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- <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 20. 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 20. 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>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span>
- </p></blockquote></div>
- <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 20. 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 20. 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>
- <div class="table">
- <a name="math_toolkit.ellint.heuman_lambda.table_heuman_lambda"></a><p class="title"><b>Table 8.69. Error rates for heuman_lambda</b></p>
- <div class="table-contents"><table class="table" summary="Error rates for heuman_lambda">
- <colgroup>
- <col>
- <col>
- <col>
- <col>
- <col>
- </colgroup>
- <thead><tr>
- <th>
- </th>
- <th>
- <p>
- GNU C++ version 7.1.0<br> linux<br> double
- </p>
- </th>
- <th>
- <p>
- GNU C++ version 7.1.0<br> linux<br> long double
- </p>
- </th>
- <th>
- <p>
- Sun compiler version 0x5150<br> Sun Solaris<br> long double
- </p>
- </th>
- <th>
- <p>
- Microsoft Visual C++ version 14.1<br> Win32<br> double
- </p>
- </th>
- </tr></thead>
- <tbody>
- <tr>
- <td>
- <p>
- Elliptic Integral Jacobi Zeta: Mathworld Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.89ε (Mean = 0.887ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.89ε (Mean = 0.887ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.08ε (Mean = 0.734ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Elliptic Integral Heuman Lambda: Random Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 3.82ε (Mean = 0.609ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 3.82ε (Mean = 0.608ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2.12ε (Mean = 0.588ε)</span>
- </p>
- </td>
- </tr>
- </tbody>
- </table></div>
- </div>
- <br class="table-break"><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 <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, 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>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span>
- </p></blockquote></div>
- <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 © 2006-2019 Nikhar
- Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
- Hubert Holin, Bruno Lalande, John Maddock, 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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