sub>ν</sub>(x) and K<sub>ν</sub>(x).
</p>
<p>
The algorithm is proposed by Temme in
@@ -1124,7 +1124,7 @@
<p>
When <span class="emphasis"><em>x</em></span> is large (<span class="emphasis"><em>x</em></span> &gt; 2), both
continued fractions converge (CF1 may be slow for really large <span class="emphasis"><em>x</em></span>).
K<sub>&#956;</sub> and K<sub>&#956;+1</sub>
K<sub>μ</sub> and K<sub>μ+1</sub>
can be calculated by
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
@@ -1173,20 +1173,19 @@ are also computed by recursions (involving gamma functions), but
vol 19, 324 (1975).
</p></blockquote></div>
<p>
Note: Temme's series converge only for |&#956;| &lt;= 1/2.
Note: Temme's series converge only for |μ| &lt;= 1/2.
</p>
<p>
K<sub>&#957;</sub>(x) is then calculated from the forward recurrence, as is K<sub>&#957;+1</sub>(x). With these
two values and f<sub>&#957;</sub>, the Wronskian yields I<sub>&#957;</sub>(x) directly.
K<sub>ν</sub>(x) is then calculated from the forward recurrence, as is K<sub>ν+1</sub>(x). With these
two values and f<sub>ν</sub>, the Wronskian yields I<sub>ν</sub>(x) directly.
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

57
doc/html/math_toolkit/bessel/sph_bessel.html
View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Spherical Bessel Functions of the First and Second Kinds</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../bessel.html" title="Bessel Functions">
<link rel="prev" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">
<link rel="next" href="bessel_derivatives.html" title="Derivatives of the Bessel Functions">
@@ -37,14 +37,14 @@
<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>
<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">&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;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<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>
<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">&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;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<h5>
<a name="math_toolkit.bessel.sph_bessel.h1"></a>
@@ -74,9 +74,9 @@
type calculation rules</em></span></a> for the single argument type T.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. 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&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">policy
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
documentation for more details</a>.
</p>
<p>
@@ -91,7 +91,7 @@
</p></blockquote></div>
<p>
Likewise y<sub>v</sub> is also cyclic for large x, but tends to -&#8734;
Likewise y<sub>v</sub> is also cyclic for large x, but tends to -∞
for small <span class="emphasis"><em>x</em></span>:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
@@ -112,7 +112,7 @@ for small <span class="emphasis"><em>x</em></span>:
<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&#160;8.48.&#160;Error rates for sph_bessel</b></p>
<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>
<div class="table-contents"><table class="table" summary="Error rates for sph_bessel">
<colgroup>
<col>
@@ -153,31 +153,31 @@ for small <span class="emphasis"><em>x</em></span>:
</td>
<td>
<p>
<span class="blue">Max = 243&#949; (Mean = 13.3&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.91e+06&#949; (Mean = 1.09e+05&#949;))
<span class="blue">Max = 243ε (Mean = 13.3ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.91e+06ε (Mean = 1.09e+05ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.978&#949; (Mean = 0.0445&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.79e+03&#949; (Mean = 107&#949;))
<span class="blue">Max = 0.978ε (Mean = 0.0445ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.79e+03ε (Mean = 107ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 243&#949; (Mean = 33.7&#949;)</span>
<span class="blue">Max = 243ε (Mean = 33.7ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 245&#949; (Mean = 16.3&#949;)</span>
<span class="blue">Max = 245ε (Mean = 16.3ε)</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&#160;8.49.&#160;Error rates for sph_neumann</b></p>
<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>
<div class="table-contents"><table class="table" summary="Error rates for sph_neumann">
<colgroup>
<col>
@@ -218,24 +218,24 @@ for small <span class="emphasis"><em>x</em></span>:
</td>
<td>
<p>
<span class="blue">Max = 234&#949; (Mean = 19.5&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.6e+06&#949; (Mean = 1.4e+05&#949;))
<span class="blue">Max = 234ε (Mean = 19.5ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.6e+06ε (Mean = 1.4e+05ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.995&#949; (Mean = 0.0665&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 8.5e+04&#949; (Mean = 5.33e+03&#949;))
<span class="blue">Max = 0.995ε (Mean = 0.0665ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 8.5e+04ε (Mean = 5.33e+03ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 234&#949; (Mean = 19.8&#949;)</span>
<span class="blue">Max = 234ε (Mean = 19.8ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 281&#949; (Mean = 31.1&#949;)</span>
<span class="blue">Max = 281ε (Mean = 31.1ε)</span>
</p>
</td>
</tr></tbody>
@@ -269,16 +269,15 @@ for small <span class="emphasis"><em>x</em></span>:
</p></blockquote></div>
<p>
which neatly avoids the problem of calculating 0/0 that can occur with the
main definition as x &#8594; 0.
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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Bivariate Statistics</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../statistics.html" title="Chapter&#160;6.&#160;Statistics">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../statistics.html" title="Chapter 6. Statistics">
<link rel="prev" href="univariate_statistics.html" title="Univariate Statistics">
<link rel="next" href="signal_statistics.html" title="Signal Statistics">
</head>
@@ -115,11 +115,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

17
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Locating Function Minima using Brent's algorithm</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../root_finding.html" title="Chapter&#160;10.&#160;Root Finding &amp; Minimization Algorithms">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../root_finding.html" title="Chapter 10. Root Finding &amp; Minimization Algorithms">
<link rel="prev" href="bad_roots.html" title="Examples Where Root Finding Goes Wrong">
<link rel="next" href="root_comparison.html" title="Comparison of Root Finding Algorithms">
</head>
@@ -74,7 +74,7 @@
<dd><p>
The number of bits precision to which the minima should be found.<br>
Note that in principle, the minima can not be located to greater accuracy
than the square root of machine epsilon (for 64-bit double, sqrt(1e-16)&#8773;1e-8),
than the square root of machine epsilon (for 64-bit double, sqrt(1e-16)≅1e-8),
therefore the value of <span class="emphasis"><em>bits</em></span> will be ignored if it's
greater than half the number of bits in the mantissa of T.
</p></dd>
@@ -651,11 +651,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

17
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>If and How to Build a Boost.Math Library, and its Examples and Tests</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../overview.html" title="Chapter&#160;1.&#160;Overview">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../overview.html" title="Chapter 1. Overview">
<link rel="prev" href="perf_over1.html" title="Performance">
<link rel="next" href="history1.html" title="History and What's New">
</head>
@@ -42,7 +42,7 @@
and program size.
</p>
<p>
Refer to <a class="link" href="../extern_c.html" title='Chapter&#160;9.&#160;TR1 and C99 external "C" Functions'>C99 and C++ TR1 C-style Functions</a>
Refer to <a class="link" href="../extern_c.html" title='Chapter 9. TR1 and C99 external "C" Functions'>C99 and C++ TR1 C-style Functions</a>
for pros and cons of using the TR1 components as opposed to the header only
ones.
</p>
@@ -147,11 +147,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>C99 C Functions</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../extern_c.html" title='Chapter&#160;9.&#160;TR1 and C99 external "C" Functions'>
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../extern_c.html" title='Chapter 9. TR1 and C99 external "C" Functions'>
<link rel="prev" href="main_tr1.html" title="C99 and TR1 C Functions Overview">
<link rel="next" href="tr1_ref.html" title="TR1 C Functions Quick Reference">
</head>
@@ -463,11 +463,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

25
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@@ -1,12 +1,12 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Cardinal Cubic B-spline interpolation</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../interpolation.html" title="Chapter&#160;12.&#160;Interpolation">
<link rel="prev" href="../interpolation.html" title="Chapter&#160;12.&#160;Interpolation">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../interpolation.html" title="Chapter 12. Interpolation">
<link rel="prev" href="../interpolation.html" title="Chapter 12. Interpolation">
<link rel="next" href="cardinal_quadratic_b.html" title="Cardinal Quadratic B-spline interpolation">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
@@ -156,9 +156,9 @@
and Performance</a>
</h4>
<p>
The call to the constructor requires &#119926;(<span class="emphasis"><em>n</em></span>) operations, where
The call to the constructor requires 𝑶(<span class="emphasis"><em>n</em></span>) operations, where
<span class="emphasis"><em>n</em></span> is the number of points to interpolate. Each call the
the interpolant is &#119926;(1) (constant time). On the author's Intel Xeon E3-1230,
the interpolant is 𝑶(1) (constant time). On the author's Intel Xeon E3-1230,
this takes 21ns as long as the vector is small enough to fit in cache.
</p>
<h4>
@@ -167,8 +167,8 @@
</h4>
<p>
Let <span class="emphasis"><em>h</em></span> be the stepsize. If <span class="emphasis"><em>f</em></span> is four-times
continuously differentiable, then the interpolant is <span class="emphasis"><em>&#119926;(h<sup>4</sup>)</em></span>
accurate and the derivative is <span class="emphasis"><em>&#119926;(h<sup>3</sup>)</em></span> accurate.
continuously differentiable, then the interpolant is <span class="emphasis"><em>𝑶(h<sup>4</sup>)</em></span>
accurate and the derivative is <span class="emphasis"><em>𝑶(h<sup>3</sup>)</em></span> accurate.
</p>
<h4>
<a name="math_toolkit.cardinal_cubic_b.h4"></a>
@@ -313,11 +313,10 @@ Found in 3 iterations
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Cardinal Quadratic B-spline interpolation</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../interpolation.html" title="Chapter&#160;12.&#160;Interpolation">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../interpolation.html" title="Chapter 12. Interpolation">
<link rel="prev" href="cardinal_cubic_b.html" title="Cardinal Cubic B-spline interpolation">
<link rel="next" href="cardinal_quintic_b.html" title="Cardinal Quintic B-spline interpolation">
</head>
@@ -85,11 +85,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Cardinal Quintic B-spline interpolation</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../interpolation.html" title="Chapter&#160;12.&#160;Interpolation">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../interpolation.html" title="Chapter 12. Interpolation">
<link rel="prev" href="cardinal_quadratic_b.html" title="Cardinal Quadratic B-spline interpolation">
<link rel="next" href="whittaker_shannon.html" title="Whittaker-Shannon interpolation">
</head>
@@ -122,11 +122,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

23
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@@ -1,11 +1,11 @@
<html>
<head>
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<title>Cardinal Trigonometric interpolation</title>
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<link rel="up" href="../interpolation.html" title="Chapter 12. Interpolation">
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<link rel="next" href="cubic_hermite.html" title="Cubic Hermite interpolation">
</head>
@@ -70,10 +70,10 @@
or <span class="emphasis"><em>n = 2m+1</em></span>, we hope to have
</p>
<p>
<span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>t</em></span>) &#8776; <span class="emphasis"><em>a</em></span><sub>0</sub>/2
+ &#8721;<sub><span class="emphasis"><em>k</em></span> = 1</sub><sup><span class="emphasis"><em>m</em></span></sup> <span class="emphasis"><em>a</em></span><sub><span class="emphasis"><em>k</em></span></sub> cos(2&#960;
<span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>t</em></span>) ≈ <span class="emphasis"><em>a</em></span><sub>0</sub>/2
+ ∑<sub><span class="emphasis"><em>k</em></span> = 1</sub><sup><span class="emphasis"><em>m</em></span></sup> <span class="emphasis"><em>a</em></span><sub><span class="emphasis"><em>k</em></span></sub> cos(2π
<span class="emphasis"><em>k</em></span> (<span class="emphasis"><em>t</em></span>-<span class="emphasis"><em>t</em></span><sub>0</sub>) /T)
+ <span class="emphasis"><em>b</em></span><sub><span class="emphasis"><em>k</em></span></sub> sin(2&#960; <span class="emphasis"><em>k</em></span>
+ <span class="emphasis"><em>b</em></span><sub><span class="emphasis"><em>k</em></span></sub> sin(2π <span class="emphasis"><em>k</em></span>
(<span class="emphasis"><em>t</em></span>-<span class="emphasis"><em>t</em></span><sub>0</sub>)/T)
</p>
<p>
@@ -108,7 +108,7 @@
of the L<sup>2</sup> norm is trivial in this basis; it is computed by the <code class="computeroutput"><span class="special">.</span><span class="identifier">squared_l2</span><span class="special">()</span></code> member function.
</p>
<p>
Below is a graph of a <span class="emphasis"><em>C</em></span><sup>&#8734;</sup> bump function approximated
Below is a graph of a <span class="emphasis"><em>C</em></span><sup>∞</sup> bump function approximated
by trigonometric series. The graphs are visually indistinguishable at 20 samples.
</p>
<p>
@@ -150,11 +150,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

23
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Catmull-Rom Splines</title>
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../interpolation.html" title="Chapter 12. Interpolation">
<link rel="prev" href="vector_barycentric.html" title="Vector-valued Barycentric Rational Interpolation">
<link rel="next" href="cardinal_trigonometric.html" title="Cardinal Trigonometric interpolation">
</head>
@@ -75,7 +75,7 @@
</li>
<li class="listitem">
Interpolation of control points-this means the curve passes through the
control points. Many curves (such as B&#233;zier) are <span class="emphasis"><em>approximating</em></span>
control points. Many curves (such as Bézier) are <span class="emphasis"><em>approximating</em></span>
- they do not pass through their control points. This makes them more difficult
to use than interpolating splines.
</li>
@@ -199,7 +199,7 @@
<p>
The following performance numbers were generated for a call to the Catmull-Rom
interpolation method. The number that follows the slash is the number of points
passed to the interpolant. We see that evaluation of the interpolant is &#119926;(<span class="emphasis"><em>log</em></span>(<span class="emphasis"><em>N</em></span>)).
passed to the interpolant. We see that evaluation of the interpolant is 𝑶(<span class="emphasis"><em>log</em></span>(<span class="emphasis"><em>N</em></span>)).
</p>
<pre class="programlisting"><span class="identifier">Run</span> <span class="identifier">on</span> <span class="number">2700</span> <span class="identifier">MHz</span> <span class="identifier">CPU</span>
<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
@@ -334,8 +334,8 @@
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Cem Yuksel, Scott Schaefer, and John Keyser, <span class="emphasis"><em>Parameterization
and applications of Catmull&#8211;Rom curves</em></span>, Computer-Aided Design
43 (2011) 747&#8211;755.
and applications of Catmull–Rom curves</em></span>, Computer-Aided Design
43 (2011) 747–755.
</li>
<li class="listitem">
Phillip J. Barry and Ronald N. Goldman, <span class="emphasis"><em>A Recursive Evaluation
@@ -346,11 +346,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

19
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Comparing Different Compilers</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
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<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../perf.html" title="Chapter 22. Performance">
<link rel="prev" href="tuning.html" title="Performance Tuning Macros">
<link rel="next" href="comparisons.html" title="Comparisons to Other Open Source Libraries">
</head>
@@ -34,7 +34,7 @@
to be main, particularly from some of the commercial offerings:
</p>
<div class="table">
<a name="math_toolkit.comp_compilers.table_Compiler_Comparison_on_Windows_x64"></a><p class="title"><b>Table&#160;22.12.&#160;Compiler Comparison on Windows x64</b></p>
<a name="math_toolkit.comp_compilers.table_Compiler_Comparison_on_Windows_x64"></a><p class="title"><b>Table 22.12. Compiler Comparison on Windows x64</b></p>
<div class="table-contents"><table class="table" summary="Compiler Comparison on Windows x64">
<colgroup>
<col>
@@ -1213,7 +1213,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.comp_compilers.table_Compiler_Comparison_on_linux"></a><p class="title"><b>Table&#160;22.13.&#160;Compiler Comparison on linux</b></p>
<a name="math_toolkit.comp_compilers.table_Compiler_Comparison_on_linux"></a><p class="title"><b>Table 22.13. Compiler Comparison on linux</b></p>
<div class="table-contents"><table class="table" summary="Compiler Comparison on linux">
<colgroup>
<col>
@@ -3194,11 +3194,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

21
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Comparisons to Other Open Source Libraries</title>
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
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<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../perf.html" title="Chapter 22. Performance">
<link rel="prev" href="comp_compilers.html" title="Comparing Different Compilers">
<link rel="next" href="perf_test_app.html" title="The Performance Test Applications">
</head>
@@ -54,7 +54,7 @@
MSVC-14.0:
</p>
<div class="table">
<a name="math_toolkit.comparisons.table_Library_Comparison_with_Microsoft_Visual_C_version_14_2_on_Windows_x64"></a><p class="title"><b>Table&#160;22.14.&#160;Library Comparison with Microsoft Visual C++ version 14.2 on Windows
<a name="math_toolkit.comparisons.table_Library_Comparison_with_Microsoft_Visual_C_version_14_2_on_Windows_x64"></a><p class="title"><b>Table 22.14. Library Comparison with Microsoft Visual C++ version 14.2 on Windows
x64</b></p>
<div class="table-contents"><table class="table" summary="Library Comparison with Microsoft Visual C++ version 14.2 on Windows
x64">
@@ -242,7 +242,7 @@
RMath:
</p>
<div class="table">
<a name="math_toolkit.comparisons.table_Library_Comparison_with_GNU_C_version_9_2_1_20191008_on_linux"></a><p class="title"><b>Table&#160;22.15.&#160;Library Comparison with GNU C++ version 9.2.1 20191008 on linux</b></p>
<a name="math_toolkit.comparisons.table_Library_Comparison_with_GNU_C_version_9_2_1_20191008_on_linux"></a><p class="title"><b>Table 22.15. Library Comparison with GNU C++ version 9.2.1 20191008 on linux</b></p>
<div class="table-contents"><table class="table" summary="Library Comparison with GNU C++ version 9.2.1 20191008 on linux">
<colgroup>
<col>
@@ -2223,7 +2223,7 @@
DCDFLIB:
</p>
<div class="table">
<a name="math_toolkit.comparisons.table_Distribution_performance_comparison_with_GNU_C_version_9_2_1_20191008_on_linux"></a><p class="title"><b>Table&#160;22.16.&#160;Distribution performance comparison with GNU C++ version 9.2.1 20191008
<a name="math_toolkit.comparisons.table_Distribution_performance_comparison_with_GNU_C_version_9_2_1_20191008_on_linux"></a><p class="title"><b>Table 22.16. Distribution performance comparison with GNU C++ version 9.2.1 20191008
on linux</b></p>
<div class="table-contents"><table class="table" summary="Distribution performance comparison with GNU C++ version 9.2.1 20191008
on linux">
@@ -4720,11 +4720,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

19
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Compilers</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../overview.html" title="Chapter&#160;1.&#160;Overview">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../overview.html" title="Chapter 1. Overview">
<link rel="prev" href="error_handling.html" title="Error Handling">
<link rel="next" href="config_macros.html" title="Configuration Macros">
</head>
@@ -33,7 +33,7 @@
to achieve this on all systems.
</p>
<div class="table">
<a name="math_toolkit.compilers_overview.supported_tested_compilers"></a><p class="title"><b>Table&#160;1.9.&#160;Supported/Tested Compilers</b></p>
<a name="math_toolkit.compilers_overview.supported_tested_compilers"></a><p class="title"><b>Table 1.9. Supported/Tested Compilers</b></p>
<div class="table-contents"><table class="table" summary="Supported/Tested Compilers">
<colgroup>
<col>
@@ -601,7 +601,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.compilers_overview.unsupported_compilers"></a><p class="title"><b>Table&#160;1.10.&#160;Unsupported Compilers</b></p>
<a name="math_toolkit.compilers_overview.unsupported_compilers"></a><p class="title"><b>Table 1.10. Unsupported Compilers</b></p>
<div class="table-contents"><table class="table" summary="Unsupported Compilers">
<colgroup>
<col>
@@ -666,11 +666,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

17
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@@ -1,13 +1,13 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>History</title>
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../inverse_complex.html" title="Chapter&#160;15.&#160;Complex Number Functions">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../inverse_complex.html" title="Chapter 15. Complex Number Functions">
<link rel="prev" href="atanh.html" title="atanh">
<link rel="next" href="../quaternions.html" title="Chapter&#160;16.&#160;Quaternions">
<link rel="next" href="../quaternions.html" title="Chapter 16. Quaternions">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
@@ -37,11 +37,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

