ceph/src/boost/libs/math/doc/sf/ellint_introduction.qbk

   1 [/
   2 Copyright (c) 2006 Xiaogang Zhang
   3 Use, modification and distribution are subject to the
   4 Boost Software License, Version 1.0. (See accompanying file
   5 LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
   6 ]
   7
   8 [section:ellint_intro Elliptic Integral Overview]
   9
  10 The main reference for the elliptic integrals is:
  11
  12 [:M. Abramowitz and I. A. Stegun (Eds.) (1964)
  13 Handbook of Mathematical Functions with Formulas, Graphs, and
  14 Mathematical Tables,
  15 National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C.]
  16
  17 Mathworld also contain a lot of useful background information:
  18
  19 [:[@http://mathworld.wolfram.com/EllipticIntegral.html Weisstein, Eric W.
  20 "Elliptic Integral." From MathWorld--A Wolfram Web Resource.]]
  21
  22 As does [@http://en.wikipedia.org/wiki/Elliptic_integral Wikipedia Elliptic integral].
  23
  24 [h4 Notation]
  25
  26 All variables are real numbers unless otherwise noted.
  27
  28 [h4 Definition]
  29
  30 [equation ellint1]
  31
  32 is called elliptic integral if ['R(t, s)] is a rational function
  33 of ['t] and ['s], and ['s[super 2]] is a cubic or quartic polynomial
  34 in ['t].
  35
  36 Elliptic integrals generally can not be expressed in terms of
  37 elementary functions. However, Legendre showed that all elliptic
  38 integrals can be reduced to the following three canonical forms:
  39
  40 Elliptic Integral of the First Kind (Legendre form)
  41
  42 [equation ellint2]
  43
  44 Elliptic Integral of the Second Kind (Legendre form)
  45
  46 [equation ellint3]
  47
  48 Elliptic Integral of the Third Kind (Legendre form)
  49
  50 [equation ellint4]
  51
  52 where
  53
  54 [equation ellint5]
  55
  56 [note ['[phi]] is called the amplitude.
  57
  58 ['k] is called the modulus.
  59
  60 ['[alpha]] is called the modular angle.
  61
  62 ['n] is called the characteristic.]
  63
  64 [caution Perhaps more than any other special functions the elliptic
  65 integrals are expressed in a variety of different ways.  In particular,
  66 the final parameter /k/ (the modulus) may be expressed using a modular
  67 angle [alpha], or a parameter /m/.  These are related by:
  68
  69 k = sin[alpha]
  70
  71 m = k[super 2] = sin[super 2][alpha]
  72
  73 So that the integral of the third kind (for example) may be expressed as
  74 either:
  75
  76 [Pi](n, [phi], k)
  77
  78 [Pi](n, [phi] \\ [alpha])
  79
  80 [Pi](n, [phi]| m)
  81
  82 To further complicate matters, some texts refer to the ['complement
  83 of the parameter m], or 1 - m, where:
  84
  85 1 - m = 1 - k[super 2] = cos[super 2][alpha]
  86
  87 This implementation uses /k/ throughout: this matches the requirements
  88 of the [@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf
  89 Technical Report on C++ Library Extensions].  However, you should
  90 be extra careful when using these functions!]
  91
  92 When ['[phi]] = ['[pi]] / 2, the elliptic integrals are called ['complete].
  93
  94 Complete Elliptic Integral of the First Kind (Legendre form)
  95
  96 [equation ellint6]
  97
  98 Complete Elliptic Integral of the Second Kind (Legendre form)
  99
 100 [equation ellint7]
 101
 102 Complete Elliptic Integral of the Third Kind (Legendre form)
 103
 104 [equation ellint8]
 105
 106 Legendre also defined a forth integral D([phi],k) which is a combination of the other three:
 107
 108 [equation ellint_d]
 109
 110 Like the other Legendre integrals this comes in both complete and incomplete forms.
 111
 112 [h4 Carlson Elliptic Integrals]
 113
 114 Carlson [[link ellint_ref_carlson77 Carlson77]] [[link ellint_ref_carlson78  Carlson78]] gives an alternative definition of
 115 elliptic integral's canonical forms:
 116
 117 Carlson's Elliptic Integral of the First Kind
 118
 119 [equation ellint9]
 120
 121 where ['x], ['y], ['z] are nonnegative and at most one of them
 122 may be zero.
 123
 124 Carlson's Elliptic Integral of the Second Kind
 125
 126 [equation ellint10]
 127
 128 where ['x], ['y] are nonnegative, at most one of them may be zero,
 129 and ['z] must be positive.
 130
 131 Carlson's Elliptic Integral of the Third Kind
 132
 133 [equation ellint11]
 134
 135 where ['x], ['y], ['z] are nonnegative, at most one of them may be
 136 zero, and ['p] must be nonzero.
 137
 138 Carlson's Degenerate Elliptic Integral
 139
 140 [equation ellint12]
 141
 142 where ['x] is nonnegative and ['y] is nonzero.
