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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.jacobi.jacobi_elliptic"></a><a class="link" href="jacobi_elliptic.html" title="Jacobi Elliptic SN, CN and DN">Jacobi Elliptic
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SN, CN and DN</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.jacobi.jacobi_elliptic.h0"></a>
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<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>
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</h5>
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<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>
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</pre>
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<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>
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<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>
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<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>
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<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>
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<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>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h5>
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<a name="math_toolkit.jacobi.jacobi_elliptic.h1"></a>
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<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>
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</h5>
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<p>
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The function <a class="link" href="jacobi_elliptic.html" title="Jacobi Elliptic SN, CN and DN">jacobi_elliptic</a>
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calculates the three copolar Jacobi elliptic functions <span class="emphasis"><em>sn(u, k)</em></span>,
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<span class="emphasis"><em>cn(u, k)</em></span> and <span class="emphasis"><em>dn(u, k)</em></span>. The returned
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value is <span class="emphasis"><em>sn(u, k)</em></span>, and if provided, <code class="computeroutput"><span class="special">*</span><span class="identifier">pcn</span></code> is set to <span class="emphasis"><em>cn(u, k)</em></span>,
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and <code class="computeroutput"><span class="special">*</span><span class="identifier">pdn</span></code>
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is set to <span class="emphasis"><em>dn(u, k)</em></span>.
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</p>
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<p>
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The functions are defined as follows, given:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/jacobi1.svg"></span>
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</p></blockquote></div>
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<p>
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The the angle <span class="emphasis"><em>φ</em></span> is called the <span class="emphasis"><em>amplitude</em></span>
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and:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/jacobi2.svg"></span>
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</p></blockquote></div>
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<div class="note"><table border="0" summary="Note">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
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<th align="left">Note</th>
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</tr>
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<tr><td align="left" valign="top"><p>
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<span class="emphasis"><em>φ</em></span> is called the amplitude. <span class="emphasis"><em>k</em></span> is
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called the elliptic modulus.
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</p></td></tr>
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</table></div>
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<div class="caution"><table border="0" summary="Caution">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
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<th align="left">Caution</th>
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</tr>
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<tr><td align="left" valign="top">
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<p>
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Rather like other elliptic functions, the Jacobi functions are expressed
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in a variety of different ways. In particular, the parameter <span class="emphasis"><em>k</em></span>
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(the modulus) may also be expressed using a modular angle α, or a parameter
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<span class="emphasis"><em>m</em></span>. These are related by:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">k = sin α</span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">m = k<sup>2</sup> = sin<sup>2</sup>α</span>
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</p></blockquote></div>
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<p>
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So that the function <span class="emphasis"><em>sn</em></span> (for example) may be expressed
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as either:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">sn(u, k)</span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">sn(u \ α)</span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">sn(u | m)</span>
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</p></blockquote></div>
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<p>
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To further complicate matters, some texts refer to the <span class="emphasis"><em>complement
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of the parameter m</em></span>, or 1 - m, where:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">1 - m = 1 - k<sup>2</sup> = cos<sup>2</sup>α</span>
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</p></blockquote></div>
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<p>
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This implementation uses <span class="emphasis"><em>k</em></span> throughout, and makes this
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the first argument to the functions: this is for alignment with the elliptic
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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
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Report on C++ Library Extensions</a>. However, you should be extra
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careful when using these functions!
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</p>
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</td></tr>
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</table></div>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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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
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documentation for more details</a>.
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</p>
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<p>
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The following graphs illustrate how these functions change as <span class="emphasis"><em>k</em></span>
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changes: for small <span class="emphasis"><em>k</em></span> these are sine waves, while as
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<span class="emphasis"><em>k</em></span> tends to 1 they become hyperbolic functions:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/jacobi_sn.svg" align="middle"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/jacobi_cn.svg" align="middle"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/jacobi_dn.svg" align="middle"></span>
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</p></blockquote></div>
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<h5>
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<a name="math_toolkit.jacobi.jacobi_elliptic.h2"></a>
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<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>
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</h5>
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<p>
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These functions are computed using only basic arithmetic operations and trigomometric
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functions, so there isn't much variation in accuracy over differing platforms.
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Typically errors are trivially small for small angles, and as is typical
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for cyclic functions, grow as the angle increases. Note that only results
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for the widest floating-point type on the system are given as narrower types
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have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero
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error</a>. All values are relative errors in units of epsilon.
