<title>Jacobi Elliptic SN, CN and DN</title>
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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>.
+ 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>,
+ and <code class="computeroutput"><span class="special">*</span><span class="identifier">pdn</span></code>
+ is set to <span class="emphasis"><em>dn(u, k)</em></span>.
</p>
<p>
The functions are defined as follows, given:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/jacobi1.svg"></span>
+
+ </p></blockquote></div>
<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>
+ The the angle <span class="emphasis"><em>φ</em></span> is called the <span class="emphasis"><em>amplitude</em></span>
+ and:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/jacobi2.svg"></span>
+
+ </p></blockquote></div>
<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>
+<tr><td align="left" valign="top"><p>
+ <span class="emphasis"><em>φ</em></span> is called the amplitude. <span class="emphasis"><em>k</em></span> is
+ called the elliptic modulus.
+ </p></td></tr>
</table></div>
<div class="caution"><table border="0" summary="Caution">
<tr>
(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>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">k = sin α</span>
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">m = k<sup>2</sup> = sin<sup>2</sup>α</span>
+ </p></blockquote></div>
<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>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">sn(u, k)</span>
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">sn(u \ α)</span>
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">sn(u | m)</span>
+ </p></blockquote></div>
<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>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">1 - m = 1 - k<sup>2</sup> = cos<sup>2</sup>α</span>
+ </p></blockquote></div>
<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
</td></tr>
</table></div>
<p>
- The final <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+ The final <a class="link" href="../../policy.html" title="Chapter 20. 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 19. Policies: Controlling Precision, Error Handling etc">policy
+ 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
documentation for more details</a>.
</p>
<p>
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>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/jacobi_sn.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/jacobi_cn.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/jacobi_dn.svg" align="middle"></span>
+
+ </p></blockquote></div>
<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>
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
+ 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.table_jacobi_cn"></a><p class="title"><b>Table 7.70. Error rates for jacobi_cn</b></p>
+<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>
<div class="table-contents"><table class="table" summary="Error rates for jacobi_cn">
<colgroup>
<col>
</table></div>
</div>
<br class="table-break"><div class="table">
-<a name="math_toolkit.jacobi.jacobi_elliptic.table_jacobi_dn"></a><p class="title"><b>Table 7.71. Error rates for jacobi_dn</b></p>
+<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>
<div class="table-contents"><table class="table" summary="Error rates for jacobi_dn">
<colgroup>
<col>
</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 7.72. Error rates for jacobi_sn</b></p>
+<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>
<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>
+<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>
-<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>
+<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>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/jacobi4.svg"></span>
- </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>.
projects / platform / upstream / boost.git / blobdiff