diff options
Diffstat (limited to 'libs/math/doc/html/math_toolkit/ellint/heuman_lambda.html')
| -rw-r--r-- | libs/math/doc/html/math_toolkit/ellint/heuman_lambda.html | 130 |
1 files changed, 130 insertions, 0 deletions
diff --git a/libs/math/doc/html/math_toolkit/ellint/heuman_lambda.html b/libs/math/doc/html/math_toolkit/ellint/heuman_lambda.html new file mode 100644 index 000000000..96fce53b1 --- /dev/null +++ b/libs/math/doc/html/math_toolkit/ellint/heuman_lambda.html @@ -0,0 +1,130 @@ +<html> +<head> +<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> +<title>Heuman Lambda Function</title> +<link rel="stylesheet" href="../../math.css" type="text/css"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> +<link rel="up" href="../ellint.html" title="Elliptic Integrals"> +<link rel="prev" href="jacobi_zeta.html" title="Jacobi Zeta Function"> +<link rel="next" href="../jacobi.html" title="Jacobi Elliptic Functions"> +</head> +<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> +<table cellpadding="2" width="100%"><tr> +<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> +<td align="center"><a href="../../../../../../index.html">Home</a></td> +<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> +<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> +<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> +<td align="center"><a href="../../../../../../more/index.htm">More</a></td> +</tr></table> +<hr> +<div class="spirit-nav"> +<a accesskey="p" href="jacobi_zeta.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../ellint.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="../jacobi.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +<div class="section"> +<div class="titlepage"><div><div><h3 class="title"> +<a name="math_toolkit.ellint.heuman_lambda"></a><a class="link" href="heuman_lambda.html" title="Heuman Lambda Function">Heuman Lambda Function</a> +</h3></div></div></div> +<h5> +<a name="math_toolkit.ellint.heuman_lambda.h0"></a> + <span class="phrase"><a name="math_toolkit.ellint.heuman_lambda.synopsis"></a></span><a class="link" href="heuman_lambda.html#math_toolkit.ellint.heuman_lambda.synopsis">Synopsis</a> + </h5> +<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">heuman_lambda</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></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"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> +<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">heuman_lambda</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span> + +<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> +<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">heuman_lambda</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> + +<span class="special">}}</span> <span class="comment">// namespaces</span> +</pre> +<h5> +<a name="math_toolkit.ellint.heuman_lambda.h1"></a> + <span class="phrase"><a name="math_toolkit.ellint.heuman_lambda.description"></a></span><a class="link" href="heuman_lambda.html#math_toolkit.ellint.heuman_lambda.description">Description</a> + </h5> +<p> + This function evaluates the Heuman Lambda Function <span class="emphasis"><em>Λ<sub>0</sub>(φ, k)</em></span> + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span> + </p> +<p> + The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result + type calculation rules</em></span></a> when the arguments are of different + types: when they are the same type then the result is the same type as the + arguments. + </p> +<p> + Requires <span class="emphasis"><em>-1 <= k <= 1</em></span>, otherwise returns the result + of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> + (outside this range the result would be complex). + </p> +<p> + The final <a class="link" href="../../policy.html" title="Chapter 14. 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 14. Policies: Controlling Precision, Error Handling etc">policy + documentation for more details</a>. + </p> +<p> + Note that there is no complete analogue of this function (where φ = π / 2) as + this takes the value 1 for all <span class="emphasis"><em>k</em></span>. + </p> +<h5> +<a name="math_toolkit.ellint.heuman_lambda.h2"></a> + <span class="phrase"><a name="math_toolkit.ellint.heuman_lambda.accuracy"></a></span><a class="link" href="heuman_lambda.html#math_toolkit.ellint.heuman_lambda.accuracy">Accuracy</a> + </h5> +<p> + These functions are trivially computed in terms of other elliptic integrals + and generally have very low error rates (a few epsilon) unless parameter + φ +is very large, in which case the usual trigonometric function argument-reduction + issues apply. + </p> +<h5> +<a name="math_toolkit.ellint.heuman_lambda.h3"></a> + <span class="phrase"><a name="math_toolkit.ellint.heuman_lambda.testing"></a></span><a class="link" href="heuman_lambda.html#math_toolkit.ellint.heuman_lambda.testing">Testing</a> + </h5> +<p> + The tests use a mixture of spot test values calculated using values calculated + at wolframalpha.com, and random test data generated using MPFR at 1000-bit + precision and a deliberately naive implementation in terms of the Legendre + integrals. + </p> +<h5> +<a name="math_toolkit.ellint.heuman_lambda.h4"></a> + <span class="phrase"><a name="math_toolkit.ellint.heuman_lambda.implementation"></a></span><a class="link" href="heuman_lambda.html#math_toolkit.ellint.heuman_lambda.implementation">Implementation</a> + </h5> +<p> + The function is then implemented in terms of Carlson's integrals R<sub>J</sub> and R<sub>F</sub> +using + the relation: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span> + </p> +<p> + This relation fails for <span class="emphasis"><em>|φ| >= π/2</em></span> in which case the + definition in terms of the Jacobi Zeta is used. + </p> +</div> +<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> +<td align="left"></td> +<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, + Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert + Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, + Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> + Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) + </p> +</div></td> +</tr></table> +<hr> +<div class="spirit-nav"> +<a accesskey="p" href="jacobi_zeta.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../ellint.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="../jacobi.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +</body> +</html> |