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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.ellint.ellint_intro"></a><a class="link" href="ellint_intro.html" title="Elliptic Integral Overview"> Elliptic |
| Integral Overview</a> |
| </h4></div></div></div> |
| <p> |
| The main reference for the elliptic integrals is: |
| </p> |
| <div class="blockquote"><blockquote class="blockquote"><p> |
| M. Abramowitz and I. A. Stegun (Eds.) (1964) Handbook of Mathematical |
| Functions with Formulas, Graphs, and Mathematical Tables, National Bureau |
| of Standards Applied Mathematics Series, U.S. Government Printing Office, |
| Washington, D.C. |
| </p></blockquote></div> |
| <p> |
| Mathworld also contain a lot of useful background information: |
| </p> |
| <div class="blockquote"><blockquote class="blockquote"><p> |
| <a href="http://mathworld.wolfram.com/EllipticIntegral.html" target="_top">Weisstein, |
| Eric W. "Elliptic Integral." From MathWorld--A Wolfram Web |
| Resource.</a> |
| </p></blockquote></div> |
| <p> |
| As does <a href="http://en.wikipedia.org/wiki/Elliptic_integral" target="_top">Wikipedia |
| Elliptic integral</a>. |
| </p> |
| <a name="math_toolkit.special.ellint.ellint_intro.notation"></a><h5> |
| <a name="id1130431"></a> |
| <a class="link" href="ellint_intro.html#math_toolkit.special.ellint.ellint_intro.notation">Notation</a> |
| </h5> |
| <p> |
| All variables are real numbers unless otherwise noted. |
| </p> |
| <a name="ellint_def"></a><a name="math_toolkit.special.ellint.ellint_intro.definition"></a><h5> |
| <a name="id1130451"></a> |
| <a class="link" href="ellint_intro.html#math_toolkit.special.ellint.ellint_intro.definition">Definition</a> |
| </h5> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint1.png"></span> |
| </p> |
| <p> |
| is called elliptic integral if <span class="emphasis"><em>R(t, s)</em></span> is a rational |
| function of <span class="emphasis"><em>t</em></span> and <span class="emphasis"><em>s</em></span>, and <span class="emphasis"><em>s<sup>2</sup></em></span> |
| is a cubic or quartic polynomial in <span class="emphasis"><em>t</em></span>. |
| </p> |
| <p> |
| Elliptic integrals generally can not be expressed in terms of elementary |
| functions. However, Legendre showed that all elliptic integrals can be |
| reduced to the following three canonical forms: |
| </p> |
| <p> |
| Elliptic Integral of the First Kind (Legendre form) |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint2.png"></span> |
| </p> |
| <p> |
| Elliptic Integral of the Second Kind (Legendre form) |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint3.png"></span> |
| </p> |
| <p> |
| Elliptic Integral of the Third Kind (Legendre form) |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint4.png"></span> |
| </p> |
| <p> |
| where |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint5.png"></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> |
| <p> |
| <span class="emphasis"><em>α</em></span> is called the modular angle. |
| </p> |
| <p> |
| <span class="emphasis"><em>n</em></span> is called the characteristic. |
| </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> |
| Perhaps more than any other special functions the elliptic integrals |
| are expressed in a variety of different ways. In particular, the final |
| parameter <span class="emphasis"><em>k</em></span> (the modulus) may 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 integral of the third kind (for example) may be expressed |
| as either: |
| </p> |
| <p> |
| Π(n, φ, k) |
| </p> |
| <p> |
| Π(n, φ \ α) |
| </p> |
| <p> |
| Π(n, φ| 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: this matches |
| 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> |
| When <span class="emphasis"><em>φ</em></span> = <span class="emphasis"><em>π</em></span> / 2, the elliptic integrals |
| are called <span class="emphasis"><em>complete</em></span>. |
| </p> |
| <p> |
| Complete Elliptic Integral of the First Kind (Legendre form) |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint6.png"></span> |
| </p> |
| <p> |
| Complete Elliptic Integral of the Second Kind (Legendre form) |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint7.png"></span> |
| </p> |
| <p> |