19
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@@ -1,12 +1,12 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Implementation and Accuracy</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../inverse_complex.html" title="Chapter&#160;15.&#160;Complex Number Functions">
<link rel="prev" href="../inverse_complex.html" title="Chapter&#160;15.&#160;Complex Number Functions">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../inverse_complex.html" title="Chapter 15. Complex Number Functions">
<link rel="prev" href="../inverse_complex.html" title="Chapter 15. Complex Number Functions">
<link rel="next" href="asin.html" title="asin">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
@@ -39,18 +39,17 @@
extremes of that range, where as a naive formula would cause overflow or underflow
to occur during the calculation, even though the result is actually a representable
value. The maximum theoretical relative error for all of these functions is
less than 9.5&#949; for every machine-representable point in the complex plane. Please
less than 9.5ε for every machine-representable point in the complex plane. Please
refer to comments in the header files themselves and to the above mentioned
paper for more information on the implementation methodology.
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

21
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@@ -1,13 +1,13 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Condition Numbers</title>
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
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<link rel="up" href="../utils.html" title="Chapter 2. Floating Point Utilities">
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@@ -20,7 +20,7 @@
</tr></table>
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@@ -168,11 +168,10 @@
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<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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>
@@ -180,7 +179,7 @@
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<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Configuration Macros</title>
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@@ -38,7 +38,7 @@
briefly below.
</p>
<div class="table">
<a name="math_toolkit.config_macros.boost_math_macros"></a><p class="title"><b>Table&#160;1.11.&#160;Boost.Math Macros</b></p>
<a name="math_toolkit.config_macros.boost_math_macros"></a><p class="title"><b>Table 1.11. Boost.Math Macros</b></p>
<div class="table-contents"><table class="table" summary="Boost.Math Macros">
<colgroup>
<col>
@@ -279,7 +279,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.config_macros.boost_math_tuning"></a><p class="title"><b>Table&#160;1.12.&#160;Boost.Math Tuning</b></p>
<a name="math_toolkit.config_macros.boost_math_tuning"></a><p class="title"><b>Table 1.12. Boost.Math Tuning</b></p>
<div class="table-contents"><table class="table" summary="Boost.Math Tuning">
<colgroup>
<col>
@@ -366,11 +366,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

249
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>The Mathematical Constants</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
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<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../constants.html" title="Chapter 4. Mathematical Constants">
<link rel="prev" href="tutorial/user_def.html" title="Use With User-Defined Types">
<link rel="next" href="new_const.html" title="Defining New Constants">
</head>
@@ -31,7 +31,7 @@
rationale for their inclusion).
</p>
<div class="table">
<a name="math_toolkit.constants.mathematical_constants"></a><p class="title"><b>Table&#160;4.1.&#160;Mathematical Constants</b></p>
<a name="math_toolkit.constants.mathematical_constants"></a><p class="title"><b>Table 4.1. Mathematical Constants</b></p>
<div class="table-contents"><table class="table" summary="Mathematical Constants">
<colgroup>
<col>
@@ -172,7 +172,7 @@
</td>
<td>
<p>
&#8730;2
√2
</p>
</td>
<td>
@@ -194,7 +194,7 @@
</td>
<td>
<p>
&#8730;3
√3
</p>
</td>
<td>
@@ -213,7 +213,7 @@
</td>
<td>
<p>
&#8730;2 /2
√2 /2
</p>
</td>
<td>
@@ -298,7 +298,7 @@
</td>
<td>
<p>
&#8730;ln(4)
√ln(4)
</p>
</td>
<td>
@@ -317,7 +317,7 @@
</td>
<td>
<p>
1/&#8730;2
1/√2
</p>
</td>
<td>
@@ -334,7 +334,7 @@
<tr>
<td>
<p>
<span class="bold"><strong>&#960; and related</strong></span>
<span class="bold"><strong>π and related</strong></span>
</p>
</td>
<td>
@@ -352,7 +352,7 @@
</td>
<td>
<p>
pi
π
</p>
</td>
<td>
@@ -362,7 +362,7 @@
</td>
<td>
<p>
Ubiquitous. Archimedes constant <a href="http://en.wikipedia.org/wiki/Pi" target="_top">&#960;</a>.
Ubiquitous. Archimedes constant <a href="http://en.wikipedia.org/wiki/Pi" target="_top">π</a>.
Equivalent to POSIX constant M_PI
</p>
</td>
@@ -375,7 +375,7 @@
</td>
<td>
<p>
&#960;/2
π/2
</p>
</td>
<td>
@@ -397,7 +397,7 @@
</td>
<td>
<p>
&#960;/3
π/3
</p>
</td>
<td>
@@ -416,7 +416,7 @@
</td>
<td>
<p>
&#960;/4
π/4
</p>
</td>
<td>
@@ -438,7 +438,7 @@
</td>
<td>
<p>
&#960;/6
π/6
</p>
</td>
<td>
@@ -457,7 +457,7 @@
</td>
<td>
<p>
2&#960;
2π
</p>
</td>
<td>
@@ -479,7 +479,7 @@
</td>
<td>
<p>
2/3 &#960;
2/3 π
</p>
</td>
<td>
@@ -490,7 +490,7 @@
<td>
<p>
<a href="http://en.wikipedia.org/wiki/Sphere#Volume_of_a_sphere" target="_top">volume
of a hemi-sphere</a> = 4/3 &#960; r&#179;
of a hemi-sphere</a> = 4/3 π r³
</p>
</td>
</tr>
@@ -502,7 +502,7 @@
</td>
<td>
<p>
3/4 &#960;
3/4 π
</p>
</td>
<td>
@@ -512,7 +512,7 @@
</td>
<td>
<p>
= 3/4 &#960;
= 3/4 π
</p>
</td>
</tr>
@@ -524,7 +524,7 @@
</td>
<td>
<p>
4/3 &#960;
4/3 π
</p>
</td>
<td>
@@ -535,7 +535,7 @@
<td>
<p>
<a href="http://en.wikipedia.org/wiki/Sphere#Volume_of_a_sphere" target="_top">volume
of a sphere</a> = 4/3 &#960; r&#179;
of a sphere</a> = 4/3 π r³
</p>
</td>
</tr>
@@ -547,7 +547,7 @@
</td>
<td>
<p>
1/(2&#960;)
1/(2π)
</p>
</td>
<td>
@@ -569,7 +569,7 @@
</td>
<td>
<p>
&#8730;&#960;
√π
</p>
</td>
<td>
@@ -591,7 +591,7 @@
</td>
<td>
<p>
&#8730; &#960;/2
√ π/2
</p>
</td>
<td>
@@ -613,7 +613,7 @@
</td>
<td>
<p>
&#8730; &#960;*2
√ π*2
</p>
</td>
<td>
@@ -635,7 +635,7 @@
</td>
<td>
<p>
1/&#960;
1/π
</p>
</td>
<td>
@@ -657,7 +657,7 @@
</td>
<td>
<p>
2/&#960;
2/π
</p>
</td>
<td>
@@ -679,7 +679,7 @@
</td>
<td>
<p>
1/&#8730;&#960;
1/√π
</p>
</td>
<td>
@@ -698,7 +698,7 @@
</td>
<td>
<p>
2/&#8730;&#960;
2/√π
</p>
</td>
<td>
@@ -720,7 +720,7 @@
</td>
<td>
<p>
1/&#8730;(2&#960;)
1/√(2π)
</p>
</td>
<td>
@@ -739,7 +739,7 @@
</td>
<td>
<p>
&#8730;(1/&#960;
√(1/π
</p>
</td>
<td>
@@ -758,7 +758,7 @@
</td>
<td>
<p>
&#960;-3
π-3
</p>
</td>
<td>
@@ -777,7 +777,7 @@
</td>
<td>
<p>
4 -&#960;
4 -π
</p>
</td>
<td>
@@ -796,7 +796,7 @@
</td>
<td>
<p>
&#960;<sup>e</sup>
π<sup>e</sup>
</p>
</td>
<td>
@@ -815,7 +815,7 @@
</td>
<td>
<p>
&#960;<sup>2</sup>
π<sup>2</sup>
</p>
</td>
<td>
@@ -834,7 +834,7 @@
</td>
<td>
<p>
&#960;<sup>2</sup>/6
π<sup>2</sup>/6
</p>
</td>
<td>
@@ -853,7 +853,7 @@
</td>
<td>
<p>
&#960;<sup>3</sup>
π<sup>3</sup>
</p>
</td>
<td>
@@ -872,7 +872,7 @@
</td>
<td>
<p>
&#8730;<sup>3</sup> &#960;
√<sup>3</sup> π
</p>
</td>
<td>
@@ -891,7 +891,7 @@
</td>
<td>
<p>
1/&#8730;<sup>3</sup> &#960;
1/√<sup>3</sup> π
</p>
</td>
<td>
@@ -965,7 +965,7 @@
</td>
<td>
<p>
e <sup>&#960;</sup>
e <sup>π</sup>
</p>
</td>
<td>
@@ -984,7 +984,7 @@
</td>
<td>
<p>
&#8730; e
√ e
</p>
</td>
<td>
@@ -1079,7 +1079,7 @@
</td>
<td>
<p>
radians = &#960; / 180
radians = π / 180
</p>
</td>
<td>
@@ -1098,7 +1098,7 @@
</td>
<td>
<p>
degrees = 180 / &#960;
degrees = 180 / π
</p>
</td>
<td>
@@ -1213,7 +1213,7 @@
</td>
<td>
<p>
(1 + &#8730;5) /2
(1 + √5) /2
</p>
</td>
<td>
@@ -1235,7 +1235,7 @@
</td>
<td>
<p>
ln(&#966;)
ln(φ)
</p>
</td>
<td>
@@ -1254,7 +1254,7 @@
</td>
<td>
<p>
1/ln(&#966;)
1/ln(φ)
</p>
</td>
<td>
@@ -1360,7 +1360,7 @@
</td>
<td>
<p>
&#950;(2)
ζ(2)
</p>
</td>
<td>
@@ -1383,7 +1383,7 @@
</td>
<td>
<p>
&#950;(3)
ζ(3)
</p>
</td>
<td>
@@ -1475,7 +1475,7 @@
</td>
<td>
<p>
12&#8730;6 &#950;(3)/ &#960;<sup>3</sup>
12√6 ζ(3)/ π<sup>3</sup>
</p>
</td>
<td>
@@ -1497,7 +1497,7 @@
</td>
<td>
<p>
2&#8730;&#960;(&#960;-3)/(4 - &#960;)<sup>3/2</sup>
2√π(π-3)/(4 - π)<sup>3/2</sup>
</p>
</td>
<td>
@@ -1519,7 +1519,7 @@
</td>
<td>
<p>
-(6&#960;<sup>2</sup>-24&#960;+16)/(4-&#960;)<sup>2</sup>
-(6π<sup>2</sup>-24π+16)/(4-π)<sup>2</sup>
</p>
</td>
<td>
@@ -1542,7 +1542,7 @@
</td>
<td>
<p>
3+(6&#960;<sup>2</sup>-24&#960;+16)/(4-&#960;)<sup>2</sup>
3+(6π<sup>2</sup>-24π+16)/(4-π)<sup>2</sup>
</p>
</td>
<td>
@@ -1556,6 +1556,138 @@
</p>
</td>
</tr>
<tr>
<td>
<p>
first_feigenbaum
</p>
</td>
<td>
</td>
<td>
<p>
4.6692016
</p>
</td>
<td>
<p>
<a href="https://en.wikipedia.org/wiki/Feigenbaum_constants" target="_top">First
Feigenbaum constant</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
plastic
</p>
</td>
<td>
<p>
Real solution of x<sup>3</sup> = x + 1
</p>
</td>
<td>
<p>
1.324717957
</p>
</td>
<td>
<p>
<a href="https://en.wikipedia.org/wiki/Plastic_number" target="_top">Plastic
constant</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
gauss
</p>
</td>
<td>
<p>
Reciprocal of agm(1, √2)
</p>
</td>
<td>
<p>
0.8346268
</p>
</td>
<td>
<p>
<a href="https://en.wikipedia.org/wiki/Gauss%27s_constant" target="_top">Gauss's
constant</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
dottie
</p>
</td>
<td>
<p>
Solution of cos(x) = x
</p>
</td>
<td>
<p>
0.739085
</p>
</td>
<td>
<p>
<a href="https://en.wikipedia.org/wiki/Dottie_number" target="_top">Dottie's
number</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
reciprocal_fibonacci
</p>
</td>
<td>
<p>
Sum of reciprocals of Fibonacci numbers
</p>
</td>
<td>
<p>
3.359885666
</p>
</td>
<td>
<p>
<a href="https://en.wikipedia.org/wiki/Reciprocal_Fibonacci_constant" target="_top">Reciprocal
Fibonacci constant</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
laplace_limit
</p>
</td>
<td>
</td>
<td>
<p>
.6627434193
</p>
</td>
<td>
<p>
<a href="https://en.wikipedia.org/wiki/Laplace_limit" target="_top">Laplace
Limit</a>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
@@ -1602,11 +1734,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

71
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@@ -1,13 +1,13 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Math Constants FAQs</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
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<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../constants.html" title="Chapter 4. Mathematical Constants">
<link rel="prev" href="new_const.html" title="Defining New Constants">
<link rel="next" href="../dist.html" title="Chapter&#160;5.&#160;Statistical Distributions and Functions">
<link rel="next" href="../dist.html" title="Chapter 5. Statistical Distributions and Functions">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
@@ -107,7 +107,7 @@
cubed for cubed n<sup>3</sup>.
</li>
<li class="listitem">
words for greek, like &#960;, &#950; and &#915;.
words for greek, like π, ζ and Γ.
</li>
<li class="listitem">
words like half, third, three_quarters, sixth for fractions. (Digit(s)
@@ -267,29 +267,14 @@
<h6>
<a name="math_toolkit.constants_faq.h8"></a>
<span class="phrase"><a name="math_toolkit.constants_faq.boost_multiprecision_by_christop"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.boost_multiprecision_by_christop">Boost.Multiprecision
by Christopher Kormanyos</a>
by Christopher Kormanyos and John Maddock</a>
</h6>
<p>
This work is based on an earlier work called e-float: Algorithm 910: A Portable
C++ Multiple-Precision System for Special-Function Calculations, in ACM TOMS,
{VOL 37, ISSUE 4, (February 2011)} (C) ACM, 2011. <a href="http://doi.acm.org/10.1145/1916461.1916469" target="_top">http://doi.acm.org/10.1145/1916461.1916469</a>
<a href="https://svn.boost.org/svn/boost/sandbox/e_float/" target="_top">e_float</a>
but is now re-factored and available under the Boost license in the Boost-sandbox
at <a href="https://svn.boost.org/svn/boost/sandbox/multiprecision/" target="_top">multiprecision</a>
where it is being refined and prepared for review.
This is a well-maintained and fully featured library which provides numerous
high-precision types, as well as wrappers to other popular libraries like MPFR.
</p>
<h6>
<a name="math_toolkit.constants_faq.h9"></a>
<span class="phrase"><a name="math_toolkit.constants_faq.boost_cpp_float_by_john_maddock_"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.boost_cpp_float_by_john_maddock_">Boost.cpp_float
by John Maddock using Expression Templates</a>
</h6>
<p>
<a href="https://svn.boost.org/svn/boost/sandbox/big_number/" target="_top">Big Number</a>
which is a reworking of <a href="https://svn.boost.org/svn/boost/sandbox/e_float/" target="_top">e_float</a>
by Christopher Kormanyos to use expression templates for faster execution.
</p>
<h6>
<a name="math_toolkit.constants_faq.h10"></a>
<span class="phrase"><a name="math_toolkit.constants_faq.ntl_class_quad_float"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.ntl_class_quad_float">NTL
class quad_float</a>
</h6>
@@ -333,7 +318,7 @@
<span class="bold"><strong>New projects should use <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span>
</p>
<h6>
<a name="math_toolkit.constants_faq.h11"></a>
<a name="math_toolkit.constants_faq.h10"></a>
<span class="phrase"><a name="math_toolkit.constants_faq.ntl_class_rr"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.ntl_class_rr">NTL
class RR</a>
</h6>
@@ -354,7 +339,7 @@
<span class="bold"><strong>New projects should use <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span>
</p>
<h6>
<a name="math_toolkit.constants_faq.h12"></a>
<a name="math_toolkit.constants_faq.h11"></a>
<span class="phrase"><a name="math_toolkit.constants_faq.gmp_and_mpfr"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.gmp_and_mpfr">GMP
and MPFR</a>
</h6>
@@ -364,7 +349,7 @@
are <span class="bold"><strong>not licensed for commercial use</strong></span>.
</p>
<h5>
<a name="math_toolkit.constants_faq.h13"></a>
<a name="math_toolkit.constants_faq.h12"></a>
<span class="phrase"><a name="math_toolkit.constants_faq.what_happened_to_a_previous_coll"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.what_happened_to_a_previous_coll">What
happened to a previous collection of constants proposed for Boost?</a>
</h5>
@@ -373,20 +358,21 @@
meet many peoples needs. None of the methods proposed met many users' essential
requirement to allow writing simply <code class="computeroutput"><span class="identifier">pi</span></code>
rather than <code class="computeroutput"><span class="identifier">pi</span><span class="special">()</span></code>.
Many science and engineering equations look difficult to read when because
function call brackets can be confused with the many other brackets often needed.
All the methods then proposed of avoiding the brackets failed to meet all needs,
Many science and engineering equations look difficult to read because function
call brackets can be confused with the many other brackets often needed. All
the methods then proposed of avoiding the brackets failed to meet all needs,
often on grounds of complexity and lack of applicability to various realistic
scenarios.
</p>
<p>
So the simple namespace method, proposed on its own, but rejected at the first
review, has been added to allow users to have convenient access to float, double
and long double values, but combined with template struct and functions to
allow simultaneous use with other non-built-in floating-point types.
review, has been added to allow users to have convenient access to <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>
and <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
values, but combined with template struct and functions to allow simultaneous
use with other non-built-in floating-point types.
</p>
<h5>
<a name="math_toolkit.constants_faq.h14"></a>
<a name="math_toolkit.constants_faq.h13"></a>
<span class="phrase"><a name="math_toolkit.constants_faq.why_do_the_constants_internally_"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.why_do_the_constants_internally_">Why do
the constants (internally) have a struct rather than a simple function?</a>
</h5>
@@ -398,7 +384,7 @@
a constant above. It should also allow use with other packages like <a href="http://www.ttmath.org/" target="_top">ttmath Bignum C++ library.</a>
</p>
<h5>
<a name="math_toolkit.constants_faq.h15"></a>
<a name="math_toolkit.constants_faq.h14"></a>
<span class="phrase"><a name="math_toolkit.constants_faq.where_can_i_find_other_high_prec"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.where_can_i_find_other_high_prec">Where
can I find other high precision constants?</a>
</h5>
@@ -439,22 +425,22 @@
</li>
</ol></div>
<h5>
<a name="math_toolkit.constants_faq.h16"></a>
<a name="math_toolkit.constants_faq.h15"></a>
<span class="phrase"><a name="math_toolkit.constants_faq.where_are_physical_constants"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.where_are_physical_constants">Where are
Physical Constants?</a>
</h5>
<p>
Not here in this Boost.Math collection, because physical constants:
Not here, because physical constants:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Are measurements, not truly constants.
</li>
<li class="listitem">
Are not truly constant and keeping changing as mensuration technology improves.
Are not truly constant and keeping changing as metrology improves.
</li>
<li class="listitem">
Have a intrinsic uncertainty.
Have intrinsic uncertainty.
</li>
<li class="listitem">
Mathematical constants are stored and represented at varying precision,
@@ -467,11 +453,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

17
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@@ -1,12 +1,12 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Introduction</title>
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@@ -122,11 +122,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

17
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@@ -1,13 +1,13 @@
<html>
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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<link rel="up" href="../overview.html" title="Chapter 1. Overview">
<link rel="prev" href="main_faq.html" title="Boost.Math Frequently Asked Questions (FAQs)">
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@@ -56,11 +56,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

