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250 lines
8.0 KiB
Plaintext
[/
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Copyright (c) 2006 Xiaogang Zhang
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Use, modification and distribution are subject to the
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Boost Software License, Version 1.0. (See accompanying file
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LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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]
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[section:ellint_intro Elliptic Integral Overview]
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The main reference for the elliptic integrals is:
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[:M. Abramowitz and I. A. Stegun (Eds.) (1964)
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Handbook of Mathematical Functions with Formulas, Graphs, and
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Mathematical Tables,
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National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C.]
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Mathworld also contain a lot of useful background information:
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[:[@http://mathworld.wolfram.com/EllipticIntegral.html Weisstein, Eric W.
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"Elliptic Integral." From MathWorld--A Wolfram Web Resource.]]
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As does [@http://en.wikipedia.org/wiki/Elliptic_integral Wikipedia Elliptic integral].
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[h4 Notation]
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All variables are real numbers unless otherwise noted.
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[h4 Definition]
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[equation ellint1]
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is called elliptic integral if ['R(t, s)] is a rational function
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of ['t] and ['s], and ['s[super 2]] is a cubic or quartic polynomial
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in ['t].
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Elliptic integrals generally can not be expressed in terms of
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elementary functions. However, Legendre showed that all elliptic
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integrals can be reduced to the following three canonical forms:
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Elliptic Integral of the First Kind (Legendre form)
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[equation ellint2]
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Elliptic Integral of the Second Kind (Legendre form)
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[equation ellint3]
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Elliptic Integral of the Third Kind (Legendre form)
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[equation ellint4]
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where
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[equation ellint5]
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[note ['[phi]] is called the amplitude.
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['k] is called the modulus.
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['[alpha]] is called the modular angle.
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['n] is called the characteristic.]
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[caution Perhaps more than any other special functions the elliptic
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integrals are expressed in a variety of different ways. In particular,
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the final parameter /k/ (the modulus) may be expressed using a modular
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angle [alpha], or a parameter /m/. These are related by:
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k = sin[alpha]
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m = k[super 2] = sin[super 2][alpha]
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So that the integral of the third kind (for example) may be expressed as
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either:
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[Pi](n, [phi], k)
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[Pi](n, [phi] \\ [alpha])
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[Pi](n, [phi]| m)
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To further complicate matters, some texts refer to the ['complement
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of the parameter m], or 1 - m, where:
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1 - m = 1 - k[super 2] = cos[super 2][alpha]
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This implementation uses /k/ throughout: this matches the requirements
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of the [@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf
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Technical Report on C++ Library Extensions]. However, you should
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be extra careful when using these functions!]
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When ['[phi]] = ['[pi]] / 2, the elliptic integrals are called ['complete].
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Complete Elliptic Integral of the First Kind (Legendre form)
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[equation ellint6]
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Complete Elliptic Integral of the Second Kind (Legendre form)
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[equation ellint7]
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Complete Elliptic Integral of the Third Kind (Legendre form)
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[equation ellint8]
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Legendre also defined a forth integral D([phi],k) which is a combination of the other three:
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[equation ellint_d]
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Like the other Legendre integrals this comes in both complete and incomplete forms.
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[h4 Carlson Elliptic Integrals]
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Carlson [[link ellint_ref_carlson77 Carlson77]] [[link ellint_ref_carlson78 Carlson78]] gives an alternative definition of
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elliptic integral's canonical forms:
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Carlson's Elliptic Integral of the First Kind
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[equation ellint9]
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where ['x], ['y], ['z] are nonnegative and at most one of them
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may be zero.
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Carlson's Elliptic Integral of the Second Kind
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[equation ellint10]
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where ['x], ['y] are nonnegative, at most one of them may be zero,
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and ['z] must be positive.
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Carlson's Elliptic Integral of the Third Kind
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[equation ellint11]
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where ['x], ['y], ['z] are nonnegative, at most one of them may be
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zero, and ['p] must be nonzero.
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Carlson's Degenerate Elliptic Integral
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[equation ellint12]
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where ['x] is nonnegative and ['y] is nonzero.
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[note ['R[sub C](x, y) = R[sub F](x, y, y)]
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['R[sub D](x, y, z) = R[sub J](x, y, z, z)]]
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Carlson's Symmetric Integral
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[equation ellint27]
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[h4 Duplication Theorem]
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Carlson proved in [[link ellint_ref_carlson78 Carlson78]] that
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[equation ellint13]
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[h4 Carlson's Formulas]
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The Legendre form and Carlson form of elliptic integrals are related
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by equations:
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[equation ellint14]
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In particular,
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[equation ellint15]
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[h4 Miscellaneous Elliptic Integrals]
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There are two functions related to the elliptic integrals which otherwise
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defy categorisation, these are the Jacobi Zeta function:
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[equation jacobi_zeta]
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and the Heuman Lambda function:
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[equation heuman_lambda]
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Both of these functions are easily implemented in terms of Carlson's integrals, and are
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provided in this library as __jacobi_zeta and __heuman_lambda.
