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| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.jacobi.jacobi_elliptic"></a><a class="link" href="jacobi_elliptic.html" title="Jacobi Elliptic SN, CN and DN">Jacobi Elliptic |
| SN, CN and DN</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.jacobi.jacobi_elliptic.h0"></a> |
| <span class="phrase"><a name="math_toolkit.jacobi.jacobi_elliptic.synopsis"></a></span><a class="link" href="jacobi_elliptic.html#math_toolkit.jacobi.jacobi_elliptic.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">jacobi_elliptic</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">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</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">jacobi_elliptic</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">V</span><span class="special">*</span> <span class="identifier">pcn</span><span class="special">,</span> <span class="identifier">V</span><span class="special">*</span> <span class="identifier">pdn</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> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</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">jacobi_elliptic</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">V</span><span class="special">*</span> <span class="identifier">pcn</span><span class="special">,</span> <span class="identifier">V</span><span class="special">*</span> <span class="identifier">pdn</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&);</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.jacobi.jacobi_elliptic.h1"></a> |
| <span class="phrase"><a name="math_toolkit.jacobi.jacobi_elliptic.description"></a></span><a class="link" href="jacobi_elliptic.html#math_toolkit.jacobi.jacobi_elliptic.description">Description</a> |
| </h5> |
| <p> |
| The function <a class="link" href="jacobi_elliptic.html" title="Jacobi Elliptic SN, CN and DN">jacobi_elliptic</a> |
| calculates the three copolar Jacobi elliptic functions <span class="emphasis"><em>sn(u, k)</em></span>, |
| <span class="emphasis"><em>cn(u, k)</em></span> and <span class="emphasis"><em>dn(u, k)</em></span>. The returned |
| value is <span class="emphasis"><em>sn(u, k)</em></span>, and if provided, <span class="emphasis"><em>*pcn</em></span> |
| is set to <span class="emphasis"><em>cn(u, k)</em></span>, and <span class="emphasis"><em>*pdn</em></span> is |
| set to <span class="emphasis"><em>dn(u, k)</em></span>. |
| </p> |
| <p> |
| The functions are defined as follows, given: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/jacobi1.svg"></span> |
| </p> |
| <p> |
| The the angle φ is called the <span class="emphasis"><em>amplitude</em></span> and: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/jacobi2.svg"></span> |
| </p> |
| <div class="note"><table border="0" summary="Note"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td> |
| <th align="left">Note</th> |
| </tr> |
| <tr><td align="left" valign="top"> |
| <p> |
| <span class="emphasis"><em>φ</em></span> is called the amplitude. |
| </p> |
| <p> |
| <span class="emphasis"><em>k</em></span> is called the modulus. |
| </p> |
| </td></tr> |
| </table></div> |
| <div class="caution"><table border="0" summary="Caution"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td> |
| <th align="left">Caution</th> |
| </tr> |
| <tr><td align="left" valign="top"> |
| <p> |
| Rather like other elliptic functions, the Jacobi functions are expressed |
| in a variety of different ways. In particular, the parameter <span class="emphasis"><em>k</em></span> |
| (the modulus) may also be expressed using a modular angle α, or a parameter |
| <span class="emphasis"><em>m</em></span>. These are related by: |
| </p> |
| <p> |
| k = sinα |
| </p> |
| <p> |
| m = k<sup>2</sup> = sin<sup>2</sup>α |
| </p> |
| <p> |
| So that the function <span class="emphasis"><em>sn</em></span> (for example) may be expressed |
| as either: |
| </p> |
| <p> |
| sn(u, k) |
| </p> |
| <p> |
| sn(u \ α) |
| </p> |
| <p> |
| sn(u| m) |
| </p> |
| <p> |
| To further complicate matters, some texts refer to the <span class="emphasis"><em>complement |
| of the parameter m</em></span>, or 1 - m, where: |
| </p> |
| <p> |
| 1 - m = 1 - k<sup>2</sup> = cos<sup>2</sup>α |
| </p> |
| <p> |
| This implementation uses <span class="emphasis"><em>k</em></span> throughout, and makes this |
