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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 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> |
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| 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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