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[h4 Numerical Algorithms]
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The conventional methods for computing elliptic integrals are Gauss
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and Landen transformations, which converge quadratically and work
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well for elliptic integrals of the first and second kinds.
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Unfortunately they suffer from loss of significant digits for the
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third kind. Carlson's algorithm [[link ellint_ref_carlson79 Carlson79]] [[link ellint_ref_carlson78 Carlson78]], by contrast,
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provides a unified method for all three kinds of elliptic integrals
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with satisfactory precisions.
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[h4 References]
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Special mention goes to:
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[:A. M. Legendre, ['Traitd des Fonctions Elliptiques et des Integrales
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Euleriennes], Vol. 1. Paris (1825).]
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However the main references are:
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# [#ellint_ref_AS]M. Abramowitz and I. A. Stegun (Eds.) (1964)
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Handbook of Mathematical Functions with Formulas, Graphs, and
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Mathematical Tables,
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National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C.
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# [#ellint_ref_carlson79]B.C. Carlson, ['Computing elliptic integrals by duplication],
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Numerische Mathematik, vol 33, 1 (1979).
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# [#ellint_ref_carlson77]B.C. Carlson, ['Elliptic Integrals of the First Kind],
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SIAM Journal on Mathematical Analysis, vol 8, 231 (1977).
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# [#ellint_ref_carlson78]B.C. Carlson, ['Short Proofs of Three Theorems on Elliptic Integrals],
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SIAM Journal on Mathematical Analysis, vol 9, 524 (1978).
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# [#ellint_ref_carlson81]B.C. Carlson and E.M. Notis, ['ALGORITHM 577: Algorithms for Incomplete
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Elliptic Integrals], ACM Transactions on Mathematmal Software,
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vol 7, 398 (1981).
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# B. C. Carlson, ['On computing elliptic integrals and functions]. J. Math. and
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Phys., 44 (1965), pp. 36-51.
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# B. C. Carlson, ['A table of elliptic integrals of the second kind]. Math. Comp., 49
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(1987), pp. 595-606. (Supplement, ibid., pp. S13-S17.)
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# B. C. Carlson, ['A table of elliptic integrals of the third kind]. Math. Comp., 51 (1988),
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pp. 267-280. (Supplement, ibid., pp. S1-S5.)
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# B. C. Carlson, ['A table of elliptic integrals: cubic cases]. Math. Comp., 53 (1989), pp.
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327-333.
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# B. C. Carlson, ['A table of elliptic integrals: one quadratic factor]. Math. Comp., 56 (1991),
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pp. 267-280.
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# B. C. Carlson, ['A table of elliptic integrals: two quadratic factors]. Math. Comp., 59
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(1992), pp. 165-180.
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# B. C. Carlson, ['[@http://arxiv.org/abs/math.CA/9409227
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Numerical computation of real or complex elliptic integrals]]. Numerical Algorithms,
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Volume 10, Number 1 / March, 1995, p13-26.
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# B. C. Carlson and John L. Gustafson, ['[@http://arxiv.org/abs/math.CA/9310223
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Asymptotic Approximations for Symmetric Elliptic Integrals]],
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SIAM Journal on Mathematical Analysis, Volume 25, Issue 2 (March 1994), 288-303.
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The following references, while not directly relevent to our implementation,
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may also be of interest:
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# R. Burlisch, ['Numerical Compuation of Elliptic Integrals and Elliptic Functions.]
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Numerical Mathematik 7, 78-90.
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# R. Burlisch, ['An extension of the Bartky Transformation to Incomplete
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Elliptic Integrals of the Third Kind]. Numerical Mathematik 13, 266-284.
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# R. Burlisch, ['Numerical Compuation of Elliptic Integrals and Elliptic Functions. III].
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Numerical Mathematik 13, 305-315.
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# T. Fukushima and H. Ishizaki, ['[@http://adsabs.harvard.edu/abs/1994CeMDA..59..237F
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Numerical Computation of Incomplete Elliptic Integrals of a General Form.]]
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Celestial Mechanics and Dynamical Astronomy, Volume 59, Number 3 / July, 1994,
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237-251.
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[endsect]
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