| Complete Elliptic Integral of the Third Kind (Legendre form) |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint8.png"></span> |
| </p> |
| <p> |
| Carlson [<a class="link" href="ellint_intro.html#ellint_ref_carlson77">Carlson77</a>] [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>] gives an alternative definition |
| of elliptic integral's canonical forms: |
| </p> |
| <p> |
| Carlson's Elliptic Integral of the First Kind |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint9.png"></span> |
| </p> |
| <p> |
| where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>, <span class="emphasis"><em>z</em></span> |
| are nonnegative and at most one of them may be zero. |
| </p> |
| <p> |
| Carlson's Elliptic Integral of the Second Kind |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint10.png"></span> |
| </p> |
| <p> |
| where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span> are nonnegative, at |
| most one of them may be zero, and <span class="emphasis"><em>z</em></span> must be positive. |
| </p> |
| <p> |
| Carlson's Elliptic Integral of the Third Kind |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint11.png"></span> |
| </p> |
| <p> |
| where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>, <span class="emphasis"><em>z</em></span> |
| are nonnegative, at most one of them may be zero, and <span class="emphasis"><em>p</em></span> |
| must be nonzero. |
| </p> |
| <p> |
| Carlson's Degenerate Elliptic Integral |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint12.png"></span> |
| </p> |
| <p> |
| where <span class="emphasis"><em>x</em></span> is nonnegative and <span class="emphasis"><em>y</em></span> |
| is nonzero. |
| </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>R<sub>C</sub>(x, y) = R<sub>F</sub>(x, y, y)</em></span> |
| </p> |
| <p> |
| <span class="emphasis"><em>R<sub>D</sub>(x, y, z) = R<sub>J</sub>(x, y, z, z)</em></span> |
| </p> |
| </td></tr> |
| </table></div> |
| <a name="ellint_theorem"></a><a name="math_toolkit.special.ellint.ellint_intro.duplication_theorem"></a><h5> |
| <a name="id1131051"></a> |
| <a class="link" href="ellint_intro.html#math_toolkit.special.ellint.ellint_intro.duplication_theorem">Duplication |
| Theorem</a> |
| </h5> |
| <p> |
| Carlson proved in [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>] |
| that |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint13.png"></span> |
| </p> |
| <a name="ellint_formula"></a><a name="math_toolkit.special.ellint.ellint_intro.carlson_s_formulas"></a><h5> |
| <a name="id1131105"></a> |
| <a class="link" href="ellint_intro.html#math_toolkit.special.ellint.ellint_intro.carlson_s_formulas">Carlson's |
| Formulas</a> |
| </h5> |
| <p> |
| The Legendre form and Carlson form of elliptic integrals are related by |
| equations: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint14.png"></span> |
| </p> |
| <p> |
| In particular, |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ellint15.png"></span> |
| </p> |
| <a name="math_toolkit.special.ellint.ellint_intro.numerical_algorithms"></a><h5> |
| <a name="id1131178"></a> |
| <a class="link" href="ellint_intro.html#math_toolkit.special.ellint.ellint_intro.numerical_algorithms">Numerical |
| Algorithms</a> |
| </h5> |
| <p> |
| The conventional methods for computing elliptic integrals are Gauss and |
| Landen transformations, which converge quadratically and work well for |
| elliptic integrals of the first and second kinds. Unfortunately they suffer |
| from loss of significant digits for the third kind. Carlson's algorithm |
| [<a class="link" href="ellint_intro.html#ellint_ref_carlson79">Carlson79</a>] [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>], |
| by contrast, provides a unified method for all three kinds of elliptic |
| integrals with satisfactory precisions. |
| </p> |
| <a name="ellint_refs"></a><a name="math_toolkit.special.ellint.ellint_intro.references"></a><h5> |
| <a name="id1131211"></a> |
| <a class="link" href="ellint_intro.html#math_toolkit.special.ellint.ellint_intro.references">References</a> |
| </h5> |
| <p> |
| Special mention goes to: |
| </p> |
| <div class="blockquote"><blockquote class="blockquote"><p> |
| A. M. Legendre, <span class="emphasis"><em>Traitd des Fonctions Elliptiques et des Integrales |
