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| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.ellint.ellint_d"></a><a class="link" href="ellint_d.html" title="Elliptic Integral D - Legendre Form">Elliptic Integral D - Legendre |
| Form</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.ellint.ellint_d.h0"></a> |
| <span class="phrase"><a name="math_toolkit.ellint.ellint_d.synopsis"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.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">ellint_d</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">ellint_d</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">ellint_d</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="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</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">ellint_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</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> <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">ellint_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</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.ellint_d.h1"></a> |
| <span class="phrase"><a name="math_toolkit.ellint.ellint_d.description"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.description">Description</a> |
| </h5> |
| <p> |
| These two functions evaluate the incomplete elliptic integral <span class="emphasis"><em>D(φ, |
| k)</em></span> and its complete counterpart <span class="emphasis"><em>D(k) = D(π/2, k)</em></span>. |
| </p> |
| <p> |
| The return type of these functions 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> |
| <pre class="programlisting"><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">ellint_d</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">ellint_3</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> |
| </pre> |
| <p> |
| Returns the incomplete elliptic integral: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span> |
| </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> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</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">ellint_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</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> <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">ellint_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</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> |
| </pre> |
| <p> |
| Returns the complete elliptic integral <span class="emphasis"><em>D(k) = D(π/2, k)</em></span> |
| </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> |
| <h5> |
| <a name="math_toolkit.ellint.ellint_d.h2"></a> |
| <span class="phrase"><a name="math_toolkit.ellint.ellint_d.accuracy"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.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.ellint_d.h3"></a> |
| <span class="phrase"><a name="math_toolkit.ellint.ellint_d.testing"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.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.ellint_d.h4"></a> |
| <span class="phrase"><a name="math_toolkit.ellint.ellint_d.implementation"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.implementation">Implementation</a> |
| </h5> |
| <p> |
| The implementation for D(φ, k) first performs argument reduction using the |
| relations: |
| </p> |
| <p> |
| <span class="emphasis"><em>D(-φ, k) = -D(φ, k)</em></span> |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| <span class="emphasis"><em>D(nπ+φ, k) = 2nD(k) + D(φ, k)</em></span> |
| </p> |
| <p> |
| to move φ   to the range [0, π/2]. |
| </p> |
| <p> |
| The functions are then implemented in terms of Carlson's integral R<sub>D</sub> |
| using |
| the relation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span> |
| </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> |
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