17
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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@@ -27,7 +27,7 @@
<a name="math_toolkit.conventions"></a><a class="link" href="conventions.html" title="Document Conventions">Document Conventions</a>
</h2></div></div></div>
<p>
<a class="indexterm" name="id890140"></a>
<a class="indexterm" name="id877671"></a>
</p>
<p>
This documentation aims to use of the following naming and formatting conventions.
@@ -70,11 +70,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Quaternion Creation Functions</title>
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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<link rel="up" href="../quaternions.html" title="Chapter 16. Quaternions">
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@@ -104,11 +104,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

21
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@@ -1,13 +1,13 @@
<html>
<head>
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@@ -66,7 +66,7 @@
code.
</p>
<p>
Johan R&#229;de wrote the optimised floating-point classification and manipulation
Johan Råde wrote the optimised floating-point classification and manipulation
code, and nonfinite facets to permit C99 output of infinities and NaNs. (nonfinite
facets were not added until Boost 1.47 but had been in use with Boost.Spirit).
This library was based on a suggestion from Robert Ramey, author of Boost.Serialization.
@@ -79,7 +79,7 @@
no longer necessary to use a globale C99 facet to handle nonfinites.
</p>
<p>
H&#229;kan Ard&#246;, Boris Gubenko, John Maddock, Markus Sch&#246;pflin and Olivier Verdier
Håkan Ardö, Boris Gubenko, John Maddock, Markus Schöpflin and Olivier Verdier
tested the floating-point library and Martin Bonner, Peter Dimov and John Maddock
provided valuable advice.
</p>
@@ -191,11 +191,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

263
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.cubic_hermite"></a><a class="link" href="cubic_hermite.html" title="Cubic Hermite interpolation">Cubic Hermite interpolation</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.cubic_hermite.h0"></a>
<span class="phrase"><a name="math_toolkit.cubic_hermite.synopsis"></a></span><a class="link" href="cubic_hermite.html#math_toolkit.cubic_hermite.synopsis">Synopsis</a>
</h4>
<pre class="programlisting"><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">interpolators</span><span class="special">/</span><span class="identifier">cubic_hermite</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">interpolators</span> <span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">cubic_hermite</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
<span class="identifier">cubic_hermite</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span><span class="special">&amp;&amp;</span> <span class="identifier">abscissas</span><span class="special">,</span> <span class="identifier">RandomAccessContainer</span><span class="special">&amp;&amp;</span> <span class="identifier">ordinates</span><span class="special">,</span> <span class="identifier">RandomAccessContainer</span><span class="special">&amp;&amp;</span> <span class="identifier">derivatives</span><span class="special">);</span>
<span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="keyword">void</span> <span class="identifier">push_back</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">dydx</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">domain</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="keyword">friend</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">&lt;&lt;(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span> <span class="special">&amp;</span> <span class="identifier">os</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">cubic_hermite</span> <span class="special">&amp;</span> <span class="identifier">m</span><span class="special">);</span>
<span class="special">};</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">cardinal_cubic_hermite</span> <span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
<span class="identifier">cardinal_cubic_hermite</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="special">&amp;&amp;</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">RandomAccessContainer</span> <span class="special">&amp;&amp;</span> <span class="identifier">dydx</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x0</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">dx</span><span class="special">)</span>
<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">domain</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">cardinal_cubic_hermite_aos</span> <span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="keyword">using</span> <span class="identifier">Point</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">Point</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
<span class="identifier">cardinal_cubic_hermite_aos</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="special">&amp;&amp;</span> <span class="identifier">data</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x0</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">dx</span><span class="special">);</span>
<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">domain</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
<span class="special">}</span> <span class="comment">// namespaces</span>
</pre>
<h4>
<a name="math_toolkit.cubic_hermite.h1"></a>
<span class="phrase"><a name="math_toolkit.cubic_hermite.cubic_hermite_interpolation"></a></span><a class="link" href="cubic_hermite.html#math_toolkit.cubic_hermite.cubic_hermite_interpolation">Cubic Hermite
Interpolation</a>
</h4>
<p>
The cubic Hermite interpolant takes non-equispaced data and interpolates between
them via cubic Hermite polynomials whose slopes must be provided. This is a
very nice interpolant for solution skeletons of ODEs steppers, since numerically
solving <span class="emphasis"><em>y</em></span>' = <span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>,
<span class="emphasis"><em>y</em></span>) produces a list of positions, values, and their derivatives,
i.e. (<span class="emphasis"><em>x</em></span>,<span class="emphasis"><em>y</em></span>,<span class="emphasis"><em>y</em></span>')=(<span class="emphasis"><em>x</em></span>,<span class="emphasis"><em>y</em></span>,<span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>,<span class="emphasis"><em>y</em></span>))-which
is exactly what is required for the cubic Hermite interpolator. The interpolant
is <span class="emphasis"><em>C</em></span><sup>1</sup> and evaluation has 𝑶(log(<span class="emphasis"><em>N</em></span>))
complexity. An example usage is as follows:
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">x</span><span class="special">{</span><span class="number">1</span><span class="special">,</span> <span class="number">5</span><span class="special">,</span> <span class="number">9</span> <span class="special">,</span> <span class="number">12</span><span class="special">};</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">y</span><span class="special">{</span><span class="number">8</span><span class="special">,</span><span class="number">17</span><span class="special">,</span> <span class="number">4</span><span class="special">,</span> <span class="special">-</span><span class="number">3</span><span class="special">};</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span> <span class="identifier">dydx</span><span class="special">{</span><span class="number">5</span><span class="special">,</span> <span class="special">-</span><span class="number">2</span><span class="special">,</span> <span class="special">-</span><span class="number">1</span><span class="special">};</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">interpolators</span><span class="special">::</span><span class="identifier">cubic_hermite</span><span class="special">;</span>
<span class="keyword">auto</span> <span class="identifier">spline</span> <span class="special">=</span> <span class="identifier">cubic_hermite</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">y</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">dydx</span><span class="special">));</span>
<span class="comment">// evaluate at a point:</span>
<span class="keyword">double</span> <span class="identifier">z</span> <span class="special">=</span> <span class="identifier">spline</span><span class="special">(</span><span class="number">3.4</span><span class="special">);</span>
<span class="comment">// evaluate derivative at a point:</span>
<span class="keyword">double</span> <span class="identifier">zprime</span> <span class="special">=</span> <span class="identifier">spline</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">3.4</span><span class="special">);</span>
</pre>
<p>
Sanity checking of the interpolator can be achieved via:
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">spline</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
Note that the interpolator is pimpl'd, so that copying the class is cheap,
and hence it can be shared between threads. (The call operator and <code class="computeroutput"><span class="special">.</span><span class="identifier">prime</span><span class="special">()</span></code>
are threadsafe; <code class="computeroutput"><span class="identifier">push_back</span></code> is
not.)
</p>
<p>
This interpolant can be updated in constant time. Hence we can use <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span></code>
to do real-time interpolation:
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">circular_buffer</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="special">...</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">initial_x</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">};</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">initial_y</span><span class="special">{</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">,</span><span class="number">6</span><span class="special">,</span><span class="number">7</span><span class="special">};</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">initial_dydx</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">};</span>
<span class="keyword">auto</span> <span class="identifier">circular_hermite</span> <span class="special">=</span> <span class="identifier">cubic_hermite</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">initial_x</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">initial_y</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">initial_dydx</span><span class="special">));</span>
<span class="comment">// interpolate via call operation:</span>
<span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">circular_hermite</span><span class="special">(</span><span class="number">3.5</span><span class="special">);</span>
<span class="comment">// add new data:</span>
<span class="identifier">circular_hermite</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="number">5</span><span class="special">,</span> <span class="number">8</span><span class="special">);</span>
<span class="comment">// interpolate at 4.5:</span>
<span class="identifier">y</span> <span class="special">=</span> <span class="identifier">circular_hermite</span><span class="special">(</span><span class="number">4.5</span><span class="special">);</span>
</pre>
<p>
For the equispaced case, we can either use <code class="computeroutput"><span class="identifier">cardinal_cubic_hermite</span></code>,
which accepts two separate arrays of <code class="computeroutput"><span class="identifier">y</span></code>
and <code class="computeroutput"><span class="identifier">dydx</span></code>, or we can use <code class="computeroutput"><span class="identifier">cardinal_cubic_hermite_aos</span></code>, which takes a
vector of <code class="computeroutput"><span class="special">(</span><span class="identifier">y</span><span class="special">,</span> <span class="identifier">dydx</span><span class="special">)</span></code>,
i.e., and array of structs (<code class="computeroutput"><span class="identifier">aos</span></code>).
The array of structs should be preferred as it uses cache more effectively.
</p>
<p>
Usage is as follows:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">interpolators</span><span class="special">::</span><span class="identifier">cardinal_cubic_hermite</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">x0</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">dx</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">y</span><span class="special">(</span><span class="number">128</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">dydx</span><span class="special">(</span><span class="number">128</span><span class="special">,</span> <span class="number">0</span><span class="special">);</span>
<span class="keyword">auto</span> <span class="identifier">ch</span> <span class="special">=</span> <span class="identifier">cardinal_cubic_hermite</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">y</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">dydx</span><span class="special">),</span> <span class="identifier">x0</span><span class="special">,</span> <span class="identifier">dx</span><span class="special">);</span>
</pre>
<p>
For the "array of structs" version:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">interpolators</span><span class="special">::</span><span class="identifier">cardinal_cubic_hermite_aos</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">&gt;&gt;</span> <span class="identifier">data</span><span class="special">(</span><span class="number">128</span><span class="special">);</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">auto</span> <span class="special">&amp;</span> <span class="identifier">datum</span> <span class="special">:</span> <span class="identifier">data</span><span class="special">)</span> <span class="special">{</span>
<span class="identifier">datum</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="comment">// y</span>
<span class="identifier">datum</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="comment">// y'</span>
<span class="special">}</span>
<span class="keyword">auto</span> <span class="identifier">ch</span> <span class="special">=</span> <span class="identifier">cardinal_cubic_hermite_aos</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">data</span><span class="special">),</span> <span class="identifier">x0</span><span class="special">,</span> <span class="identifier">dx</span><span class="special">);</span>
</pre>
<p>
For a quick sanity check, we can call <code class="computeroutput"><span class="identifier">ch</span><span class="special">.</span><span class="identifier">domain</span><span class="special">()</span></code>:
</p>
<pre class="programlisting"><span class="keyword">auto</span> <span class="special">[</span><span class="identifier">x_min</span><span class="special">,</span> <span class="identifier">x_max</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">ch</span><span class="special">.</span><span class="identifier">domain</span><span class="special">();</span>
</pre>
<h4>
<a name="math_toolkit.cubic_hermite.h2"></a>
<span class="phrase"><a name="math_toolkit.cubic_hermite.performance"></a></span><a class="link" href="cubic_hermite.html#math_toolkit.cubic_hermite.performance">Performance</a>
</h4>
<p>
Google benchmark was used to evaluate the performance.
</p>
<pre class="programlisting"><span class="identifier">Run</span> <span class="identifier">on</span> <span class="special">(</span><span class="number">16</span> <span class="identifier">X</span> <span class="number">4300</span> <span class="identifier">MHz</span> <span class="identifier">CPU</span> <span class="identifier">s</span><span class="special">)</span>
<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
<span class="identifier">L1</span> <span class="identifier">Data</span> <span class="number">32</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
<span class="identifier">L1</span> <span class="identifier">Instruction</span> <span class="number">32</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
<span class="identifier">L2</span> <span class="identifier">Unified</span> <span class="number">1024</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
<span class="identifier">L3</span> <span class="identifier">Unified</span> <span class="number">11264</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x1</span><span class="special">)</span>
<span class="identifier">Load</span> <span class="identifier">Average</span><span class="special">:</span> <span class="number">1.16</span><span class="special">,</span> <span class="number">1.11</span><span class="special">,</span> <span class="number">0.99</span>
<span class="special">-----------------------------------------------------</span>
<span class="identifier">Benchmark</span> <span class="identifier">Time</span>
<span class="special">-----------------------------------------------------</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span> <span class="number">9.67</span> <span class="identifier">ns</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span> <span class="number">10.4</span> <span class="identifier">ns</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span> <span class="number">10.9</span> <span class="identifier">ns</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span> <span class="number">12.3</span> <span class="identifier">ns</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span> <span class="number">13.4</span> <span class="identifier">ns</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span> <span class="number">14.9</span> <span class="identifier">ns</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span> <span class="number">15.5</span> <span class="identifier">ns</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span> <span class="number">18.0</span> <span class="identifier">ns</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span> <span class="number">19.9</span> <span class="identifier">ns</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span> <span class="number">23.0</span> <span class="identifier">ns</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span> <span class="number">25.3</span> <span class="identifier">ns</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span> <span class="number">28.9</span> <span class="identifier">ns</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span> <span class="number">31.0</span> <span class="identifier">ns</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span> <span class="number">1.32</span> <span class="identifier">lgN</span>
<span class="identifier">CubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_RMS</span> <span class="number">13</span> <span class="special">%</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span> <span class="number">3.14</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span> <span class="number">3.13</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span> <span class="number">3.19</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span> <span class="number">3.14</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span> <span class="number">3.38</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span> <span class="number">3.54</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span> <span class="number">3.51</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span> <span class="number">3.76</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span> <span class="number">5.82</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span> <span class="number">5.76</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span> <span class="number">5.85</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span> <span class="number">5.69</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span> <span class="number">5.77</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span> <span class="number">4.28</span> <span class="special">(</span><span class="number">1</span><span class="special">)</span>
<span class="identifier">CardinalCubicHermite</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_RMS</span> <span class="number">28</span> <span class="special">%</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span> <span class="number">3.22</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span> <span class="number">3.22</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span> <span class="number">3.26</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span> <span class="number">3.23</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span> <span class="number">3.49</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span> <span class="number">3.57</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span> <span class="number">3.73</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span> <span class="number">4.12</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span> <span class="number">4.13</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span> <span class="number">4.12</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span> <span class="number">4.12</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span> <span class="number">4.12</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span> <span class="number">4.19</span> <span class="identifier">ns</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span> <span class="number">3.73</span> <span class="special">(</span><span class="number">1</span><span class="special">)</span>
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_RMS</span> <span class="number">11</span> <span class="special">%</span>
</pre>
<p>
The logarithmic complexity of the non-equispaced version is evident, as is
the better cache utilization of the "array of structs" version as
the problem size gets larger.
</p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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<a name="math_toolkit.daubechies"></a><a class="link" href="daubechies.html" title="Daubechies Wavelets and Scaling Functions">Daubechies Wavelets and Scaling
Functions</a>
</h2></div></div></div>
<h5>
<a name="math_toolkit.daubechies.h0"></a>
<span class="phrase"><a name="math_toolkit.daubechies.synopsis"></a></span><a class="link" href="daubechies.html#math_toolkit.daubechies.synopsis">Synopsis</a>
</h5>
<pre class="programlisting"><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">daubechies_scaling</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span> <span class="special">{</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">p</span><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">daubechies_scaling</span> <span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="identifier">daubechies_scaling</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">grid_refinements</span> <span class="special">=</span> <span class="special">-</span><span class="number">1</span><span class="special">);</span>
<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="identifier">double_prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">support</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">int64_t</span> <span class="identifier">bytes</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">order</span><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">dyadic_grid</span><span class="special">(</span><span class="identifier">int64_t</span> <span class="identifier">j_max</span><span class="special">);</span>
<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">daubechies_wavelet</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">p</span><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">daubechies_wavelet</span> <span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="identifier">daubechies_wavelet</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">grid_refinements</span> <span class="special">=</span> <span class="special">-</span><span class="number">1</span><span class="special">);</span>
<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="identifier">double_prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">support</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">int64_t</span> <span class="identifier">bytes</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
<span class="special">}</span> <span class="comment">// namespaces</span>
</pre>
<p>
Daubechies wavelets and scaling functions are a family of compactly supported
functions indexed by an integer <span class="emphasis"><em>p</em></span> which have <span class="emphasis"><em>p</em></span>
vanishing moments and an associated filter of length <span class="emphasis"><em>2p</em></span>.
They are used in signal denoising, Galerkin methods for PDEs, and compression.
</p>
<p>
The canonical reference on these functions is Daubechies' monograph <span class="emphasis"><em>Ten
Lectures on Wavelets</em></span>, whose notational conventions we attempt to
follow here.
</p>
<p>
A basic usage is as follows:
</p>
<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">phi</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">daubechies_scaling</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">8</span><span class="special">&gt;();</span>
<span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">phi</span><span class="special">(</span><span class="number">0.38</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">dydx</span> <span class="special">=</span> <span class="identifier">phi</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">0.38</span><span class="special">);</span>
<span class="keyword">auto</span> <span class="identifier">psi</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">daubechies_wavelet</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">8</span><span class="special">&gt;();</span>
<span class="identifier">y</span> <span class="special">=</span> <span class="identifier">psi</span><span class="special">(</span><span class="number">0.38</span><span class="special">);</span>
</pre>
<p>
Note that the constructor call is expensive, as it must assemble a <span class="emphasis"><em>dyadic
grid</em></span>--values of <sub><span class="emphasis"><em>p</em></span></sub>φ at dyadic rationals,
i.e., numbers of the form n/2<sup><span class="emphasis"><em>j</em></span></sup>. You should only instantiate
this class once in the duration of a program. The class is pimpl'd and all
its member functions are threadsafe, so it can be copied cheaply and shared
between threads. The default number of grid refinements is chosen so that the
relative error is controlled to ~2-3 ULPs away from the right-hand side of
the support, where superexponential growth of the condition number of function
evaluation makes this impossible. However, controlling relative error of Daubechies
wavelets and scaling functions is much more difficult than controlling absolute
error, and the memory consumption is much higher in relative mode. The memory
consumption of the class can be queried via
</p>
<pre class="programlisting"><span class="identifier">int64_t</span> <span class="identifier">mem</span> <span class="special">=</span> <span class="identifier">phi</span><span class="special">.</span><span class="identifier">bytes</span><span class="special">();</span>
</pre>
<p>
and if this is deemed unacceptably large, the user may choose to control absolute
error via calling the constructor with the <code class="computeroutput"><span class="identifier">grid_refinements</span></code>
parameter set to -2, so
</p>
<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">phi</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">daubechies_scaling</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">8</span><span class="special">&gt;(-</span><span class="number">2</span><span class="special">);</span>
</pre>
<p>
gives a scaling function which keeps the absolute error bounded by roughly
the double precision unit roundoff.
</p>
<p>
If context precludes the ability to reuse the class throughout the program,
it makes sense to reduce the accuracy even further. This can be done by specifying
the grid refinements, for example,
</p>
<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">phi</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">daubechies_scaling</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">8</span><span class="special">&gt;(</span><span class="number">12</span><span class="special">);</span>
</pre>
<p>
creates a Daubechies scaling function interpolated from a dyadic grid computed
down to depth <span class="emphasis"><em>j</em></span> = 12. The call to the constructor is exponential
time in the number of grid refinements, and the call operator, <code class="computeroutput"><span class="special">.</span><span class="identifier">prime</span></code>, and
<code class="computeroutput"><span class="special">.</span><span class="identifier">double_prime</span></code>
are constant time.
</p>
<p>
Note that the only reason that this is a class, rather than a free function
is that the dyadic grids would make the Boost source download extremely large.
Hence, it may make sense to precompute the dyadic grid and dump it in a <code class="computeroutput"><span class="special">.</span><span class="identifier">cpp</span></code> file;
this can be achieved via
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">float128</span><span class="special">;</span>
<span class="keyword">int</span> <span class="identifier">grid_refinements</span> <span class="special">=</span> <span class="number">12</span><span class="special">;</span>
<span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">derivative</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>
<span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">p</span> <span class="special">=</span> <span class="number">8</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">float128</span><span class="special">&gt;</span> <span class="identifier">v</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">dyadic_grid</span><span class="special">&lt;</span><span class="identifier">float128</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">derivative</span><span class="special">&gt;(</span><span class="identifier">grid_refinements</span><span class="special">);</span>
</pre>
<p>
Note that quad precision is the most accurate precision provided, for both
the dyadic grid and for the scaling function. 1ULP accuracy can only be achieved
for float and double precision, in well-conditioned regions.
</p>
<p>
Derivatives are only available if the wavelet and scaling function has sufficient
smoothness. The compiler will gladly inform you of your error if you try to
call <code class="computeroutput"><span class="special">.</span><span class="identifier">prime</span></code>
on <sub>2</sub>φ, which is not differentiable, but be aware that smoothness increases
with the number of vanishing moments.
</p>
<p>
The axioms of a multiresolution analysis ensure that integer shifts of the
scaling functions are elements of the multiresolution analysis; a side effect
is that the supports of the (unshifted) wavelet and scaling functions are arbitrary.
For this reason, we have provided <code class="computeroutput"><span class="special">.</span><span class="identifier">support</span><span class="special">()</span></code>
so that you can check our conventions:
</p>
<pre class="programlisting"><span class="keyword">auto</span> <span class="special">[</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">phi</span><span class="special">.</span><span class="identifier">support</span><span class="special">();</span>
</pre>
<p>
For definiteness though, for the scaling function, the support is always [0,
<span class="emphasis"><em>2p</em></span> - 1], and the support of the wavelet is [ -<span class="emphasis"><em>p</em></span>
+ 1, <span class="emphasis"><em>p</em></span>].
</p>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/daubechies_2_scaling.svg"></object></span> The 2 vanishing
moment scaling function.
</p>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/daubechies_8_scaling.svg"></object></span> The 8 vanishing
moment scaling function.
</p>
<h4>
<a name="math_toolkit.daubechies.h1"></a>
<span class="phrase"><a name="math_toolkit.daubechies.references"></a></span><a class="link" href="daubechies.html#math_toolkit.daubechies.references">References</a>
</h4>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Daubechies, Ingrid. <span class="emphasis"><em>Ten Lectures on Wavelets.</em></span> Vol.
61. Siam, 1992.
</li>
<li class="listitem">
Mallat, Stephane. <span class="emphasis"><em>A Wavelet Tour of Signal Processing: the sparse
way</em></span> Academic press, 2008.
</li>
<li class="listitem">
Thompson, Nicholas, Maddock, John et al. <span class="emphasis"><em>Towards 1ULP Evaluation
of Daubechies Wavelets</em></span> https://arxiv.org/ftp/arxiv/papers/2005/2005.05424.pdf
</li>
</ul></div>
</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 © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>
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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.daubechies_filters"></a><a class="link" href="daubechies_filters.html" title="Daubechies Filters">Daubechies Filters</a>
</h2></div></div></div>
<h5>
<a name="math_toolkit.daubechies_filters.h0"></a>
<span class="phrase"><a name="math_toolkit.daubechies_filters.synopsis"></a></span><a class="link" href="daubechies_filters.html#math_toolkit.daubechies_filters.synopsis">Synopsis</a>
</h5>
<pre class="programlisting"><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">filters</span><span class="special">/</span><span class="identifier">daubechies</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">:</span><span class="identifier">filters</span> <span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">p</span><span class="special">&gt;</span>
<span class="keyword">constexpr</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">2</span><span class="special">*</span><span class="identifier">p</span><span class="special">&gt;</span> <span class="identifier">daubechies_scaling_filter</span><span class="special">();</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">p</span><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">2</span><span class="special">*</span><span class="identifier">p</span><span class="special">&gt;</span> <span class="identifier">daubechies_wavelet_filter</span><span class="special">();</span>
<span class="special">}</span> <span class="comment">// namespaces</span>
</pre>
<p>
The Daubechies filters provided by Boost.Math return the filter coefficients
of the compactly-supported family discovered by Daubechies.
</p>
<p>
A basic usage is as follows:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">filters</span><span class="special">::</span><span class="identifier">daubechies_scaling_filter</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">filters</span><span class="special">::</span><span class="identifier">daubechies_wavelet_filter</span><span class="special">;</span>
<span class="keyword">auto</span> <span class="identifier">h</span> <span class="special">=</span> <span class="identifier">daubechies_scaling_filter</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">8</span><span class="special">&gt;();</span>
<span class="keyword">auto</span> <span class="identifier">g</span> <span class="special">=</span> <span class="identifier">daubechies_wavelet_filter</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">8</span><span class="special">&gt;();</span>
</pre>
<p>
The filters take the number of vanishing moments as a template argument, returning
a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span></code> which has twice as many entries as
vanishing moments.
</p>
<h4>
<a name="math_toolkit.daubechies_filters.h1"></a>
<span class="phrase"><a name="math_toolkit.daubechies_filters.caveats_and_ambiguities"></a></span><a class="link" href="daubechies_filters.html#math_toolkit.daubechies_filters.caveats_and_ambiguities">Caveats and
Ambiguities</a>
</h4>
<p>
Numerous notational conventions for the filter coefficients exist. The first
ambiguity is whether they should be indexed by the number of vanishing moments,
or the number of filter taps. <span class="emphasis"><em>Boost.Math indexes the family by the
number of vanishing moments.</em></span> This indexing convention is the same
as PyWavelets and Mathematica, but Numerical Recipes and Wikipedia index by
the number of filter taps.
</p>
<p>
The next ambiguity is the overall scale. Boost (and PyWavelets) normalize the
filters so that their L<sub>2</sub> norm is 1. Others normalize so that the L<sub>2</sub> norm is √2.
</p>
<p>
Finally, we can use a convolutional representation of the filters, or a dot
product representation. The dot product has all elements reversed relative
to the convolutional representation.
</p>
<p>
The Boost representation agrees with Table 1 of Daubechies 1988 paper "Orthonormal
Bases of Compactly Supported Wavelets", and Mallat's "A Wavelet Tour
of Signal Processing", Table 7.2.
</p>
<p>
The Boost convention:
</p>
<pre class="programlisting"><span class="comment">// ... </span>
<span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">p</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span>
<span class="keyword">auto</span> <span class="identifier">h</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">filters</span><span class="special">::</span><span class="identifier">daubechies_scaling_filter</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="identifier">p</span><span class="special">&gt;();</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"h = {"</span> <span class="special">&lt;&lt;</span> <span class="identifier">h</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span> <span class="special">&lt;&lt;</span> <span class="identifier">h</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span> <span class="special">&lt;&lt;</span> <span class="identifier">h</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span> <span class="special">&lt;&lt;</span> <span class="identifier">h</span><span class="special">[</span><span class="number">3</span><span class="special">]</span> <span class="special">&lt;&lt;</span> <span class="string">"}\n"</span><span class="special">;</span>
<span class="comment">// output:</span>
<span class="identifier">h</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.48296291314453416</span><span class="special">,</span> <span class="number">0.83651630373780794</span><span class="special">,</span> <span class="number">0.22414386804201339</span><span class="special">,</span> <span class="special">-</span><span class="number">0.12940952255126037</span><span class="special">}</span>
</pre>
<p>
Mathematica conventions:
</p>
<pre class="programlisting"><span class="identifier">WaveletFilterCoefficients</span><span class="special">[</span><span class="identifier">DaubechiesWavelet</span><span class="special">[</span><span class="number">2</span><span class="special">],</span> <span class="string">"PrimalLowpass"</span><span class="special">]</span>
<span class="special">{{</span><span class="number">0</span><span class="special">,</span> <span class="number">0.341506</span><span class="special">},</span> <span class="special">{</span><span class="number">1</span><span class="special">,</span> <span class="number">0.591506</span><span class="special">},</span> <span class="special">{</span><span class="number">2</span><span class="special">,</span> <span class="number">0.158494</span><span class="special">},</span> <span class="special">{</span><span class="number">3</span><span class="special">,</span> <span class="special">-</span><span class="number">0.0915064</span><span class="special">}}</span>
</pre>
<p>
(Multiplying all elements of the Mathematica filter by √2 gives the Boost filter.)
</p>
<p>
PyWavelet conventions:
</p>
<pre class="programlisting"><span class="special">&gt;&gt;&gt;</span> <span class="identifier">import</span> <span class="identifier">pywt</span>
<span class="special">&gt;&gt;&gt;</span> <span class="identifier">print</span><span class="special">(</span><span class="identifier">pywt</span><span class="special">.</span><span class="identifier">Wavelet</span><span class="special">(</span><span class="char">'db2'</span><span class="special">).</span><span class="identifier">dec_lo</span><span class="special">)</span>
<span class="special">[-</span><span class="number">0.12940952255126037</span><span class="special">,</span> <span class="number">0.2241438680420134</span><span class="special">,</span> <span class="number">0.8365163037378079</span><span class="special">,</span> <span class="number">0.48296291314453416</span><span class="special">]</span>
<span class="special">&gt;&gt;&gt;</span> <span class="identifier">np</span><span class="special">.</span><span class="identifier">linalg</span><span class="special">.</span><span class="identifier">norm</span><span class="special">(</span><span class="identifier">pywt</span><span class="special">.</span><span class="identifier">Wavelet</span><span class="special">(</span><span class="char">'db2'</span><span class="special">).</span><span class="identifier">dec_lo</span><span class="special">)</span>
<span class="number">1.0</span>
</pre>
<p>
(Reversing the PyWavelet filter gives the Boost filter.)
</p>
<p>
The wavelet filters bring additional ambiguity, because they are defined only
by being orthogonal to the scaling filter. So both sign and scale are determined
by convention.
</p>
<p>
For this reason, we demo some other conventions for the wavelet filter as well:
</p>
<p>
Boost:
</p>
<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">g</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">filters</span><span class="special">::</span><span class="identifier">daubechies_wavelet_filter</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="identifier">p</span><span class="special">&gt;();</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"g = {"</span> <span class="special">&lt;&lt;</span> <span class="identifier">g</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span> <span class="special">&lt;&lt;</span> <span class="identifier">g</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span> <span class="special">&lt;&lt;</span> <span class="identifier">g</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span> <span class="special">&lt;&lt;</span> <span class="identifier">g</span><span class="special">[</span><span class="number">3</span><span class="special">]</span> <span class="special">&lt;&lt;</span> <span class="string">"}\n"</span><span class="special">;</span>
<span class="comment">// output:</span>
<span class="identifier">g</span> <span class="special">=</span> <span class="special">{-</span><span class="number">0.12940952255126037</span><span class="special">,</span> <span class="special">-</span><span class="number">0.22414386804201339</span><span class="special">,</span> <span class="number">0.83651630373780794</span><span class="special">,</span> <span class="special">-</span><span class="number">0.48296291314453416</span><span class="special">}</span>
</pre>
<p>
Mathematica:
</p>
<pre class="programlisting"><span class="identifier">WaveletFilterCoefficients</span><span class="special">[</span><span class="identifier">DaubechiesWavelet</span><span class="special">[</span><span class="number">2</span><span class="special">],</span> <span class="string">"PrimalHighpass"</span><span class="special">]</span>
<span class="special">{{-</span><span class="number">2</span><span class="special">,</span> <span class="special">-</span><span class="number">0.0915064</span><span class="special">},</span> <span class="special">{-</span><span class="number">1</span><span class="special">,</span> <span class="special">-</span><span class="number">0.158494</span><span class="special">},</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span> <span class="number">0.591506</span><span class="special">},</span> <span class="special">{</span><span class="number">1</span><span class="special">,</span> <span class="special">-</span><span class="number">0.341506</span><span class="special">}}</span>
</pre>
<p>
(Note how Mathematica indexes the filter starting at -2; Boost starts at 0,
and the scaling convention is different.)
</p>
<p>
PyWavelets:
</p>
<pre class="programlisting"><span class="special">&gt;&gt;&gt;</span> <span class="identifier">print</span><span class="special">(</span><span class="identifier">pywt</span><span class="special">.</span><span class="identifier">Wavelet</span><span class="special">(</span><span class="char">'db2'</span><span class="special">).</span><span class="identifier">dec_hi</span><span class="special">)</span>
<span class="special">[-</span><span class="number">0.48296291314453416</span><span class="special">,</span> <span class="number">0.8365163037378079</span><span class="special">,</span> <span class="special">-</span><span class="number">0.2241438680420134</span><span class="special">,</span> <span class="special">-</span><span class="number">0.12940952255126037</span><span class="special">]</span>
</pre>
<p>
(Again, reversing the PyWavelets filter gives the Boost filter.)
</p>
<h4>
<a name="math_toolkit.daubechies_filters.h2"></a>
<span class="phrase"><a name="math_toolkit.daubechies_filters.accuracy"></a></span><a class="link" href="daubechies_filters.html#math_toolkit.daubechies_filters.accuracy">Accuracy</a>
</h4>
<p>
The filters are accurate to octuple precision.
</p>
<h4>
<a name="math_toolkit.daubechies_filters.h3"></a>
<span class="phrase"><a name="math_toolkit.daubechies_filters.references"></a></span><a class="link" href="daubechies_filters.html#math_toolkit.daubechies_filters.references">References</a>
</h4>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Ingrid Daubechies, <span class="emphasis"><em>Ten Lectures on Wavelets</em></span>, SIAM
Volume 61, 1992
</li>
<li class="listitem">
Ingrid Daubechies, <span class="emphasis"><em>Orthonormal bases of compactly supported wavelets</em></span>,
Communications on pure and applied mathematics 41.7 (1988): 909-996.
</li>
<li class="listitem">
Stéphane Mallat. <span class="emphasis"><em>A wavelet tour of signal processing.</em></span>
Elsevier, 1999.
</li>
</ul></div>
</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 © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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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@@ -67,7 +67,7 @@
There are two sources of error from finite differences: One, the truncation
error arising from using a finite number of samples to cancel out higher order
terms in the Taylor series. The second is the roundoff error involved in evaluating
the function. The truncation error goes to zero as <span class="emphasis"><em>h</em></span> &#8594;
the function. The truncation error goes to zero as <span class="emphasis"><em>h</em></span> →
0, but the roundoff error becomes unbounded. By balancing these two sources
of error, we can choose a value of <span class="emphasis"><em>h</em></span> that minimizes the
maximum total error. For this reason boost's <code class="computeroutput"><span class="identifier">finite_difference_derivative</span></code>
@@ -118,12 +118,12 @@
</p>
<p>
For a finite-difference method of order <span class="emphasis"><em>k</em></span>, the error is
<span class="emphasis"><em>C</em></span> &#949;<sup>k/k+1</sup>. In the limit <span class="emphasis"><em>k</em></span> &#8594;
&#8734;, we see that the error tends to &#949;, recovering the full precision
<span class="emphasis"><em>C</em></span> ε<sup>k/k+1</sup>. In the limit <span class="emphasis"><em>k</em></span> →
∞, we see that the error tends to ε, recovering the full precision
for the type. However, this ignores the fact that higher-order methods require
subtracting more nearly-equal (perhaps noisy) terms, so the constant <span class="emphasis"><em>C</em></span>
grows with <span class="emphasis"><em>k</em></span>. Since <span class="emphasis"><em>C</em></span> grows quickly
and &#949;<sup>k/k+1</sup> approaches &#949; slowly, we can see there is a compromise
and ε<sup>k/k+1</sup> approaches ε slowly, we can see there is a compromise
between high-order accuracy and conditioning of the difference quotient. In
practice we have found that <span class="emphasis"><em>k=6</em></span> seems to be a good compromise
between the two (and have made this the default), but users are encouraged
@@ -131,7 +131,7 @@
given problem.
</p>
<div class="table">
<a name="math_toolkit.diff.id"></a><p class="title"><b>Table&#160;13.1.&#160;Cost of Finite-Difference Numerical Differentiation</b></p>
<a name="math_toolkit.diff.id"></a><p class="title"><b>Table 13.1. Cost of Finite-Difference Numerical Differentiation</b></p>
<div class="table-contents"><table class="table" summary="Cost of Finite-Difference Numerical Differentiation">
<colgroup>
<col>
@@ -181,7 +181,7 @@
</td>
<td>
<p>
&#949;<sup>1/2</sup>
ε<sup>1/2</sup>
</p>
</td>
<td>
@@ -208,7 +208,7 @@
</td>
<td>
<p>
&#949;<sup>2/3</sup>
ε<sup>2/3</sup>
</p>
</td>
<td>
@@ -235,7 +235,7 @@
</td>
<td>
<p>
&#949;<sup>4/5</sup>
ε<sup>4/5</sup>
</p>
</td>
<td>
@@ -262,7 +262,7 @@
</td>
<td>
<p>
&#949;<sup>6/7</sup>
ε<sup>6/7</sup>
</p>
</td>
<td>
@@ -289,7 +289,7 @@
</td>
<td>
<p>
&#949;<sup>8/9</sup>
ε<sup>8/9</sup>
</p>
</td>
<td>
@@ -314,9 +314,9 @@
real line of a holomorphic function which takes real values at real argument,
then the <span class="bold"><strong>complex step derivative</strong></span> can be used.
The idea is very simple: Since <span class="emphasis"><em>f</em></span> is complex-differentiable,
<span class="emphasis"><em>f(x+&#8520; h) = f(x) + &#8520; hf'(x) - h<sup>2</sup>f''(x) + &#119926;(h<sup>3</sup>)</em></span>.
As long as <span class="emphasis"><em>f(x)</em></span> &#8712; &#8477;, then <span class="emphasis"><em>f'(x)
= &#8465; f(x+&#8520; h)/h + &#119926;(h<sup>2</sup>)</em></span>. This method requires a single
<span class="emphasis"><em>f(x+ⅈ h) = f(x) + ⅈ hf'(x) - h<sup>2</sup>f''(x) + 𝑶(h<sup>3</sup>)</em></span>.
As long as <span class="emphasis"><em>f(x)</em></span> ∈ ℝ, then <span class="emphasis"><em>f'(x)
= ℑ f(x+ⅈ h)/h + 𝑶(h<sup>2</sup>)</em></span>. This method requires a single
complex function evaluation and is not subject to the catastrophic subtractive
cancellation that plagues finite-difference calculations.
</p>
@@ -348,11 +348,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

