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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.ellint.ellint_1"></a><a class="link" href="ellint_1.html" title="Elliptic Integrals of the First Kind - Legendre Form"> Elliptic Integrals |
| of the First Kind - Legendre Form</a> |
| </h4></div></div></div> |
| <a name="math_toolkit.special.ellint.ellint_1.synopsis"></a><h6> |
| <a name="id1137461"></a> |
| <a class="link" href="ellint_1.html#math_toolkit.special.ellint.ellint_1.synopsis">Synopsis</a> |
| </h6> |
| <p> |
| |
| </p> |
| <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_1</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| </pre> |
| <p> |
| </p> |
| <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="../../main_overview/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_1</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="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/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_1</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="Policies">Policy</a><span class="special">&);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../../main_overview/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_1</span><span class="special">(</span><span class="identifier">T</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">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/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_1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <a name="math_toolkit.special.ellint.ellint_1.description"></a><h6> |
| <a name="id1137865"></a> |
| <a class="link" href="ellint_1.html#math_toolkit.special.ellint.ellint_1.description">Description</a> |
| </h6> |
| <p> |
| These two functions evaluate the incomplete elliptic integral of the first |
| kind <span class="emphasis"><em>F(φ, k)</em></span> and its complete counterpart <span class="emphasis"><em>K(k) |
| = F(π/2, k)</em></span>. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/ellint_1.png" align="middle"></span> |
| </p> |
| <p> |
| The return type of these functions is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result |
| type calculation rules</em></span></a> when T1 and T2 are 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="../../main_overview/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_1</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="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/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_1</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="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the incomplete elliptic integral of the first kind <span class="emphasis"><em>F(φ, |
| k)</em></span>: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint2.png"></span> |
| </p> |
| <p> |
| Requires -1 <= k <= 1, otherwise returns the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>. |
| </p> |
| <p> |
| </p> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Policies">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="Policies">policy documentation for more details</a>. |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../../main_overview/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_1</span><span class="special">(</span><span class="identifier">T</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">T</span><span class="special">></span> |
| <a class="link" href="../../main_overview/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_1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the complete elliptic integral of the first kind <span class="emphasis"><em>K(k)</em></span>: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint6.png"></span> |
| </p> |
| <p> |
| Requires -1 <= k <= 1, otherwise returns the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>. |
| </p> |
| <p> |
| </p> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Policies">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="Policies">policy documentation for more details</a>. |
| </p> |
| <p> |
| </p> |
| <a name="math_toolkit.special.ellint.ellint_1.accuracy"></a><h6> |
| <a name="id1138330"></a> |
| <a class="link" href="ellint_1.html#math_toolkit.special.ellint.ellint_1.accuracy">Accuracy</a> |
| </h6> |
| <p> |
| These functions are computed using only basic arithmetic operations, so |
| there isn't much variation in accuracy over differing platforms. Note that |
| only results for the widest floating point type on the system are given |
| as narrower types have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively zero error</a>. |
| All values are relative errors in units of epsilon. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.special.ellint.ellint_1.errors_rates_in_the_elliptic_integrals_of_the_first_kind"></a><p class="title"><b>Table 41. Errors Rates in the Elliptic Integrals of the First Kind</b></p> |
| <div class="table-contents"><table class="table" summary="Errors Rates in the Elliptic Integrals of the First Kind"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| F(φ, k) |
| </p> |
| </th> |
| <th> |
| <p> |
| K(k) |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32 / Visual C++ 8.0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=3 Mean=0.8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.8 Mean=0.7 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Red Hat Linux / G++ 3.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2.6 Mean=1.7 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2.2 Mean=1.8 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HP-UX / HP aCC 6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4.6 Mean=1.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=3.7 Mean=1.5 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><a name="math_toolkit.special.ellint.ellint_1.testing"></a><h6> |
| <a name="id1138516"></a> |
| <a class="link" href="ellint_1.html#math_toolkit.special.ellint.ellint_1.testing">Testing</a> |
| </h6> |
| <p> |
| The tests use a mixture of spot test values calculated using the online |
| calculator at <a href="http://functions.wolfram.com/" target="_top">functions.wolfram.com</a>, |
| and random test data generated using NTL::RR at 1000-bit precision and |
| this implementation. |
| </p> |
| <a name="math_toolkit.special.ellint.ellint_1.implementation"></a><h6> |
| <a name="id1138537"></a> |
| <a class="link" href="ellint_1.html#math_toolkit.special.ellint.ellint_1.implementation">Implementation</a> |
| </h6> |
| <p> |
| These functions are implemented in terms of Carlson's integrals using the |
| relations: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint19.png"></span> |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint20.png"></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 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<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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