 143
 144 [note ['R[sub C](x, y) = R[sub F](x, y, y)]
 145
 146 ['R[sub D](x, y, z) = R[sub J](x, y, z, z)]]
 147
 148 Carlson's Symmetric Integral
 149
 150 [equation ellint27]
 151
 152 [h4 Duplication Theorem]
 153
 154 Carlson proved in [[link ellint_ref_carlson78  Carlson78]] that
 155
 156 [equation ellint13]
 157
 158 [h4 Carlson's Formulas]
 159
 160 The Legendre form and Carlson form of elliptic integrals are related
 161 by equations:
 162
 163 [equation ellint14]
 164
 165 In particular,
 166
 167 [equation ellint15]
 168
 169 [h4 Miscellaneous Elliptic Integrals]
 170
 171 There are two functions related to the elliptic integrals which otherwise
 172 defy categorisation, these are the Jacobi Zeta function:
 173
 174 [equation jacobi_zeta]
 175
 176 and the Heuman Lambda function:
 177
 178 [equation heuman_lambda]
 179
 180 Both of these functions are easily implemented in terms of Carlson's integrals, and are
 181 provided in this library as __jacobi_zeta and __heuman_lambda.
 182
 183 [h4 Numerical Algorithms]
 184
 185 The conventional methods for computing elliptic integrals are Gauss
 186 and Landen transformations, which converge quadratically and work
 187 well for elliptic integrals of the first and second kinds.
 188 Unfortunately they suffer from loss of significant digits for the
 189 third kind. Carlson's algorithm [[link ellint_ref_carlson79  Carlson79]] [[link ellint_ref_carlson78  Carlson78]], by contrast,
 190 provides a unified method for all three kinds of elliptic integrals
 191 with satisfactory precisions.
 192
 193 [h4 References]
 194
 195 Special mention goes to:
 196
 197 [:A. M. Legendre, ['Traitd des Fonctions Elliptiques et des Integrales
 198 Euleriennes], Vol. 1. Paris (1825).]
 199
 200 However the main references are:
 201
 202 # [#ellint_ref_AS]M. Abramowitz and I. A. Stegun (Eds.) (1964)
 203 Handbook of Mathematical Functions with Formulas, Graphs, and
 204 Mathematical Tables,
 205 National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C.
 206 # [#ellint_ref_carlson79]B.C. Carlson, ['Computing elliptic integrals by duplication],
 207     Numerische Mathematik, vol 33, 1 (1979).
 208 # [#ellint_ref_carlson77]B.C. Carlson, ['Elliptic Integrals of the First Kind],
 209     SIAM Journal on Mathematical Analysis, vol 8, 231 (1977).
 210 # [#ellint_ref_carlson78]B.C. Carlson, ['Short Proofs of Three Theorems on Elliptic Integrals],
 211     SIAM Journal on Mathematical Analysis, vol 9, 524 (1978).
 212 # [#ellint_ref_carlson81]B.C. Carlson and E.M. Notis, ['ALGORITHM 577: Algorithms for Incomplete
 213     Elliptic Integrals], ACM Transactions on Mathematmal Software,
 214     vol 7, 398 (1981).
 215 # B. C. Carlson, ['On computing elliptic integrals and functions]. J. Math. and
 216 Phys., 44 (1965), pp. 36-51.
 217 # B. C. Carlson, ['A table of elliptic integrals of the second kind]. Math. Comp., 49
 218 (1987), pp. 595-606. (Supplement, ibid., pp. S13-S17.)
 219 # B. C. Carlson, ['A table of elliptic integrals of the third kind]. Math. Comp., 51 (1988),
 220 pp. 267-280. (Supplement, ibid., pp. S1-S5.)
 221 # B. C. Carlson, ['A table of elliptic integrals: cubic cases]. Math. Comp., 53 (1989), pp.
 222 327-333.
 223 # B. C. Carlson, ['A table of elliptic integrals: one quadratic factor]. Math. Comp., 56 (1991),
 224 pp. 267-280.
 225 # B. C. Carlson, ['A table of elliptic integrals: two quadratic factors]. Math. Comp., 59
 226 (1992), pp. 165-180.
 227 # B. C. Carlson, ['[@http://arxiv.org/abs/math.CA/9409227
 228 Numerical computation of real or complex elliptic integrals]]. Numerical Algorithms,
 229 Volume 10, Number 1 / March, 1995, p13-26.
 230 # B. C. Carlson and John L. Gustafson, ['[@http://arxiv.org/abs/math.CA/9310223
 231 Asymptotic Approximations for Symmetric Elliptic Integrals]],
 232 SIAM Journal on Mathematical Analysis, Volume 25, Issue 2 (March 1994), 288-303.
 233
 234
 235 The following references, while not directly relevent to our implementation,
 236 may also be of interest:
 237
 238 # R. Burlisch, ['Numerical Compuation of Elliptic Integrals and Elliptic Functions.]
 239 Numerical Mathematik 7, 78-90.
 240 # R. Burlisch, ['An extension of the Bartky Transformation to Incomplete
 241 Elliptic Integrals of the Third Kind]. Numerical Mathematik 13, 266-284.
 242 # R. Burlisch, ['Numerical Compuation of Elliptic Integrals and Elliptic Functions. III].
 243 Numerical Mathematik 13, 305-315.
 244 # T. Fukushima and H. Ishizaki, ['[@http://adsabs.harvard.edu/abs/1994CeMDA..59..237F
 245 Numerical Computation of Incomplete Elliptic Integrals of a General Form.]]
 246 Celestial Mechanics and Dynamical Astronomy, Volume 59, Number 3 / July, 1994,
 247 237-251.
 248
 249 [endsect]