23
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View File

@@ -1,13 +1,13 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Lanczos Smoothing Derivatives</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../quadrature.html" title="Chapter&#160;13.&#160;Quadrature and Differentiation">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../quadrature.html" title="Chapter 13. Quadrature and Differentiation">
<link rel="prev" href="autodiff.html" title="Automatic Differentiation">
<link rel="next" href="../filters.html" title="Chapter&#160;14.&#160;Filters">
<link rel="next" href="../filters.html" title="Chapter 14. Filters">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
@@ -98,12 +98,12 @@
<span class="identifier">lanczos</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">dvdt</span><span class="special">);</span>
</pre>
<p>
If the data has variance &#963;<sup>2</sup>, then the variance of the computed derivative
is roughly &#963;<sup>2</sup><span class="emphasis"><em>p</em></span><sup>3</sup> <span class="emphasis"><em>n</em></span><sup>-3</sup> &#916;
If the data has variance σ<sup>2</sup>, then the variance of the computed derivative
is roughly σ<sup>2</sup><span class="emphasis"><em>p</em></span><sup>3</sup> <span class="emphasis"><em>n</em></span><sup>-3</sup> Δ
<span class="emphasis"><em>t</em></span><sup>-2</sup>, i.e., it increases cubically with the approximation
order <span class="emphasis"><em>p</em></span>, linearly with the data variance, and decreases
at the cube of the filter length <span class="emphasis"><em>n</em></span>. In addition, we must
not forget the discretization error which is <span class="emphasis"><em>O</em></span>(&#916;
not forget the discretization error which is <span class="emphasis"><em>O</em></span>(Δ
<span class="emphasis"><em>t</em></span><sup><span class="emphasis"><em>p</em></span></sup>). You can play around with the
approximation order <span class="emphasis"><em>p</em></span> and the filter length <span class="emphasis"><em>n</em></span>:
</p>
@@ -232,11 +232,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
doc/html/math_toolkit/directories.html
View File