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</p>
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<div class="table">
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<a name="math_toolkit.jacobi.jacobi_elliptic.table_jacobi_cn"></a><p class="title"><b>Table 8.70. Error rates for jacobi_cn</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for jacobi_cn">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5150<br> Sun Solaris<br> long double
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</p>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 14.1<br> Win32<br> double
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Jacobi Elliptic: Mathworld Data
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 17.3ε (Mean = 4.29ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_cn_GSL_2_1_Jacobi_Elliptic_Mathworld_Data">And
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other failures.</a>)
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 71.6ε (Mean = 19.3ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 71.6ε (Mean = 19.4ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 45.8ε (Mean = 11.4ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Jacobi Elliptic: Random Data
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.816ε (Mean = 0.0563ε)</span><br>
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<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.43ε (Mean = 0.803ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.68ε (Mean = 0.443ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.68ε (Mean = 0.454ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.83ε (Mean = 0.455ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Jacobi Elliptic: Random Small Values
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 55.2ε (Mean = 1.64ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_cn_GSL_2_1_Jacobi_Elliptic_Random_Small_Values">And
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other failures.</a>)
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 10.4ε (Mean = 0.594ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 10.4ε (Mean = 0.602ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 26.2ε (Mean = 1.17ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Jacobi Elliptic: Modulus near 1
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.919ε (Mean = 0.127ε)</span><br> <br>
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(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 0ε (Mean = 0ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_cn_GSL_2_1_Jacobi_Elliptic_Modulus_near_1">And
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other failures.</a>)
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 675ε (Mean = 87.1ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 675ε (Mean = 86.8ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 513ε (Mean = 126ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Jacobi Elliptic: Large Phi
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 14.2ε (Mean = 0.927ε)</span><br> <br>
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(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 5.92e+03ε (Mean = 477ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2.97e+04ε (Mean = 1.9e+03ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2.97e+04ε (Mean = 1.9e+03ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 3.27e+04ε (Mean = 1.93e+03ε)</span>
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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</div>
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<br class="table-break"><div class="table">
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<a name="math_toolkit.jacobi.jacobi_elliptic.table_jacobi_dn"></a><p class="title"><b>Table 8.71. Error rates for jacobi_dn</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for jacobi_dn">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5150<br> Sun Solaris<br> long double
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</p>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 14.1<br> Win32<br> double
|
|
</p>
|
|
</th>
|
|
</tr></thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Jacobi Elliptic: Mathworld Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
|
2.1:</em></span> Max = 2.82ε (Mean = 1.18ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_dn_GSL_2_1_Jacobi_Elliptic_Mathworld_Data">And
|
|
other failures.</a>)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 49ε (Mean = 14ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 49ε (Mean = 14ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 34.3ε (Mean = 8.71ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Jacobi Elliptic: Random Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
|
2.1:</em></span> Max = 3ε (Mean = 0.61ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.53ε (Mean = 0.473ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.53ε (Mean = 0.481ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.52ε (Mean = 0.466ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Jacobi Elliptic: Random Small Values
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0.5ε (Mean = 0.0122ε)</span><br> <br>
|
|
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.5ε (Mean = 0.391ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_dn_GSL_2_1_Jacobi_Elliptic_Random_Small_Values">And
|
|
other failures.</a>)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 22.4ε (Mean = 0.777ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 22.4ε (Mean = 0.763ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 16.1ε (Mean = 0.685ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Jacobi Elliptic: Modulus near 1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 2.28ε (Mean = 0.194ε)</span><br> <br>
|
|
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 0ε (Mean = 0ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_dn_GSL_2_1_Jacobi_Elliptic_Modulus_near_1">And
|
|
other failures.</a>)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 3.75e+03ε (Mean = 293ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 3.75e+03ε (Mean = 293ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 6.24e+03ε (Mean = 482ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Jacobi Elliptic: Large Phi
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 14.1ε (Mean = 0.897ε)</span><br> <br>
|
|
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 121ε (Mean = 22ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 2.82e+04ε (Mean = 1.79e+03ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 2.82e+04ε (Mean = 1.79e+03ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.67e+04ε (Mean = 1e+03ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table></div>
|
|
</div>
|
|
<br class="table-break"><div class="table">
|
|
<a name="math_toolkit.jacobi.jacobi_elliptic.table_jacobi_sn"></a><p class="title"><b>Table 8.72. Error rates for jacobi_sn</b></p>
|
|
<div class="table-contents"><table class="table" summary="Error rates for jacobi_sn">
|
|
<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>
|
|
Jacobi Elliptic: Mathworld Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
|
2.1:</em></span> Max = 588ε (Mean = 146ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_sn_GSL_2_1_Jacobi_Elliptic_Mathworld_Data">And
|
|
other failures.</a>)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 341ε (Mean = 80.7ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 341ε (Mean = 80.7ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 481ε (Mean = 113ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Jacobi Elliptic: Random Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
|
2.1:</em></span> Max = 4.02ε (Mean = 1.07ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 2.01ε (Mean = 0.584ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 2.01ε (Mean = 0.593ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.92ε (Mean = 0.567ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Jacobi Elliptic: Random Small Values
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
|
2.1:</em></span> Max = 11.7ε (Mean = 1.65ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_sn_GSL_2_1_Jacobi_Elliptic_Random_Small_Values">And
|
|
other failures.</a>)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.99ε (Mean = 0.347ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.99ε (Mean = 0.347ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 2.11ε (Mean = 0.385ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Jacobi Elliptic: Modulus near 1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
|
2.1:</em></span> Max = 0ε (Mean = 0ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_jacobi_sn_GSL_2_1_Jacobi_Elliptic_Modulus_near_1">And
|
|
other failures.</a>)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 109ε (Mean = 7.35ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 109ε (Mean = 7.38ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 23.2ε (Mean = 1.85ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Jacobi Elliptic: Large Phi
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 12ε (Mean = 0.771ε)</span><br> <br>
|
|
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 4.54e+04ε (Mean = 2.63e+03ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 2.45e+04ε (Mean = 1.51e+03ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 2.45e+04ε (Mean = 1.51e+03ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.36e+04ε (Mean = 2.54e+03ε)</span>
|
|
</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>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/jacobi3.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
Then filter off the special cases:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic"><span class="emphasis"><em>sn(0, k) = 0</em></span> and <span class="emphasis"><em>cn(0,
|
|
k) = dn(0, k) = 1</em></span></span>
|
|
</p></blockquote></div>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic"><span class="emphasis"><em>sn(u, 0) = sin(u), cn(u, 0) = cos(u)
|
|
and dn(u, 0) = 1</em></span></span>
|
|
</p></blockquote></div>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic"><span class="emphasis"><em>sn(u, 1) = tanh(u), cn(u, 1) = dn(u,
|
|
1) = 1 / cosh(u)</em></span></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
And for <span class="emphasis"><em>k<sup>4</sup> < ε</em></span> we have:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/jacobi4.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<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 © 2006-2019 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>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
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