| the first argument to the functions: this is for alignment with the elliptic |
| integrals which match the requirements of the <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf" target="_top">Technical |
| Report on C++ Library Extensions</a>. However, you should be extra |
| careful when using these functions! |
| </p> |
| </td></tr> |
| </table></div> |
| <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> |
| The following graphs illustrate how these functions change as <span class="emphasis"><em>k</em></span> |
| changes: for small <span class="emphasis"><em>k</em></span> these are sine waves, while as |
| <span class="emphasis"><em>k</em></span> tends to 1 they become hyperbolic functions: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/jacobi_sn.svg" align="middle"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/jacobi_cn.svg" align="middle"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/jacobi_dn.svg" align="middle"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.jacobi.jacobi_elliptic.h2"></a> |
| <span class="phrase"><a name="math_toolkit.jacobi.jacobi_elliptic.accuracy"></a></span><a class="link" href="jacobi_elliptic.html#math_toolkit.jacobi.jacobi_elliptic.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| These functions are computed using only basic arithmetic operations and trigomometric |
| functions, so there isn't much variation in accuracy over differing platforms. |
| Typically errors are trivially small for small angles, and as is typical |
| for cyclic functions, grow as the angle increases. Note that only results |
| for the widest floating point type on the system are given as narrower types |
| have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero |
| error</a>. All values are relative errors in units of epsilon. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.jacobi.jacobi_elliptic.errors_rates_in_the_jacobi_ellip"></a><p class="title"><b>Table 6.29. Errors Rates in the Jacobi Elliptic Functions</b></p> |
| <div class="table-contents"><table class="table" summary="Errors Rates in the Jacobi Elliptic Functions"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| u < 1 |
| </p> |
| </th> |
| <th> |
| <p> |
| Large u |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32 / Visual C++ 8.0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2 Mean=0.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=44000 Mean=2500 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Ubuntu Linux / G++ 4.7 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2.0 Mean=0.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=25000 Mean=1500 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><h5> |
| <a name="math_toolkit.jacobi.jacobi_elliptic.h3"></a> |
| <span class="phrase"><a name="math_toolkit.jacobi.jacobi_elliptic.testing"></a></span><a class="link" href="jacobi_elliptic.html#math_toolkit.jacobi.jacobi_elliptic.testing">Testing</a> |
| </h5> |
| <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 MPFR at 1000-bit precision and this |
| implementation. |
| </p> |
| <h5> |
| <a name="math_toolkit.jacobi.jacobi_elliptic.h4"></a> |
| <span class="phrase"><a name="math_toolkit.jacobi.jacobi_elliptic.implementation"></a></span><a class="link" href="jacobi_elliptic.html#math_toolkit.jacobi.jacobi_elliptic.implementation">Implementation</a> |
| </h5> |
| <p> |
| For <span class="emphasis"><em>k > 1</em></span> we apply the relations: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/jacobi3.svg"></span> |
| </p> |
| <p> |
| Then filter off the special cases: |
| </p> |
| <p> |
| <span class="emphasis"><em>sn(0, k) = 0</em></span> and <span class="emphasis"><em>cn(0, k) = dn(0, k) = 1</em></span>. |
| </p> |
| <p> |
| <span class="emphasis"><em>sn(u, 0) = sin(u), cn(u, 0) = cos(u) and dn(u, 0) = 1</em></span>. |
| </p> |
| <p> |
| <span class="emphasis"><em>sn(u, 1) = tanh(u), cn(u, 1) = dn(u, 1) = 1 / cosh(u)</em></span>. |
| </p> |
| <p> |
| And for <span class="emphasis"><em>k<sup>4</sup> < ε</em></span> we have: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/jacobi4.svg"></span> |
| </p> |
| <p> |
| Otherwise the values are calculated using the method of <a href="http://dlmf.nist.gov/22.20#SS2" target="_top">arithmetic |
| geometric means</a>. |
| </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>) |
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