@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Directory and File Structure</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../overview.html" title="Chapter&#160;1.&#160;Overview">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../overview.html" title="Chapter 1. Overview">
<link rel="prev" href="hints.html" title="Other Hints and tips">
<link rel="next" href="namespaces.html" title="Namespaces">
</head>
@@ -111,11 +111,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

17
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View File

@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Conceptual Requirements for Distribution Types</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../using_udt.html" title="Chapter&#160;20.&#160;Use with User-Defined Floating-Point Types - Boost.Multiprecision and others">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../using_udt.html" title="Chapter 20. Use with User-Defined Floating-Point Types - Boost.Multiprecision and others">
<link rel="prev" href="real_concepts.html" title="Conceptual Requirements for Real Number Types">
<link rel="next" href="archetypes.html" title="Conceptual Archetypes for Reals and Distributions">
</head>
@@ -96,7 +96,7 @@
</td>
<td>
<p>
The <a class="link" href="../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> to use when evaluating functions
The <a class="link" href="../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> to use when evaluating functions
that depend on this distribution.
</p>
</td>
@@ -380,11 +380,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

17
doc/html/math_toolkit/dist_ref.html
View File

@@ -1,11 +1,11 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Statistical Distributions Reference</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.12.0">
<link rel="up" href="../dist.html" title="Chapter&#160;5.&#160;Statistical Distributions and Functions">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dist.html" title="Chapter 5. Statistical Distributions and Functions">
<link rel="prev" href="stat_tut/dist_params.html" title="Discrete Probability Distributions">
<link rel="next" href="dist_ref/nmp.html" title="Non-Member Properties">
</head>
@@ -60,6 +60,8 @@
Gamma Distribution</a></span></dt>
<dt><span class="section"><a href="dist_ref/dists/inverse_gaussian_dist.html">Inverse
Gaussian (or Inverse Normal) Distribution</a></span></dt>
<dt><span class="section"><a href="dist_ref/dists/kolmogorov_smirnov_dist.html">Kolmogorov-Smirnov
Distribution</a></span></dt>
<dt><span class="section"><a href="dist_ref/dists/laplace_dist.html">Laplace Distribution</a></span></dt>
<dt><span class="section"><a href="dist_ref/dists/logistic_dist.html">Logistic
Distribution</a></span></dt>
@@ -94,11 +96,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

21
doc/html/math_toolkit/dist_ref/dist_algorithms.html
View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Distribution Algorithms</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dist_ref.html" title="Statistical Distributions Reference">
<link rel="prev" href="dists/weibull_dist.html" title="Weibull Distribution">
<link rel="next" href="../future.html" title="Extras/Future Directions">
@@ -56,13 +56,13 @@
<pre class="programlisting"><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">distributions</span><span class="special">/</span><span class="identifier">find_location</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Dist</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span> <span class="comment">// explicit error handling policy</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Dist</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">&gt;</span> <span class="comment">// explicit error handling policy</span>
<span class="keyword">typename</span> <span class="identifier">Dist</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">find_location</span><span class="special">(</span> <span class="comment">// For example, normal mean.</span>
<span class="keyword">typename</span> <span class="identifier">Dist</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">z</span><span class="special">,</span> <span class="comment">// location of random variable z to give probability, P(X &gt; z) == p.</span>
<span class="comment">// For example, a nominal minimum acceptable z, so that p * 100 % are &gt; z</span>
<span class="keyword">typename</span> <span class="identifier">Dist</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// probability value desired at x, say 0.95 for 95% &gt; z.</span>
<span class="keyword">typename</span> <span class="identifier">Dist</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">scale</span><span class="special">,</span> <span class="comment">// scale parameter, for example, normal standard deviation.</span>
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;</span> <span class="identifier">pol</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">&amp;</span> <span class="identifier">pol</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Dist</span><span class="special">&gt;</span> <span class="comment">// with default policy.</span>
<span class="keyword">typename</span> <span class="identifier">Dist</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">find_location</span><span class="special">(</span> <span class="comment">// For example, normal mean.</span>
@@ -81,13 +81,13 @@
<pre class="programlisting"><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">distributions</span><span class="special">/</span><span class="identifier">find_scale</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Dist</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Dist</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">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">Dist</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">find_scale</span><span class="special">(</span> <span class="comment">// For example, normal mean.</span>
<span class="keyword">typename</span> <span class="identifier">Dist</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">z</span><span class="special">,</span> <span class="comment">// location of random variable z to give probability, P(X &gt; z) == p.</span>
<span class="comment">// For example, a nominal minimum acceptable weight z, so that p * 100 % are &gt; z</span>
<span class="keyword">typename</span> <span class="identifier">Dist</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// probability value desired at x, say 0.95 for 95% &gt; z.</span>
<span class="keyword">typename</span> <span class="identifier">Dist</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">location</span><span class="special">,</span> <span class="comment">// location parameter, for example, normal distribution mean.</span>
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;</span> <span class="identifier">pol</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">&amp;</span> <span class="identifier">pol</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Dist</span><span class="special">&gt;</span> <span class="comment">// with default policy.</span>
<span class="keyword">typename</span> <span class="identifier">Dist</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">find_scale</span><span class="special">(</span> <span class="comment">// For example, normal mean.</span>
@@ -119,11 +119,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

15
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Distributions</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dist_ref.html" title="Statistical Distributions Reference">
<link rel="prev" href="nmp.html" title="Non-Member Properties">
<link rel="next" href="dists/arcine_dist.html" title="Arcsine Distribution">
@@ -57,6 +57,8 @@
Gamma Distribution</a></span></dt>
<dt><span class="section"><a href="dists/inverse_gaussian_dist.html">Inverse
Gaussian (or Inverse Normal) Distribution</a></span></dt>
<dt><span class="section"><a href="dists/kolmogorov_smirnov_dist.html">Kolmogorov-Smirnov
Distribution</a></span></dt>
<dt><span class="section"><a href="dists/laplace_dist.html">Laplace Distribution</a></span></dt>
<dt><span class="section"><a href="dists/logistic_dist.html">Logistic
Distribution</a></span></dt>
@@ -89,11 +91,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

47
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Arcsine Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="../dists.html" title="Distributions">
<link rel="next" href="bernoulli_dist.html" title="Bernoulli Distribution">
@@ -30,12 +30,12 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">arcsine_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">arcsine_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">arcsine</span><span class="special">;</span> <span class="comment">// double precision standard arcsine distribution [0,1].</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</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">&gt;</span>
<span class="keyword">class</span> <span class="identifier">arcsine_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -83,7 +83,7 @@
is given by:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">f(x; x_min, x_max) = 1 /(&#960;&#8901;&#8730;((x - x_min)&#8901;(x_max
<span class="serif_italic">f(x; x_min, x_max) = 1 /(π⋅√((x - x_min)⋅(x_max
- x_min))</span>
</p></blockquote></div>
<p>
@@ -122,8 +122,8 @@
The Cumulative Distribution Function CDF is defined as
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">F(x) = 2&#8901;arcsin(&#8730;((x-x_min)/(x_max - x))) /
&#960;</span>
<span class="serif_italic">F(x) = 2⋅arcsin(√((x-x_min)/(x_max - x))) /
π</span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/arcsine_cdf.svg" align="middle"></span>
@@ -259,7 +259,7 @@
<span class="comment">// pdf has a minimum at x = 0.5</span>
</pre>
<p>
From the plot of PDF, it is clear that <span class="emphasis"><em>x</em></span> = &#189; is the
From the plot of PDF, it is clear that <span class="emphasis"><em>x</em></span> = ½ is the
<span class="bold"><strong>minimum</strong></span> of the curve, so this is the
<span class="bold"><strong>least likely</strong></span> scenario. (This is highly
counter-intuitive, considering that fair tosses must <span class="bold"><strong>eventually</strong></span>
@@ -272,7 +272,7 @@
= 1.
</p>
<p>
If fraction of time on the left is a &#188;, it is only slightly more likely
If fraction of time on the left is a ¼, it is only slightly more likely
because the curve is quite flat bottomed.
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">1.</span> <span class="special">/</span> <span class="number">4</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.735</span>
@@ -341,7 +341,7 @@
<p>
The arcsine distribution with <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max
= 1</em></span> is special case of the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta
Distribution</a> with &#945; = 1/2 and &#946; = 1/2.
Distribution</a> with α = 1/2 and β = 1/2.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.arcine_dist.h5"></a>
@@ -410,7 +410,7 @@
</td>
<td>
<p>
x &#8712; [a, b], default x &#8712; [0, 1]
x ∈ [a, b], default x ∈ [0, 1]
</p>
</td>
</tr>
@@ -422,7 +422,7 @@
</td>
<td>
<p>
f(x; a, b) = 1/(&#960;&#8901;&#8730;(x - a)&#8901;(b - x))
f(x; a, b) = 1/(π⋅√(x - a)⋅(b - x))
</p>
</td>
</tr>
@@ -434,7 +434,7 @@
</td>
<td>
<p>
F(x) = 2/&#960;&#8901;sin<sup>-1</sup>(&#8730;(x - a) / (b - a) )
F(x) = 2/π⋅sin<sup>-1</sup>(√(x - a) / (b - a) )
</p>
</td>
</tr>
@@ -446,7 +446,7 @@
</td>
<td>
<p>
2/(&#960;&#8901;cos<sup>-1</sup>(&#8730;(x - a) / (b - a)))
2/(π⋅cos<sup>-1</sup>(√(x - a) / (b - a)))
</p>
</td>
</tr>
@@ -458,7 +458,7 @@
</td>
<td>
<p>
-a&#8901;sin<sup>2</sup>(&#189;&#960;&#8901;p) + a + b&#8901;sin<sup>2</sup>(&#189;&#960;&#8901;p)
-a⋅sin<sup>2</sup>(½π⋅p) + a + b⋅sin<sup>2</sup>(½π⋅p)
</p>
</td>
</tr>
@@ -470,7 +470,7 @@
</td>
<td>
<p>
-a&#8901;cos<sup>2</sup>(&#189;&#960;&#8901;p) + a + b&#8901;cos<sup>2</sup>(&#189;&#960;&#8901;q)
-a⋅cos<sup>2</sup>(½π⋅p) + a + b⋅cos<sup>2</sup>(½π⋅q)
</p>
</td>
</tr>
@@ -482,7 +482,7 @@
</td>
<td>
<p>
&#189;(a+b)
½(a+b)
</p>
</td>
</tr>
@@ -494,7 +494,7 @@
</td>
<td>
<p>
&#189;(a+b)
½(a+b)
</p>
</td>
</tr>
@@ -506,7 +506,7 @@
</td>
<td>
<p>
x &#8712; [a, b], so raises domain_error (returning NaN).
x ∈ [a, b], so raises domain_error (returning NaN).
</p>
</td>
</tr>
@@ -630,11 +630,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

17
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Bernoulli Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="arcine_dist.html" title="Arcsine Distribution">
<link rel="next" href="beta_dist.html" title="Beta Distribution">
@@ -30,12 +30,12 @@
<pre class="programlisting"><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">distributions</span><span class="special">/</span><span class="identifier">bernoulli</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">bernoulli_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">bernoulli_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">bernoulli</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</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">&gt;</span>
<span class="keyword">class</span> <span class="identifier">bernoulli_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -350,11 +350,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

51
doc/html/math_toolkit/dist_ref/dists/beta_dist.html
View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Beta Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="bernoulli_dist.html" title="Bernoulli Distribution">
<link rel="next" href="binomial_dist.html" title="Binomial Distribution">
@@ -30,14 +30,14 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">beta_distribution</span><span class="special">;</span>
<span class="comment">// typedef beta_distribution&lt;double&gt; beta;</span>
<span class="comment">// Note that this is deliberately NOT provided,</span>
<span class="comment">// to avoid a clash with the function name beta.</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</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">&gt;</span>
<span class="keyword">class</span> <span class="identifier">beta_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -102,17 +102,17 @@
distribution</a> defined on the interval [0,1] is given by:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">f(x;&#945;,&#946;) = x<sup>&#945; - 1</sup> (1 - x)<sup>&#946; -1</sup> / B(&#945;, &#946;)</span>
<span class="serif_italic">f(x;α,β) = x<sup>α - 1</sup> (1 - x)<sup>β -1</sup> / B(α, β)</span>
</p></blockquote></div>
<p>
where <span class="serif_italic">B(&#945;, &#946;)</span> is the <a href="http://en.wikipedia.org/wiki/Beta_function" target="_top">beta
where <span class="serif_italic">B(α, β)</span> is the <a href="http://en.wikipedia.org/wiki/Beta_function" target="_top">beta
function</a>, implemented in this library as <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a>.
Division by the beta function ensures that the pdf is normalized to the
range zero to unity.
</p>
<p>
The following graph illustrates examples of the pdf for various values
of the shape parameters. Note the <span class="emphasis"><em>&#945; = &#946; = 2</em></span> (blue line)
of the shape parameters. Note the <span class="emphasis"><em>α = β = 2</em></span> (blue line)
is dome-shaped, and might be approximated by a symmetrical triangular distribution.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
@@ -120,10 +120,10 @@
</p></blockquote></div>
<p>
If &#945; = &#946; = 1, then it is a &#8195;
If α = β = 1, then it is a  
<a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
distribution</a>, equal to unity in the entire interval x = 0 to 1.
If &#945; and &#946; are &lt; 1, then the pdf is U-shaped. If &#945; != &#946;, then the shape is
If α and β are &lt; 1, then the pdf is U-shaped. If α != β, then the shape is
asymmetric and could be approximated by a triangle whose apex is away from
the centre (where x = half).
</p>
@@ -192,14 +192,14 @@
Two pairs of parameter estimators are provided.
</p>
<p>
One estimates either &#945; or &#946;
One estimates either α or β
from presumed-known mean and variance.
</p>
<p>
The other pair estimates either &#945; or &#946; from the cdf and x.
The other pair estimates either α or β from the cdf and x.
</p>
<p>
It is also possible to estimate &#945; and &#946; from 'known' mode &amp; quantile. For
It is also possible to estimate α and β from 'known' mode &amp; quantile. For
example, calculators are provided by the <a href="http://www.ausvet.com.au/pprev/content.php?page=PPscript" target="_top">Pooled
Prevalence Calculator</a> and <a href="http://www.epi.ucdavis.edu/diagnostictests/betabuster.html" target="_top">Beta
Buster</a> but this is not yet implemented here.
@@ -209,7 +209,7 @@ from presumed-known mean and variance.
<span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
</pre>
<p>
Returns the unique value of &#945; that corresponds to a beta distribution with
Returns the unique value of α that corresponds to a beta distribution with
mean <span class="emphasis"><em>mean</em></span> and variance <span class="emphasis"><em>variance</em></span>.
</p>
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
@@ -217,7 +217,7 @@ from presumed-known mean and variance.
<span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
</pre>
<p>
Returns the unique value of &#946; that corresponds to a beta distribution with
Returns the unique value of β that corresponds to a beta distribution with
mean <span class="emphasis"><em>mean</em></span> and variance <span class="emphasis"><em>variance</em></span>.
</p>
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
@@ -226,7 +226,7 @@ from presumed-known mean and variance.
<span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf</span>
</pre>
<p>
Returns the value of &#945; that gives: <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">beta_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;(</span><span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">beta</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">probability</span></code>.
Returns the value of α that gives: <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">beta_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;(</span><span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">beta</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">probability</span></code>.
</p>
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="comment">// alpha.</span>
@@ -234,7 +234,7 @@ from presumed-known mean and variance.
<span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf.</span>
</pre>
<p>
Returns the value of &#946; that gives: <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">beta_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;(</span><span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">beta</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">probability</span></code>.
Returns the value of β that gives: <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">beta_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;(</span><span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">beta</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">probability</span></code>.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.beta_dist.h4"></a>
@@ -277,7 +277,7 @@ from presumed-known mean and variance.
distributions</a>
</h5>
<p>
The beta distribution with both &#945; and &#946; = 1 follows a <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
The beta distribution with both α and β = 1 follows a <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
distribution</a>.
</p>
<p>
@@ -286,10 +286,10 @@ from presumed-known mean and variance.
</p>
<p>
The <a href="http://en.wikipedia.org/wiki/Binomial_distribution" target="_top">binomial
distribution</a> is closely related when &#945; and &#946; are integers.
distribution</a> is closely related when α and β are integers.
</p>
<p>
With integer values of &#945; and &#946; the distribution B(i, j) is that of the j-th
With integer values of α and β the distribution B(i, j) is that of the j-th
highest of a sample of i + j + 1 independent random variables uniformly
distributed between 0 and 1. The cumulative probability from 0 to x is
thus the probability that the j-th highest value is less than x. Or it
@@ -315,7 +315,7 @@ from presumed-known mean and variance.
</h5>
<p>
In the following table <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
are the parameters &#945; and &#946;, <span class="emphasis"><em>x</em></span> is the random variable,
are the parameters α and β, <span class="emphasis"><em>x</em></span> is the random variable,
<span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
</p>
<div class="informaltable"><table class="table">
@@ -344,7 +344,7 @@ from presumed-known mean and variance.
</td>
<td>
<p>
<span class="serif_italic">f(x;&#945;,&#946;) = x<sup>&#945; - 1</sup> (1 - x)<sup>&#946; -1</sup> / B(&#945;, &#946;)</span>
<span class="serif_italic">f(x;α,β) = x<sup>α - 1</sup> (1 - x)<sup>β -1</sup> / B(α, β)</span>
</p>
<p>
Implemented using <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(a,
@@ -603,11 +603,10 @@ from presumed-known mean and variance.
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

23
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View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Binomial Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="beta_dist.html" title="Beta Distribution">
<link rel="next" href="cauchy_dist.html" title="Cauchy-Lorentz Distribution">
@@ -31,12 +31,12 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">binomial_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">binomial</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</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">&gt;</span>
<span class="keyword">class</span> <span class="identifier">binomial_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -365,11 +365,11 @@
</p>
<p>
If the desired risk level that the true success fraction lies outside
the bounds is &#945;, then you pass &#945;/2 to these functions.
the bounds is α, then you pass α/2 to these functions.
</p>
<p>
So for example a two sided 95% confidence interval would be obtained
by passing &#945; = 0.025 to each of the functions.
by passing α = 0.025 to each of the functions.
</p>
<p>
<a class="link" href="../../stat_tut/weg/binom_eg/binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution">See worked
@@ -493,7 +493,7 @@
in the context of this distribution:
</p>
<div class="table">
<a name="math_toolkit.dist_ref.dists.binomial_dist.meaning_of_the_non_member_access"></a><p class="title"><b>Table&#160;5.1.&#160;Meaning of the non-member accessors</b></p>
<a name="math_toolkit.dist_ref.dists.binomial_dist.meaning_of_the_non_member_access"></a><p class="title"><b>Table 5.1. Meaning of the non-member accessors</b></p>
<div class="table-contents"><table class="table" summary="Meaning of the non-member accessors">
<colgroup>
<col>
@@ -887,11 +887,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