| Euleriennes</em></span>, Vol. 1. Paris (1825). |
| </p></blockquote></div> |
| <p> |
| However the main references are: |
| </p> |
| <a name="ellint_ref_AS"></a><a name="ellint_ref_carlson79"></a><a name="ellint_ref_carlson77"></a><a name="ellint_ref_carlson78"></a><a name="ellint_ref_carlson81"></a><div class="orderedlist"><ol type="1"> |
| <li> |
| M. Abramowitz and I. A. Stegun (Eds.) (1964) Handbook of Mathematical |
| Functions with Formulas, Graphs, and Mathematical Tables, National |
| Bureau of Standards Applied Mathematics Series, U.S. Government Printing |
| Office, Washington, D.C. |
| </li> |
| <li> |
| B.C. Carlson, <span class="emphasis"><em>Computing elliptic integrals by duplication</em></span>, |
| Numerische Mathematik, vol 33, 1 (1979). |
| </li> |
| <li> |
| B.C. Carlson, <span class="emphasis"><em>Elliptic Integrals of the First Kind</em></span>, |
| SIAM Journal on Mathematical Analysis, vol 8, 231 (1977). |
| </li> |
| <li> |
| B.C. Carlson, <span class="emphasis"><em>Short Proofs of Three Theorems on Elliptic |
| Integrals</em></span>, SIAM Journal on Mathematical Analysis, vol 9, |
| 524 (1978). |
| </li> |
| <li> |
| B.C. Carlson and E.M. Notis, <span class="emphasis"><em>ALGORITHM 577: Algorithms for |
| Incomplete Elliptic Integrals</em></span>, ACM Transactions on Mathematmal |
| Software, vol 7, 398 (1981). |
| </li> |
| <li> |
| B. C. Carlson, <span class="emphasis"><em>On computing elliptic integrals and functions</em></span>. |
| J. Math. and Phys., 44 (1965), pp. 36-51. |
| </li> |
| <li> |
| B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals of the second |
| kind</em></span>. Math. Comp., 49 (1987), pp. 595-606. (Supplement, |
| ibid., pp. S13-S17.) |
| </li> |
| <li> |
| B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals of the third |
| kind</em></span>. Math. Comp., 51 (1988), pp. 267-280. (Supplement, |
| ibid., pp. S1-S5.) |
| </li> |
| <li> |
| B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: cubic cases</em></span>. |
| Math. Comp., 53 (1989), pp. 327-333. |
| </li> |
| <li> |
| B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: one quadratic |
| factor</em></span>. Math. Comp., 56 (1991), pp. 267-280. |
| </li> |
| <li> |
| B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: two quadratic |
| factors</em></span>. Math. Comp., 59 (1992), pp. 165-180. |
| </li> |
| <li> |
| B. C. Carlson, <span class="emphasis"><em><a href="http://arxiv.org/abs/math.CA/9409227" target="_top">Numerical |
| computation of real or complex elliptic integrals</a></em></span>. |
| Numerical Algorithms, Volume 10, Number 1 / March, 1995, p13-26. |
| </li> |
| <li> |
| B. C. Carlson and John L. Gustafson, <span class="emphasis"><em><a href="http://arxiv.org/abs/math.CA/9310223" target="_top">Asymptotic |
| Approximations for Symmetric Elliptic Integrals</a></em></span>, |
| SIAM Journal on Mathematical Analysis, Volume 25, Issue 2 (March 1994), |
| 288-303. |
| </li> |
| </ol></div> |
| <p> |
| The following references, while not directly relevent to our implementation, |
| may also be of interest: |
| </p> |
| <div class="orderedlist"><ol type="1"> |
| <li> |
| R. Burlisch, <span class="emphasis"><em>Numerical Compuation of Elliptic Integrals and |
| Elliptic Functions.</em></span> Numerical Mathematik 7, 78-90. |
| </li> |
| <li> |
| R. Burlisch, <span class="emphasis"><em>An extension of the Bartky Transformation to |
| Incomplete Elliptic Integrals of the Third Kind</em></span>. Numerical |
| Mathematik 13, 266-284. |
| </li> |
| <li> |
| R. Burlisch, <span class="emphasis"><em>Numerical Compuation of Elliptic Integrals and |
| Elliptic Functions. III</em></span>. Numerical Mathematik 13, 305-315. |
| </li> |
| <li> |
| T. Fukushima and H. Ishizaki, <span class="emphasis"><em><a href="http://adsabs.harvard.edu/abs/1994CeMDA..59..237F" target="_top">Numerical |
| Computation of Incomplete Elliptic Integrals of a General Form.</a></em></span> |
| Celestial Mechanics and Dynamical Astronomy, Volume 59, Number 3 / |
| July, 1994, 237-251. |
| </li> |
| </ol></div> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
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| <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>) |
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