35
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Cauchy-Lorentz Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="binomial_dist.html" title="Binomial Distribution">
<link rel="next" href="chi_squared_dist.html" title="Chi Squared Distribution">
@@ -29,12 +29,12 @@
</h4></div></div></div>
<pre class="programlisting"><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">distributions</span><span class="special">/</span><span class="identifier">cauchy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">cauchy_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">cauchy_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">cauchy</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</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">&gt;</span>
<span class="keyword">class</span> <span class="identifier">cauchy_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -61,7 +61,7 @@
<p>
The location parameter <span class="emphasis"><em>x<sub>0</sub></em></span> is the location of the peak
of the distribution (the mode of the distribution), while the scale parameter
&#947; specifies half the width of the PDF at half the maximum height. If the
γ specifies half the width of the PDF at half the maximum height. If the
location is zero, and the scale 1, then the result is a standard Cauchy
distribution.
</p>
@@ -161,7 +161,7 @@
</h5>
<p>
In the following table x<sub>0 </sub> is the location parameter of the distribution,
&#947; is its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate,
γ is its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate,
<span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
</p>
<div class="informaltable"><table class="table">
@@ -190,8 +190,8 @@
</td>
<td>
<p>
Using the relation: <span class="emphasis"><em>pdf = 1 / (&#960; * &#947; * (1 + ((x - x<sub>0 </sub>)
/ &#947;)<sup>2</sup>) </em></span>
Using the relation: <span class="emphasis"><em>pdf = 1 / (π * γ * (1 + ((x - x<sub>0 </sub>)
/ γ)<sup>2</sup>) </em></span>
</p>
</td>
</tr>
@@ -206,15 +206,15 @@
The cdf is normally given by:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">p = 0.5 + atan(x)/&#960;</span>
<span class="serif_italic">p = 0.5 + atan(x)/π</span>
</p></blockquote></div>
<p>
But that suffers from cancellation error as x -&gt; -&#8734;. So recall
But that suffers from cancellation error as x -&gt; -∞. So recall
that for <code class="computeroutput"><span class="identifier">x</span> <span class="special">&lt;</span>
<span class="number">0</span></code>:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">atan(x) = -&#960;/2 - atan(1/x)</span>
<span class="serif_italic">atan(x) = -π/2 - atan(1/x)</span>
</p></blockquote></div>
<p>
Substituting into the above we get:
@@ -225,7 +225,7 @@
<p>
So the procedure is to calculate the cdf for -fabs(x) using the
above formula. Note that to factor in the location and scale
parameters you must substitute (x - x<sub>0 </sub>) / &#947; for x in the above.
parameters you must substitute (x - x<sub>0 </sub>) / γ for x in the above.
</p>
<p>
This procedure yields the smaller of <span class="emphasis"><em>p</em></span> and
@@ -248,7 +248,7 @@
is reduced to the range [-0.5, 0.5], then the relation
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">x = x<sub>0 </sub> &#177; &#947; / tan(&#960; * p)</span>
<span class="serif_italic">x = x<sub>0 </sub> ± γ / tan(π * p)</span>
</p></blockquote></div>
<p>
is used to obtain the result. Whether we're adding or subtracting
@@ -293,11 +293,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

27
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Chi Squared Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="cauchy_dist.html" title="Cauchy-Lorentz Distribution">
<link rel="next" href="empirical_cdf.html" title="Empirical Cumulative Distribution Function">
@@ -31,12 +31,12 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">chi_squared_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">chi_squared_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">chi_squared</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</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">&gt;</span>
<span class="keyword">class</span> <span class="identifier">chi_squared_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -62,9 +62,9 @@
</pre>
<p>
The Chi-Squared distribution is one of the most widely used distributions
in statistical tests. If &#967;<sub>i</sub> are &#957;
in statistical tests. If χ<sub>i</sub> are ν
independent, normally distributed random
variables with means &#956;<sub>i</sub> and variances &#963;<sub>i</sub><sup>2</sup>, then the random variable:
variables with means μ<sub>i</sub> and variances σ<sub>i</sub><sup>2</sup>, then the random variable:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/chi_squ_ref1.svg"></span>
@@ -75,9 +75,9 @@ independent, normally distributed random
</p>
<p>
The Chi-Squared distribution is a special case of the gamma distribution
and has a single parameter &#957; that specifies the number of degrees of freedom.
and has a single parameter ν that specifies the number of degrees of freedom.
The following graph illustrates how the distribution changes for different
values of &#957;:
values of ν:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/chi_squared_pdf.svg" align="middle"></span>
@@ -190,7 +190,7 @@ independent, normally distributed random
avoid a discontinuity in the mode as alternative code in a comment).
</p>
<p>
The domain of the random variable is [0, +&#8734;].
The domain of the random variable is [0, +∞].
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.chi_squared_dist.h2"></a>
@@ -396,11 +396,10 @@ independent, normally distributed random
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

13
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Empirical Cumulative Distribution Function</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="chi_squared_dist.html" title="Chi Squared Distribution">
<link rel="next" href="exp_dist.html" title="Exponential Distribution">
@@ -187,11 +187,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

35
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Exponential Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="empirical_cdf.html" title="Empirical Cumulative Distribution Function">
<link rel="next" href="extreme_dist.html" title="Extreme Value Distribution">
@@ -28,12 +28,12 @@
</h4></div></div></div>
<pre class="programlisting"><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">distributions</span><span class="special">/</span><span class="identifier">exponential</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">exponential_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">exponential_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">exponential</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</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">&gt;</span>
<span class="keyword">class</span> <span class="identifier">exponential_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -104,7 +104,7 @@
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The domain of the random variable is [0, +&#8734;].
The domain of the random variable is [0, +∞].
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.exp_dist.h2"></a>
@@ -121,7 +121,7 @@
<span class="phrase"><a name="math_toolkit.dist_ref.dists.exp_dist.implementation"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist_ref.dists.exp_dist.implementation">Implementation</a>
</h5>
<p>
In the following table &#955; is the parameter lambda of the distribution, <span class="emphasis"><em>x</em></span>
In the following table λ is the parameter lambda of the distribution, <span class="emphasis"><em>x</em></span>
is the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
= 1-p</em></span>.
</p>
@@ -151,7 +151,7 @@
</td>
<td>
<p>
Using the relation: pdf = &#955; * exp(-&#955; * x)
Using the relation: pdf = λ * exp(-λ * x)
</p>
</td>
</tr>
@@ -163,7 +163,7 @@
</td>
<td>
<p>
Using the relation: p = 1 - exp(-x * &#955;) = -expm1(-x * &#955;)
Using the relation: p = 1 - exp(-x * λ) = -expm1(-x * λ)
</p>
</td>
</tr>
@@ -175,7 +175,7 @@
</td>
<td>
<p>
Using the relation: q = exp(-x * &#955;)
Using the relation: q = exp(-x * λ)
</p>
</td>
</tr>
@@ -187,7 +187,7 @@
</td>
<td>
<p>
Using the relation: x = -log(1-p) / &#955; = -log1p(-p) / &#955;
Using the relation: x = -log(1-p) / λ = -log1p(-p) / λ
</p>
</td>
</tr>
@@ -199,7 +199,7 @@
</td>
<td>
<p>
Using the relation: x = -log(q) / &#955;
Using the relation: x = -log(q) / λ
</p>
</td>
</tr>
@@ -211,7 +211,7 @@
</td>
<td>
<p>
1/&#955;
1/λ
</p>
</td>
</tr>
@@ -223,7 +223,7 @@
</td>
<td>
<p>
1/&#955;
1/λ
</p>
</td>
</tr>
@@ -313,11 +313,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

19
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Extreme Value Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="exp_dist.html" title="Exponential Distribution">
<link rel="next" href="f_dist.html" title="F Distribution">
@@ -29,12 +29,12 @@
</h4></div></div></div>
<pre class="programlisting"><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">distributions</span><span class="special">/</span><span class="identifier">extreme</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">extreme_value_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">extreme_value_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">extreme_value</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</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">&gt;</span>
<span class="keyword">class</span> <span class="identifier">extreme_value_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -138,7 +138,7 @@
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The domain of the random parameter is [-&#8734;, +&#8734;].
The domain of the random parameter is [-∞, +∞].
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.extreme_dist.h2"></a>
@@ -316,11 +316,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

25
doc/html/math_toolkit/dist_ref/dists/f_dist.html
View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>F Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="extreme_dist.html" title="Extreme Value Distribution">
<link rel="next" href="gamma_dist.html" title="Gamma (and Erlang) Distribution">
@@ -30,12 +30,12 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">fisher_f_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">fisher_f_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">fisher_f</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</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">&gt;</span>
<span class="keyword">class</span> <span class="identifier">fisher_f_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -53,15 +53,15 @@
</pre>
<p>
The F distribution is a continuous distribution that arises when testing
whether two samples have the same variance. If &#967;<sup>2</sup><sub>m</sub> and &#967;<sup>2</sup><sub>n</sub> are independent
whether two samples have the same variance. If χ<sup>2</sup><sub>m</sub> and χ<sup>2</sup><sub>n</sub> are independent
variates each distributed as Chi-Squared with <span class="emphasis"><em>m</em></span> and
<span class="emphasis"><em>n</em></span> degrees of freedom, then the test statistic:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">F<sub>n,m</sub> = (&#967;<sup>2</sup><sub>n</sub> / n) / (&#967;<sup>2</sup><sub>m</sub> / m)</span>
<span class="serif_italic">F<sub>n,m</sub> = (χ<sup>2</sup><sub>n</sub> / n) / (χ<sup>2</sup><sub>m</sub> / m)</span>
</p></blockquote></div>
<p>
Is distributed over the range [0, &#8734;] with an F distribution, and has the
Is distributed over the range [0, ∞] with an F distribution, and has the
PDF:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
@@ -119,7 +119,7 @@
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The domain of the random variable is [0, +&#8734;].
The domain of the random variable is [0, +∞].
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.f_dist.h2"></a>
@@ -420,11 +420,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

41
doc/html/math_toolkit/dist_ref/dists/gamma_dist.html
View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Gamma (and Erlang) Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="f_dist.html" title="F Distribution">
<link rel="next" href="geometric_dist.html" title="Geometric Distribution">
@@ -31,7 +31,7 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">gamma_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -75,7 +75,7 @@
</td></tr>
</table></div>
<p>
For shape parameter <span class="emphasis"><em>k</em></span> and scale parameter &#952; it is defined
For shape parameter <span class="emphasis"><em>k</em></span> and scale parameter θ it is defined
by the probability density function:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
@@ -83,8 +83,8 @@
</p></blockquote></div>
<p>
Sometimes an alternative formulation is used: given parameters &#945; = k and
&#946; = 1 / &#952;, then the distribution can be defined by the PDF:
Sometimes an alternative formulation is used: given parameters α = k and
β = 1 / θ, then the distribution can be defined by the PDF:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/gamma_dist_ref2.svg"></span>
@@ -167,7 +167,7 @@
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The domain of the random variable is [0,+&#8734;].
The domain of the random variable is [0,+∞].
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.gamma_dist.h2"></a>
@@ -187,7 +187,7 @@
</h5>
<p>
In the following table <span class="emphasis"><em>k</em></span> is the shape parameter of
the distribution, &#952; is its scale parameter, <span class="emphasis"><em>x</em></span> is the
the distribution, θ is its scale parameter, <span class="emphasis"><em>x</em></span> is the
random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
= 1-p</em></span>.
</p>
@@ -218,7 +218,7 @@
<td>
<p>
Using the relation: pdf = <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(k,
x / &#952;) / &#952;
x / θ) / θ
</p>
</td>
</tr>
@@ -231,7 +231,7 @@
<td>
<p>
Using the relation: p = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(k,
x / &#952;)
x / θ)
</p>
</td>
</tr>
@@ -244,7 +244,7 @@
<td>
<p>
Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(k,
x / &#952;)
x / θ)
</p>
</td>
</tr>
@@ -256,7 +256,7 @@
</td>
<td>
<p>
Using the relation: x = &#952; * <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(k,
Using the relation: x = θ * <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(k,
p)
</p>
</td>
@@ -269,7 +269,7 @@
</td>
<td>
<p>
Using the relation: x = &#952;* <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(k,
Using the relation: x = θ* <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(k,
p)
</p>
</td>
@@ -282,7 +282,7 @@
</td>
<td>
<p>
k&#952;
kθ
</p>
</td>
</tr>
@@ -294,7 +294,7 @@
</td>
<td>
<p>
k&#952;<sup>2</sup>
kθ<sup>2</sup>
</p>
</td>
</tr>
@@ -306,7 +306,7 @@
</td>
<td>
<p>
(k-1)&#952; for <span class="emphasis"><em>k&gt;1</em></span> otherwise a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
(k-1)θ for <span class="emphasis"><em>k&gt;1</em></span> otherwise a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
@@ -351,11 +351,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

31
doc/html/math_toolkit/dist_ref/dists/geometric_dist.html
View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Geometric Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="gamma_dist.html" title="Gamma (and Erlang) Distribution">
<link rel="next" href="hyperexponential_dist.html" title="Hyperexponential Distribution">
@@ -31,12 +31,12 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">geometric_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">geometric_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">geometric</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</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">&gt;</span>
<span class="keyword">class</span> <span class="identifier">geometric_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -109,7 +109,7 @@
</p></blockquote></div>
<p>
For example, when throwing a 6-face dice the success probability <span class="emphasis"><em>p</em></span>
= 1/6 = 0.1666&#8202;&#775;. Throwing repeatedly until a <span class="emphasis"><em>three</em></span>
= 1/6 = 0.1666 ̇. Throwing repeatedly until a <span class="emphasis"><em>three</em></span>
appears, the probability distribution of the number of times <span class="emphasis"><em>not-a-three</em></span>
is thrown is geometric.
</p>
@@ -152,7 +152,7 @@
The geometric distribution is a special case of the <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative
Binomial Distribution</a> with successes parameter <span class="emphasis"><em>r</em></span>
= 1, so only one first and only success is required : thus by definition
&#8192;&#8192; <code class="computeroutput"><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">)</span> <span class="special">==</span>
   <code class="computeroutput"><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">)</span> <span class="special">==</span>
<span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span></code>
</p>
<pre class="programlisting"><span class="identifier">negative_binomial_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">);</span>
@@ -479,7 +479,7 @@
the context of this distribution:
</p>
<div class="table">
<a name="math_toolkit.dist_ref.dists.geometric_dist.meaning_of_the_non_member_access"></a><p class="title"><b>Table&#160;5.2.&#160;Meaning of the non-member accessors.</b></p>
<a name="math_toolkit.dist_ref.dists.geometric_dist.meaning_of_the_non_member_access"></a><p class="title"><b>Table 5.2. Meaning of the non-member accessors.</b></p>
<div class="table-contents"><table class="table" summary="Meaning of the non-member accessors.">
<colgroup>
<col>
@@ -706,7 +706,7 @@
</td>
<td>
<p>
(1-p)/p&#178;
(1-p)/p²
</p>
</td>
</tr>
@@ -730,7 +730,7 @@
</td>
<td>
<p>
(2-p)/&#8730;q
(2-p)/√q
</p>
</td>
</tr>
@@ -742,7 +742,7 @@
</td>
<td>
<p>
9+p&#178;/q
9+p²/q
</p>
</td>
</tr>
@@ -754,7 +754,7 @@
</td>
<td>
<p>
6 +p&#178;/q
6 +p²/q
</p>
</td>
</tr>
@@ -828,11 +828,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

131
doc/html/math_toolkit/dist_ref/dists/hyperexponential_dist.html
View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Hyperexponential Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="geometric_dist.html" title="Geometric Distribution">
<link rel="next" href="hypergeometric_dist.html" title="Hypergeometric Distribution">
@@ -31,12 +31,12 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">typename</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">typename</span> <a class="link" href="../../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">hyperexponential_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">hyperexponential_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">hyperexponential</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">typename</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">typename</span> <a class="link" href="../../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">hyperexponential_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -102,8 +102,8 @@
<p>
A <span class="emphasis"><em>k</em></span>-phase hyperexponential distribution is characterized
by two parameters, namely a <span class="emphasis"><em>phase probability vector</em></span>
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(&#945;<sub>1</sub>,...,&#945;<sub>k</sub>)</em></span> and a
<span class="emphasis"><em>rate vector</em></span> <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(&#955;<sub>1</sub>,...,&#955;<sub>k</sub>)</em></span>.
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(α<sub>1</sub>,...,α<sub>k</sub>)</em></span> and a
<span class="emphasis"><em>rate vector</em></span> <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(λ<sub>1</sub>,...,λ<sub>k</sub>)</em></span>.
</p>
<p>
The <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
@@ -120,24 +120,24 @@
</p>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(1.0)</em></span> and <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(1.0)</em></span> (which degenerates to a simple
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(1.0)</em></span> and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(1.0)</em></span> (which degenerates to a simple
exponential distribution),
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.1, 0.9)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.1, 0.9)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.9, 0.1)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.9, 0.1)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.2, 0.3, 0.5)</em></span>
and <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.0, 1.5)</em></span>,
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.2, 0.3, 0.5)</em></span>
and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.0, 1.5)</em></span>,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.5, 0.3, 0.2)</em></span>
and <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.0, 1.5)</em></span>.
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.5, 0.3, 0.2)</em></span>
and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.0, 1.5)</em></span>.
</li>
</ol></div>
<div class="blockquote"><blockquote class="blockquote"><p>
@@ -152,34 +152,34 @@
</p>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.1, 0.9)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.1, 0.9)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.6, 0.4)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.6, 0.4)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.9, 0.1)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.5, 1.5)</em></span>,
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.9, 0.1)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.5, 1.5)</em></span>,
</li>
<li class="listitem">
Exponential distribution with parameter <span class="emphasis"><em>&#955;=0.5</em></span>,
Exponential distribution with parameter <span class="emphasis"><em>λ=0.5</em></span>,
</li>
<li class="listitem">
Exponential distribution with parameter <span class="emphasis"><em>&#955;=1.5</em></span>.
Exponential distribution with parameter <span class="emphasis"><em>λ=1.5</em></span>.
</li>
</ol></div>
<p>
As expected, as the first element <span class="emphasis"><em>&#945;<sub>1</sub></em></span> of the <span class="emphasis"><em>phase
As expected, as the first element <span class="emphasis"><em>α<sub>1</sub></em></span> of the <span class="emphasis"><em>phase
probability vector</em></span> approaches to <span class="emphasis"><em>1</em></span> (or,
equivalently, <span class="emphasis"><em>&#945;<sub>2</sub></em></span> approaches to <span class="emphasis"><em>0</em></span>),
equivalently, <span class="emphasis"><em>α<sub>2</sub></em></span> approaches to <span class="emphasis"><em>0</em></span>),
the resulting hyperexponential distribution nears the exponential distribution
with parameter <span class="emphasis"><em>&#955;=0.5</em></span>. Conversely, as the first element
<span class="emphasis"><em>&#945;<sub>2</sub></em></span> of the <span class="emphasis"><em>phase probability vector</em></span>
approaches to <span class="emphasis"><em>1</em></span> (or, equivalently, <span class="emphasis"><em>&#945;<sub>1</sub></em></span>
with parameter <span class="emphasis"><em>λ=0.5</em></span>. Conversely, as the first element
<span class="emphasis"><em>α<sub>2</sub></em></span> of the <span class="emphasis"><em>phase probability vector</em></span>
approaches to <span class="emphasis"><em>1</em></span> (or, equivalently, <span class="emphasis"><em>α<sub>1</sub></em></span>
approaches to <span class="emphasis"><em>0</em></span>), the resulting hyperexponential distribution
nears the exponential distribution with parameter <span class="emphasis"><em>&#955;=1.5</em></span>.
nears the exponential distribution with parameter <span class="emphasis"><em>λ=1.5</em></span>.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf_samerate.svg" align="middle"></span>
@@ -192,16 +192,16 @@
</p>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(1.0)</em></span> and <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(2.0)</em></span> (which degenerates to a simple
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(1.0)</em></span> and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(2.0)</em></span> (which degenerates to a simple
exponential distribution),
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.5, 0.5)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.3, 1.5)</em></span>,
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.5, 0.5)</em></span> and
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.3, 1.5)</em></span>,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(1.0/3.0, 1.0/3.0, 1.0/3.0)</em></span>
and <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(0.2, 1.5, 3.0)</em></span>,
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(1.0/3.0, 1.0/3.0, 1.0/3.0)</em></span>
and <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(0.2, 1.5, 3.0)</em></span>,
</li>
</ol></div>
<div class="blockquote"><blockquote class="blockquote"><p>
@@ -227,9 +227,9 @@
theory</a> to model the distribution of the superposition of <span class="emphasis"><em>k</em></span>
independent events, like, for instance, the service time distribution of
a queueing station with <span class="emphasis"><em>k</em></span> servers in parallel where
the <span class="emphasis"><em>i</em></span>-th server is chosen with probability <span class="emphasis"><em>&#945;<sub>i</sub></em></span>
the <span class="emphasis"><em>i</em></span>-th server is chosen with probability <span class="emphasis"><em>α<sub>i</sub></em></span>
and its service time distribution is an exponential distribution with rate
<span class="emphasis"><em>&#955;<sub>i</sub></em></span> (Allen,1990; Papadopolous et al.,1993; Trivedi,2002).
<span class="emphasis"><em>λ<sub>i</sub></em></span> (Allen,1990; Papadopolous et al.,1993; Trivedi,2002).
</p>
<p>
For instance, CPUs service-time distribution in a computing system has
@@ -259,9 +259,9 @@
Distribution</a>.
</li>
<li class="listitem">
When the <span class="emphasis"><em>k</em></span> rates are all equal to <span class="emphasis"><em>&#955;</em></span>,
When the <span class="emphasis"><em>k</em></span> rates are all equal to <span class="emphasis"><em>λ</em></span>,
the hyperexponential distribution is simple an <a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential
Distribution</a> with rate <span class="emphasis"><em>&#955;</em></span>.
Distribution</a> with rate <span class="emphasis"><em>λ</em></span>.
</li>
</ul></div>
<h4>
@@ -279,8 +279,8 @@
lifetime of 12 years. Assuming the lifetime of this appliance follows an
exponential distribution, the lifetime distribution of the purchased appliance
can be modeled as a hyperexponential distribution with phase probability
vector <span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(1/2,1/2)</em></span>
and rate vector <span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(1/10,1/12)</em></span>
vector <span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(1/2,1/2)</em></span>
and rate vector <span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(1/10,1/12)</em></span>
(Wolfram,2014).
</p>
<p>
@@ -503,20 +503,20 @@
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.34561,0.08648,0.56791),
<span class="bold"><strong>&#955;</strong></span>=(0.008,0.00005,0.02894)</em></span>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.34561,0.08648,0.56791),
<span class="bold"><strong>λ</strong></span>=(0.008,0.00005,0.02894)</em></span>
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.24667,0.37948,0.37385),
<span class="bold"><strong>&#955;</strong></span>=(0.00004,0.000002,0.00059)</em></span>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.24667,0.37948,0.37385),
<span class="bold"><strong>λ</strong></span>=(0.00004,0.000002,0.00059)</em></span>
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.09325,0.22251,0.68424),
<span class="bold"><strong>&#955;</strong></span>=(0.000003,0.00109,0.00109)</em></span>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.09325,0.22251,0.68424),
<span class="bold"><strong>λ</strong></span>=(0.000003,0.00109,0.00109)</em></span>
</p>
</td>
</tr>
@@ -528,20 +528,20 @@
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.38881,0.18227,0.42892),
<span class="bold"><strong>&#955;</strong></span>=(0.000006,0.05228,0.00081)</em></span>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.38881,0.18227,0.42892),
<span class="bold"><strong>λ</strong></span>=(0.000006,0.05228,0.00081)</em></span>
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.42093,0.43960,0.13947),
<span class="bold"><strong>&#955;</strong></span>=(0.00186,0.00008,0.0000008)</em></span>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.42093,0.43960,0.13947),
<span class="bold"><strong>λ</strong></span>=(0.00186,0.00008,0.0000008)</em></span>
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.44885,0.30675,0.2444),
<span class="bold"><strong>&#955;</strong></span>=(0.00143,0.00005,0.0000004)</em></span>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.44885,0.30675,0.2444),
<span class="bold"><strong>λ</strong></span>=(0.00143,0.00005,0.0000004)</em></span>
</p>
</td>
</tr>
@@ -553,20 +553,20 @@
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.39442,0.24644,0.35914),
<span class="bold"><strong>&#955;</strong></span>=(0.00030,0.00003,0.00257)</em></span>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.39442,0.24644,0.35914),
<span class="bold"><strong>λ</strong></span>=(0.00030,0.00003,0.00257)</em></span>
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.37621,0.14838,0.47541),
<span class="bold"><strong>&#955;</strong></span>=(0.00498,0.000005,0.00022)</em></span>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.37621,0.14838,0.47541),
<span class="bold"><strong>λ</strong></span>=(0.00498,0.000005,0.00022)</em></span>
</p>
</td>
<td>
<p>
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(0.34131,0.12544,0.53325),
<span class="bold"><strong>&#955;</strong></span>=(0.000297,0.000003,0.00410)</em></span>
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(0.34131,0.12544,0.53325),
<span class="bold"><strong>λ</strong></span>=(0.000297,0.000003,0.00410)</em></span>
</p>
</td>
</tr>
@@ -1217,12 +1217,12 @@
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>&#945;</strong></span>=(&#945;<sub>1</sub>,...,&#945;<sub>k</sub>)</em></span> is
<span class="emphasis"><em><span class="bold"><strong>α</strong></span>=(α<sub>1</sub>,...,α<sub>k</sub>)</em></span> is
the <span class="emphasis"><em>phase probability vector</em></span> parameter of the
<span class="emphasis"><em>k</em></span>-phase hyperexponential distribution,
</li>
<li class="listitem">
<span class="emphasis"><em><span class="bold"><strong>&#955;</strong></span>=(&#955;<sub>1</sub>,...,&#955;<sub>k</sub>)</em></span> is
<span class="emphasis"><em><span class="bold"><strong>λ</strong></span>=(λ<sub>1</sub>,...,λ<sub>k</sub>)</em></span> is
the <span class="emphasis"><em>rate vector</em></span> parameter of the <span class="emphasis"><em>k</em></span>-phase
hyperexponential distribution,
</li>
@@ -1256,7 +1256,7 @@
</td>
<td>
<p>
<span class="emphasis"><em>x</em></span> &#8712; [0,&#8734;)
<span class="emphasis"><em>x</em></span> ∈ [0,∞)
</p>
</td>
</tr>
@@ -1454,11 +1454,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

17
doc/html/math_toolkit/dist_ref/dists/hypergeometric_dist.html
View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Hypergeometric Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="hyperexponential_dist.html" title="Hyperexponential Distribution">
<link rel="next" href="inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution">
@@ -31,7 +31,7 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">hypergeometric_distribution</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
@@ -177,7 +177,7 @@
</p>
<p>
However, note that the implementation method of the quantile function
always returns an integral value, therefore attempting to use a <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> that requires (or produces) a real valued
always returns an integral value, therefore attempting to use a <a class="link" href="../../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> that requires (or produces) a real valued
result will result in a compile time error.
</p>
</td></tr>
@@ -330,11 +330,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

79
doc/html/math_toolkit/dist_ref/dists/inverse_chi_squared_dist.html
View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Inverse Chi Squared Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="hypergeometric_dist.html" title="Hypergeometric Distribution">
<link rel="next" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">
@@ -31,7 +31,7 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">inverse_chi_squared_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -107,7 +107,7 @@
<p>
The inverse_chi_squared distribution is used in <a href="http://en.wikipedia.org/wiki/Bayesian_statistics" target="_top">Bayesian
statistics</a>: the scaled inverse chi-square is conjugate prior for
the normal distribution with known mean, model parameter &#963;&#178; (variance).
the normal distribution with known mean, model parameter σ² (variance).
</p>
<p>
See <a href="http://en.wikipedia.org/wiki/Conjugate_prior" target="_top">conjugate
@@ -120,10 +120,10 @@
</p>
<p>
The inverse_chi_squared distribution is a special case of a inverse_gamma
distribution with &#957; (degrees_of_freedom) shape (&#945;) and scale (&#946;) where
distribution with ν (degrees_of_freedom) shape (α) and scale (β) where
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">&#945;= &#957; /2 and &#946; = &#189;</span>
<span class="serif_italic">α= ν /2 and β = ½</span>
</p></blockquote></div>
<div class="note"><table border="0" summary="Note">
<tr>
@@ -147,36 +147,36 @@
</td></tr>
</table></div>
<p>
For degrees of freedom parameter &#957;, the (<span class="bold"><strong>unscaled</strong></span>)
For degrees of freedom parameter ν, the (<span class="bold"><strong>unscaled</strong></span>)
inverse chi_squared distribution is defined by the probability density
function (PDF):
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">f(x;&#957;) = 2<sup>-&#957;/2</sup> x<sup>-&#957;/2-1</sup> e<sup>-1/2x</sup> / &#915;(&#957;/2)</span>
<span class="serif_italic">f(x;ν) = 2<sup>-ν/2</sup> x<sup>-ν/2-1</sup> e<sup>-1/2x</sup> / Γ(ν/2)</span>
</p></blockquote></div>
<p>
and Cumulative Density Function (CDF)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">F(x;&#957;) = &#915;(&#957;/2, 1/2x) / &#915;(&#957;/2)</span>
<span class="serif_italic">F(x;ν) = Γ(ν/2, 1/2x) / Γ(ν/2)</span>
</p></blockquote></div>
<p>
For degrees of freedom parameter &#957; and scale parameter &#958;, the <span class="bold"><strong>scaled</strong></span>
For degrees of freedom parameter ν and scale parameter ξ, the <span class="bold"><strong>scaled</strong></span>
inverse chi_squared distribution is defined by the probability density
function (PDF):
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">f(x;&#957;, &#958;) = (&#958;&#957;/2)<sup>&#957;/2</sup> e<sup>-&#957;&#958;/2x</sup> x<sup>-1-&#957;/2</sup> / &#915;(&#957;/2)</span>
<span class="serif_italic">f(x;ν, ξ) = (ξν/2)<sup>ν/2</sup> e<sup>-νξ/2x</sup> x<sup>-1-ν/2</sup> / Γ(ν/2)</span>
</p></blockquote></div>
<p>
and Cumulative Density Function (CDF)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">F(x;&#957;, &#958;) = &#915;(&#957;/2, &#957;&#958;/2x) / &#915;(&#957;/2)</span>
<span class="serif_italic">F(x;ν, ξ) = Γ(ν/2, νξ/2x) / Γ(ν/2)</span>
</p></blockquote></div>
<p>
The following graphs illustrate how the PDF and CDF of the inverse chi_squared
distribution varies for a few values of parameters &#957; and &#958;:
distribution varies for a few values of parameters ν and ξ:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/inverse_chi_squared_pdf.svg" align="middle"></span>
@@ -195,23 +195,23 @@
<span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">);</span> <span class="comment">// Explicitly scaled.</span>
</pre>
<p>
Constructs an inverse chi_squared distribution with &#957; degrees of freedom
Constructs an inverse chi_squared distribution with ν degrees of freedom
<span class="emphasis"><em>df</em></span>, and scale <span class="emphasis"><em>scale</em></span> with default
value 1/df.
</p>
<p>
Requires that the degrees of freedom &#957; parameter is greater than zero, otherwise
Requires that the degrees of freedom ν parameter is greater than zero, otherwise
calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
</p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the degrees_of_freedom &#957; parameter of this distribution.
Returns the degrees_of_freedom ν parameter of this distribution.
</p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the scale &#958; parameter of this distribution.
Returns the scale ξ parameter of this distribution.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h1"></a>
@@ -231,7 +231,7 @@
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The domain of the random variate is [0,+&#8734;].
The domain of the random variate is [0,+∞].
</p>
<div class="note"><table border="0" summary="Note">
<tr>
@@ -264,11 +264,11 @@
<span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.implementation"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.implementation">Implementation</a>
</h5>
<p>
In the following table &#957; is the degrees of freedom parameter and &#958; is the scale
In the following table ν is the degrees of freedom parameter and ξ is the scale
parameter of the distribution, <span class="emphasis"><em>x</em></span> is the random variate,
<span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>
its complement. Parameters &#945; for shape and &#946; for scale are used for the inverse
gamma function: &#945; = &#957;/2 and &#946; = &#957; * &#958;/2.
its complement. Parameters α for shape and β for scale are used for the inverse
gamma function: α = ν/2 and β = ν * ξ/2.
</p>
<div class="informaltable"><table class="table">
<colgroup>
@@ -296,8 +296,8 @@
</td>
<td>
<p>
Using the relation: pdf = <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(&#945;,
&#946;/ x, &#946;) / x * x
Using the relation: pdf = <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(α,
β/ x, β) / x * x
</p>
</td>
</tr>
@@ -309,8 +309,8 @@
</td>
<td>
<p>
Using the relation: p = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(&#945;,
&#946; / x)
Using the relation: p = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(α,
β / x)
</p>
</td>
</tr>
@@ -322,8 +322,8 @@
</td>
<td>
<p>
Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(&#945;,
&#946; / x)
Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(α,
β / x)
</p>
</td>
</tr>
@@ -335,7 +335,7 @@
</td>
<td>
<p>
Using the relation: x = &#946;/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(&#945;,
Using the relation: x = β/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(α,
p)
</p>
</td>
@@ -348,7 +348,7 @@
</td>
<td>
<p>
Using the relation: x = &#945;/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(&#945;,
Using the relation: x = α/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(α,
q)
</p>
</td>
@@ -361,7 +361,7 @@
</td>
<td>
<p>
&#957; * &#958; / (&#957; + 2)
ν * ξ / (ν + 2)
</p>
</td>
</tr>
@@ -386,7 +386,7 @@
</td>
<td>
<p>
&#957;&#958; / (&#957; - 2) for &#957; &gt; 2, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
νξ / (ν - 2) for ν &gt; 2, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
@@ -398,7 +398,7 @@
</td>
<td>
<p>
2 &#957;&#178; &#958;&#178; / ((&#957; -2)&#178; (&#957; -4)) for &#957; &gt;4, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
2 ν² ξ² / ((ν -2)² (ν -4)) for ν &gt;4, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
@@ -410,7 +410,7 @@
</td>
<td>
<p>
4 &#8730;2 &#8730;(&#957;-4) /(&#957;-6) for &#957; &gt;6, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
4 √2 √(ν-4) /(ν-6) for ν &gt;6, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
@@ -422,7 +422,7 @@
</td>
<td>
<p>
12 * (5&#957; - 22) / ((&#957; - 6) * (&#957; - 8)) for &#957; &gt;8, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
12 * (5ν - 22) / ((ν - 6) * (ν - 8)) for ν &gt;8, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
@@ -434,7 +434,7 @@
</td>
<td>
<p>
3 + 12 * (5&#957; - 22) / ((&#957; - 6) * (&#957;-8)) for &#957; &gt;8, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
3 + 12 * (5ν - 22) / ((ν - 6) * (ν-8)) for ν &gt;8, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
@@ -458,11 +458,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

57
doc/html/math_toolkit/dist_ref/dists/inverse_gamma_dist.html
View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Inverse Gamma Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution">
<link rel="next" href="inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution">
@@ -31,7 +31,7 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">inverse_gamma_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -91,17 +91,17 @@
</td></tr>
</table></div>
<p>
For shape parameter &#945; and scale parameter &#946;, it is defined by the probability
For shape parameter α and scale parameter β, it is defined by the probability
density function (PDF):
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">f(x;&#945;, &#946;) = &#946;<sup>&#945;</sup> * (1/x) <sup>&#945;+1</sup> exp(-&#946;/x) / &#915;(&#945;)</span>
<span class="serif_italic">f(x;α, β) = β<sup>α</sup> * (1/x) <sup>α+1</sup> exp(-β/x) / Γ(α)</span>
</p></blockquote></div>
<p>
and cumulative density function (CDF)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">F(x;&#945;, &#946;) = &#915;(&#945;, &#946;/x) / &#915;(&#945;)</span>
<span class="serif_italic">F(x;α, β) = Γ(α, β/x) / Γ(α)</span>
</p></blockquote></div>
<p>
The following graphs illustrate how the PDF and CDF of the inverse gamma
@@ -123,7 +123,7 @@
<pre class="programlisting"><span class="identifier">inverse_gamma_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
Constructs an inverse gamma distribution with shape &#945; and scale &#946;.
Constructs an inverse gamma distribution with shape α and scale β.
</p>
<p>
Requires that the shape and scale parameters are greater than zero, otherwise
@@ -132,12 +132,12 @@
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the &#945; shape parameter of this inverse gamma distribution.
Returns the α shape parameter of this inverse gamma distribution.
</p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the &#946; scale parameter of this inverse gamma distribution.
Returns the β scale parameter of this inverse gamma distribution.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.h1"></a>
@@ -157,7 +157,7 @@
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The domain of the random variate is [0,+&#8734;].
The domain of the random variate is [0,+∞].
</p>
<div class="note"><table border="0" summary="Note">
<tr>
@@ -188,7 +188,7 @@
<span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.implementation"></a></span><a class="link" href="inverse_gamma_dist.html#math_toolkit.dist_ref.dists.inverse_gamma_dist.implementation">Implementation</a>
</h5>
<p>
In the following table &#945; is the shape parameter of the distribution, &#945; is its
In the following table α is the shape parameter of the distribution, α is its
scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span>
is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
</p>
@@ -218,8 +218,8 @@
</td>
<td>
<p>
Using the relation: pdf = <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(&#945;,
&#946;/ x, &#946;) / x * x
Using the relation: pdf = <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(α,
β/ x, β) / x * x
</p>
</td>
</tr>
@@ -231,8 +231,8 @@
</td>
<td>
<p>
Using the relation: p = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(&#945;,
&#946; / x)
Using the relation: p = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(α,
β / x)
</p>
</td>
</tr>
@@ -244,8 +244,8 @@
</td>
<td>
<p>
Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(&#945;,
&#946; / x)
Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(α,
β / x)
</p>
</td>
</tr>
@@ -257,7 +257,7 @@
</td>
<td>
<p>
Using the relation: x = &#946;/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(&#945;,
Using the relation: x = β/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(α,
p)
</p>
</td>
@@ -270,7 +270,7 @@
</td>
<td>
<p>
Using the relation: x = &#945;/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(&#945;,
Using the relation: x = α/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(α,
q)
</p>
</td>
@@ -283,7 +283,7 @@
</td>
<td>
<p>
&#946; / (&#945; + 1)
β / (α + 1)
</p>
</td>
</tr>
@@ -307,7 +307,7 @@
</td>
<td>
<p>
&#946; / (&#945; - 1) for &#945; &gt; 1, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
β / (α - 1) for α &gt; 1, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
@@ -319,7 +319,7 @@
</td>
<td>
<p>
(&#946; * &#946;) / ((&#945; - 1) * (&#945; - 1) * (&#945; - 2)) for &#945; &gt;2, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
(β * β) / ((α - 1) * (α - 1) * (α - 2)) for α &gt;2, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
@@ -331,7 +331,7 @@
</td>
<td>
<p>
4 * sqrt (&#945; -2) / (&#945; -3) for &#945; &gt;3, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
4 * sqrt (α -2) / (α -3) for α &gt;3, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
@@ -343,7 +343,7 @@
</td>
<td>
<p>
(30 * &#945; - 66) / ((&#945;-3)*(&#945; - 4)) for &#945; &gt;4, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
(30 * α - 66) / ((α-3)*(α - 4)) for α &gt;4, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
</p>
</td>
</tr>
@@ -352,11 +352,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

73
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@@ -1,13 +1,13 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Inverse Gaussian (or Inverse Normal) Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">
<link rel="next" href="laplace_dist.html" title="Laplace Distribution">
<link rel="next" href="kolmogorov_smirnov_dist.html" title="Kolmogorov-Smirnov Distribution">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
@@ -20,7 +20,7 @@
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="inverse_gamma_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.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="laplace_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
<a accesskey="p" href="inverse_gamma_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.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="kolmogorov_smirnov_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h4 class="title">
@@ -31,7 +31,7 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">inverse_gaussian_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -102,26 +102,26 @@
<span class="identifier">inverse_gaussian</span> <span class="identifier">my_ig</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">3</span><span class="special">);</span>
</pre>
<p>
For mean parameters &#956; and scale (also called precision) parameter &#955;, and random
For mean parameters μ and scale (also called precision) parameter λ, and random
variate x, the inverse_gaussian distribution is defined by the probability
density function (PDF):
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">f(x;&#956;, &#955;) = &#8730;(&#955;/2&#960;x<sup>3</sup>) e<sup>-&#955;(x-&#956;)&#178;/2&#956;&#178;x</sup> </span>
<span class="serif_italic">f(x;μ, λ) = √(λ/2πx<sup>3</sup>) e<sup>-λ(x-μ)²/2μ²x</sup> </span>
</p></blockquote></div>
<p>
and Cumulative Density Function (CDF):
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">F(x;&#956;, &#955;) = &#934;{&#8730;(&#955;<span class="emphasis"><em>x) (x</em></span>&#956;-1)}
+ e<sup>2&#956;/&#955;</sup> &#934;{-&#8730;(&#955;/&#956;) (1+x/&#956;)} </span>
<span class="serif_italic">F(x;μ, λ) = Φ{√(λ<span class="emphasis"><em>x) (x</em></span>μ-1)}
+ e<sup>2μ/λ</sup> Φ{-√(λ/μ) (1+x/μ)} </span>
</p></blockquote></div>
<p>
where &#934; is the standard normal distribution CDF.
where Φ is the standard normal distribution CDF.
</p>
<p>
The following graphs illustrate how the PDF and CDF of the inverse_gaussian
distribution varies for a few values of parameters &#956; and &#955;:
distribution varies for a few values of parameters μ and λ:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_pdf.svg" align="middle"></span>
@@ -132,11 +132,11 @@
</p></blockquote></div>
<p>
Tweedie also provided 3 other parameterisations where (&#956; and &#955;) are replaced
by their ratio &#966; = &#955;/&#956; and by 1/&#956;: these forms may be more suitable for Bayesian
Tweedie also provided 3 other parameterisations where (μ and λ) are replaced
by their ratio φ = λ/μ and by 1/μ: these forms may be more suitable for Bayesian
applications. These can be found on Seshadri, page 2 and are also discussed
by Chhikara and Folks on page 105. Another related parameterisation, the
__wald_distrib (where mean &#956; is unity) is also provided.
__wald_distrib (where mean μ is unity) is also provided.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h0"></a>
@@ -146,22 +146,22 @@
<pre class="programlisting"><span class="identifier">inverse_gaussian_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// optionally scaled.</span>
</pre>
<p>
Constructs an inverse_gaussian distribution with &#956; mean, and scale &#955;, with
Constructs an inverse_gaussian distribution with μ mean, and scale λ, with
both default values 1.
</p>
<p>
Requires that both the mean &#956; parameter and scale &#955; are greater than zero,
Requires that both the mean μ parameter and scale λ are greater than zero,
otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
</p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the mean &#956; parameter of this distribution.
Returns the mean μ parameter of this distribution.
</p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the scale &#955; parameter of this distribution.
Returns the scale λ parameter of this distribution.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h1"></a>
@@ -181,7 +181,7 @@
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The domain of the random variate is [0,+&#8734;).
The domain of the random variate is [0,+∞).
</p>
<div class="note"><table border="0" summary="Note">
<tr>
@@ -199,7 +199,7 @@
</h5>
<p>
The inverse_gaussian distribution is implemented in terms of the exponential
function and standard normal distribution <span class="emphasis"><em>N</em></span>0,1 &#934; : refer
function and standard normal distribution <span class="emphasis"><em>N</em></span>0,1 Φ : refer
to the accuracy data for those functions for more information. But in general,
gamma (and thus inverse gamma) results are often accurate to a few epsilon,
&gt;14 decimal digits accuracy for 64-bit double.
@@ -209,10 +209,10 @@
<span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.implementation"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.implementation">Implementation</a>
</h5>
<p>
In the following table &#956; is the mean parameter and &#955; is the scale parameter
In the following table μ is the mean parameter and λ is the scale parameter
of the inverse_gaussian distribution, <span class="emphasis"><em>x</em></span> is the random
variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>
its complement. Parameters &#956; for shape and &#955; for scale are used for the inverse
its complement. Parameters μ for shape and λ for scale are used for the inverse
gaussian function.
</p>
<div class="informaltable"><table class="table">
@@ -241,7 +241,7 @@
</td>
<td>
<p>
&#8730;(&#955;/ 2&#960;x<sup>3</sup>) e<sup>-&#955;(x - &#956;)&#178;/ 2&#956;&#178;x</sup>
√(λ/ 2πx<sup>3</sup>) e<sup>-λ(x - μ)²/ 2μ²x</sup>
</p>
</td>
</tr>
@@ -253,7 +253,7 @@
</td>
<td>
<p>
&#934;{&#8730;(&#955;<span class="emphasis"><em>x) (x</em></span>&#956;-1)} + e<sup>2&#956;/&#955;</sup> &#934;{-&#8730;(&#955;/&#956;) (1+x/&#956;)}
Φ{√(λ<span class="emphasis"><em>x) (x</em></span>μ-1)} + e<sup>2μ/λ</sup> Φ{-√(λ/μ) (1+x/μ)}
</p>
</td>
</tr>
@@ -265,7 +265,7 @@
</td>
<td>
<p>
using complement of &#934; above.
using complement of Φ above.
</p>
</td>
</tr>
@@ -303,7 +303,7 @@
</td>
<td>
<p>
&#956; {&#8730;(1+9&#956;&#178;/4&#955;&#178;)&#178; - 3&#956;/2&#955;}
μ {√(1+9μ²/4λ²)² - 3μ/2λ}
</p>
</td>
</tr>
@@ -328,7 +328,7 @@
</td>
<td>
<p>
&#956;
μ
</p>
</td>
</tr>
@@ -340,7 +340,7 @@
</td>
<td>
<p>
&#956;&#179;/&#955;
μ³/λ
</p>
</td>
</tr>
@@ -352,7 +352,7 @@
</td>
<td>
<p>
3 &#8730; (&#956;/&#955;)
3 √ (μ/λ)
</p>
</td>
</tr>
@@ -364,7 +364,7 @@
</td>
<td>
<p>
15&#956;/&#955;
15μ/λ
</p>
</td>
</tr>
@@ -376,7 +376,7 @@
</td>
<td>
<p>
12&#956;/&#955;
12μ/λ
</p>
</td>
</tr>
@@ -431,11 +431,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>
@@ -443,7 +442,7 @@
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="inverse_gamma_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.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="laplace_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
<a accesskey="p" href="inverse_gamma_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.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="kolmogorov_smirnov_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
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106
doc/html/math_toolkit/dist_ref/dists/kolmogorov_smirnov_dist.html
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@@ -3,8 +3,8 @@
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Kolmogorov-Smirnov Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.2">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution">
<link rel="next" href="laplace_dist.html" title="Laplace Distribution">
@@ -27,7 +27,6 @@
<a name="math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist"></a><a class="link" href="kolmogorov_smirnov_dist.html" title="Kolmogorov-Smirnov Distribution">Kolmogorov-Smirnov
Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><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">distributions</span><span class="special">/</span><span class="identifier">kolmogorov_smirnov</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
@@ -52,78 +51,72 @@
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
<p>
The Kolmogorov-Smirnov test in statistics compares two empirical distributions,
or an empirical distribution against any theoretical distribution.<a href="#ftn.math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f0" class="footnote" name="math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f0"><sup class="footnote">[1]</sup></a> It makes use of a specific distribution which is informally
known in the literature as the Kolmogorv-Smirnov distribution, implemented
here.
</p>
<p>
<p>
Formally, if <span class="emphasis"><em>n</em></span> observations are taken from a theoretical
distribution <span class="emphasis"><em>G(x)</em></span>, and if <span class="emphasis"><em>G</em></span><sub><span class="emphasis"><em>n</em></span></sub>(<span class="emphasis"><em>x</em></span>)
represents the empirical CDF of those <span class="emphasis"><em>n</em></span> observations,
then the test statistic
</p>
<div class="blockquote"><blockquote class="blockquote">
<p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/kolmogorov_smirnov_test_statistic.svg"></span>
</p>
</blockquote></div>
<p>
</p></blockquote></div>
<p>
will be distributed according to a Kolmogorov-Smirnov distribution parameterized
by <span class="emphasis"><em>n</em></span>.
</p>
<p>
<p>
The exact form of a Kolmogorov-Smirnov distribution is the subject of a
large, decades-old literature.<a href="#ftn.math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f1" class="footnote" name="math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f1"><sup class="footnote">[2]</sup></a> In the interest of simplicity, Boost implements the first-order,
limiting form of this distribution (the same form originally identified
by Kolmogorov<a href="#ftn.math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f2" class="footnote" name="math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f2"><sup class="footnote">[3]</sup></a>), namely
</p>
<div class="blockquote"><blockquote class="blockquote">
<p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/kolmogorov_smirnov_distribution.svg"></span>
</p>
</blockquote></div>
<p>
</p></blockquote></div>
<p>
Note that while the exact distribution only has support over [0, 1], this
limiting form has positive mass above unity, particlularly for small <span class="emphasis"><em>n</em></span>.
limiting form has positive mass above unity, particularly for small <span class="emphasis"><em>n</em></span>.
The following graph illustrations how the distribution changes for different
values of <span class="emphasis"><em>n</em></span>:
</p>
<div class="blockquote"><blockquote class="blockquote">
<p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/kolmogorov_smirnov_pdf.svg" align="middle"></span>
</p>
</blockquote></div>
<h5>
</p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.h0"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.member_functions"></a></span><a class="link" href="kolmogorov_smirnov_dist.html#math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.member_functions">Member
Functions</a>
</h5>
<pre class="programlisting"><span class="identifier">kolmogorov_smirnov_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">n</span><span class="special">);</span>
</pre>
<p>
<p>
Constructs a Kolmogorov-Smirnov distribution with <span class="emphasis"><em>n</em></span>
observations.
</p>
<p>
<p>
Requires n &gt; 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
</p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">number_of_observations</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
<p>
Returns the parameter <span class="emphasis"><em>n</em></span> from which this object was
constructed.
</p>
<h5>
<h5>
<a name="math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.h1"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.non_member_accessors"></a></span><a class="link" href="kolmogorov_smirnov_dist.html#math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.non_member_accessors">Non-member
Accessors</a>
</h5>
<p>
<p>
All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
functions</a> that are generic to all distributions are supported:
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
@@ -135,47 +128,44 @@
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
<p>
The domain of the random variable is [0, +∞].
</p>
<h5>
<h5>
<a name="math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.h2"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.accuracy"></a></span><a class="link" href="kolmogorov_smirnov_dist.html#math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.accuracy">Accuracy</a>
</h5>
<p>
<p>
The CDF of the Kolmogorov-Smirnov distribution is implemented in terms
of the fourth <a class="link" href="../../jacobi_theta/jacobi_theta_overview.html" title="Overview of the Jacobi Theta Functions">Jacobi
Theta function</a>; please refer to the accuracy ULP plots for that
function.
</p>
<p>
<p>
The PDF is implemented separately, and the following ULP plot illustrates
its accuracy:
</p>
<div class="blockquote"><blockquote class="blockquote">
<p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/kolmogorov_smirnov_pdf_ulp.svg" align="middle"></span>
</p>
</blockquote></div>
<p>
</p></blockquote></div>
<p>
Because PDF values are simply scaled out and up by the square root of
<span class="emphasis"><em>n</em></span>, the above plot is representative for all values
of <span class="emphasis"><em>n</em></span>. Note that for present purposes, "accuracy"
refers to deviations from the limiting approximation, rather than deviations
from the exact distribution.
</p>
<h5>
<h5>
<a name="math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.h3"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.implementation"></a></span><a class="link" href="kolmogorov_smirnov_dist.html#math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.implementation">Implementation</a>
</h5>
<p>
<p>
In the following table, <span class="emphasis"><em>n</em></span> is the number of observations,
<span class="emphasis"><em>x</em></span> is the random variable, π is Archimedes' constant,
and ζ(3) is Apéry's constant.
</p>
<div class="informaltable">
<table class="informaltable" border="1">
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
@@ -325,34 +315,34 @@
</td>
</tr>
</tbody>
</table>
</div>
<div class="footnotes">
</table></div>
<div class="footnotes">
<br><hr style="width:100; text-align:left;margin-left: 0">
<div id="ftn.math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f0" class="footnote">
<p><a href="#math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f0" class="para"><sup class="para">[1] </sup></a>
<a class="ulink" href="https://en.wikipedia.org/wiki/Kolmogorov%%E0%EpSmirnov_test" target="_top">Wikipedia:
<div id="ftn.math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f0" class="footnote"><p><a href="#math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f0" class="para"><sup class="para">[1] </sup></a>
<a href="https://en.wikipedia.org/wiki/Kolmogorov%%&amp;0%upSmirnov_test" target="_top">Wikipedia:
Kolmogorov-Smirnov test</a>
</p>
</div>
<div id="ftn.math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f1" class="footnote">
<p><a href="#math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f1" class="para"><sup class="para">[2] </sup></a>
<a class="ulink" href="https://www.jstatsoft.org/article/view/v039i11" target="_top">Simard, R.
</p></div>
<div id="ftn.math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f1" class="footnote"><p><a href="#math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f1" class="para"><sup class="para">[2] </sup></a>
<a href="https://www.jstatsoft.org/article/view/v039i11" target="_top">Simard, R.
and L'Ecuyer, P. (2011) "Computing the Two-Sided Kolmogorov-Smirnov
Distribution". Journal of Statistical Software, vol. 39, no. 11.</a>
</p>
</div>
<div id="ftn.math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f2" class="footnote">
<p><a href="#math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f2" class="para"><sup class="para">[3] </sup></a>
</p></div>
<div id="ftn.math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f2" class="footnote"><p><a href="#math_toolkit.dist_ref.dists.kolmogorov_smirnov_dist.f2" class="para"><sup class="para">[3] </sup></a>
Kolmogorov A (1933). "Sulla determinazione empirica di una legge
di distribuzione". G. Ist. Ital. Attuari. 4: 83–91.
</p>
</div>
</p></div>
</div>
</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"></div></td>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>
</tr></table>
<hr>
<div class="spirit-nav">

57
doc/html/math_toolkit/dist_ref/dists/laplace_dist.html
View File

@@ -1,12 +1,12 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Laplace Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution">
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</head>
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@@ -20,7 +20,7 @@
</tr></table>
<hr>
<div class="spirit-nav">
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<a accesskey="p" href="kolmogorov_smirnov_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.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="logistic_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h4 class="title">
@@ -30,12 +30,12 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">laplace_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">laplace_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">laplace</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</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">&gt;</span>
<span class="keyword">class</span> <span class="identifier">laplace_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -56,7 +56,7 @@
1972, p. 930). It is also called the double exponential distribution.
</p>
<p>
For location parameter <span class="emphasis"><em>&#956;</em></span> and scale parameter <span class="emphasis"><em>&#963;</em></span>,
For location parameter <span class="emphasis"><em>μ</em></span> and scale parameter <span class="emphasis"><em>σ</em></span>,
it is defined by the probability density function:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
@@ -68,8 +68,8 @@
deviation of the normal or Gaussian distribution.
</p>
<p>
The following graph illustrates the effect of the parameters &#956; and &#963; on the
PDF. Note that the domain of the random variable remains [-&#8734;,+&#8734;] irrespective
The following graph illustrates the effect of the parameters μ and σ on the
PDF. Note that the domain of the random variable remains [-∞,+∞] irrespective
of the value of the location parameter:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
@@ -126,7 +126,7 @@
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The domain of the random variable is [-&#8734;,+&#8734;].
The domain of the random variable is [-∞,+∞].
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.laplace_dist.h2"></a>
@@ -141,7 +141,7 @@
<span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.implementation"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.implementation">Implementation</a>
</h5>
<p>
In the following table &#956; is the location parameter of the distribution, &#963; is
In the following table μ is the location parameter of the distribution, σ is
its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span>
is the probability and its complement <span class="emphasis"><em>q = 1-p</em></span>.
</p>
@@ -171,7 +171,7 @@
</td>
<td>
<p>
Using the relation: pdf = e<sup>-abs(x-&#956;) / &#963;</sup> / (2 * &#963;)
Using the relation: pdf = e<sup>-abs(x-μ) / σ</sup> / (2 * σ)
</p>
</td>
</tr>
@@ -186,10 +186,10 @@
Using the relations:
</p>
<p>
x &lt; &#956; : p = e<sup>(x-&#956;)/&#963; </sup> / &#963;
x &lt; μ : p = e<sup>(x-μ)/σ </sup> / σ
</p>
<p>
x &gt;= &#956; : p = 1 - e<sup>(&#956;-x)/&#963; </sup> / &#963;
x &gt;= μ : p = 1 - e<sup>(μ-x)/σ </sup> / σ
</p>
</td>
@@ -205,10 +205,10 @@
Using the relation:
</p>
<p>
-x &lt; &#956; : q = e<sup>(-x-&#956;)/&#963; </sup> / &#963;
-x &lt; μ : q = e<sup>(-x-μ)/σ </sup> / σ
</p>
<p>
-x &gt;= &#956; : q = 1 - e<sup>(&#956;+x)/&#963; </sup> / &#963;
-x &gt;= μ : q = 1 - e<sup>(μ+x)/σ </sup> / σ
</p>
</td>
@@ -224,10 +224,10 @@
Using the relations:
</p>
<p>
p &lt; 0.5 : x = &#956; + &#963; * log(2*p)
p &lt; 0.5 : x = μ + σ * log(2*p)
</p>
<p>
p &gt;= 0.5 : x = &#956; - &#963; * log(2-2*p)
p &gt;= 0.5 : x = μ - σ * log(2-2*p)
</p>
</td>
</tr>
@@ -242,10 +242,10 @@
Using the relation:
</p>
<p>
q &gt; 0.5: x = &#956; + &#963;*log(2-2*q)
q &gt; 0.5: x = μ + σ*log(2-2*q)
</p>
<p>
q &lt;=0.5: x = &#956; - &#963;*log( 2*q )
q &lt;=0.5: x = μ - σ*log( 2*q )
</p>
</td>
</tr>
@@ -257,7 +257,7 @@
</td>
<td>
<p>
&#956;
μ
</p>
</td>
</tr>
@@ -269,7 +269,7 @@
</td>
<td>
<p>
2 * &#963;<sup>2</sup>
2 * σ<sup>2</sup>
</p>
</td>
</tr>
@@ -281,7 +281,7 @@
</td>
<td>
<p>
&#956;
μ
</p>
</td>
</tr>
@@ -345,11 +345,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>
@@ -357,7 +356,7 @@
</tr></table>
<hr>
<div class="spirit-nav">
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View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Logistic Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="laplace_dist.html" title="Laplace Distribution">
<link rel="next" href="lognormal_dist.html" title="Log Normal Distribution">
@@ -31,7 +31,7 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">logistic_distribution</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
@@ -112,7 +112,7 @@
</p>
<p>
The domain of the random variable is [-[max_value], +[min_value]]. However,
the pdf and cdf support inputs of +&#8734; and -&#8734;
the pdf and cdf support inputs of +∞ and -∞
as special cases if RealType permits.
</p>
<p>
@@ -273,7 +273,7 @@ as special cases if RealType permits.
</td>
<td>
<p>
(&#960;*s)<sup>2</sup> / 3
(π*s)<sup>2</sup> / 3
</p>
</td>
</tr>
@@ -282,11 +282,10 @@ as special cases if RealType permits.
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

21
doc/html/math_toolkit/dist_ref/dists/lognormal_dist.html
View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Log Normal Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="logistic_dist.html" title="Logistic Distribution">
<link rel="next" href="negative_binomial_dist.html" title="Negative Binomial Distribution">
@@ -31,12 +31,12 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</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> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">lognormal_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">lognormal_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">lognormal</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;21.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</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">&gt;</span>
<span class="keyword">class</span> <span class="identifier">lognormal_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
@@ -71,7 +71,7 @@
</p>
<p>
The following graph illustrates the effect of the location parameter on
the PDF, note that the range of the random variable remains [0,+&#8734;] irrespective
the PDF, note that the range of the random variable remains [0,+∞] irrespective
of the value of the location parameter:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
@@ -136,7 +136,7 @@
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The domain of the random variable is [0,+&#8734;].
The domain of the random variable is [0,+∞].
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.lognormal_dist.h2"></a>
@@ -316,11 +316,10 @@
</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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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>

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