Summary
Logarithms, arctangents, and elliptic integrals of all three kinds (including complete integrals) are evaluated numerically by successive applications of the duplication theorem. When the convergence is improved by including a fixed number of terms of Taylor's series, the error ultimately decreases by a factor of 4096 in each cycle of iteration. Except for Cauchy principal values there is no separation of cases according to the values of the variables, and no serious cancellations occur if the variables are real and nonnegative. Only rational operations and square roots are required. An appendix contains a recurrence relation and two new representations (in terms of elementary symmetric functions and power sums) forR-polynomials, as well as an upper bound for the error made in truncating the Taylor series of anR-function.
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References
Brent, R.P.: Fast multiple-precision evaluation of elementary functions. J. Assoc. Comput. Mach.23, 242–251 (1976)
Bulirsch, R.: Numerical calculation of elliptic integrals and elliptic functions. Numer. Math.7, 78–90, 353–354 (1965);13, 305–315 (1969)
Carlson, B.C.: On computing elliptic integrals and functions. J. Math. and Phys.44, 36–51 (1965)
Carlson, B.C.: Hidden symmetries of special functions. SIAM Rev.12, 332–345 (1970)
Carlson, B.C.: An algorithm for computing logarithms and arctangents. Math. Comp.26, 543–549 (1972)
Carlson, B.C.: Special Functions of Applied Mathematics. New York: Academic Press 1977
Cazenave, R.: Intégrales et fonctions elliptiques en vue des applications. Centre de Documentation de l'Armement, Paris, 1969
Fettis, H.E., Caslin, J.C.: Tables of elliptic integrals of the first, second and third kind. Report ARL 64-232, Wright-Patterson Air Force Base, Ohio, 1964. Errata in Math. Comp.20, 639–640 (1966)
Franke, C.H.: Numerical evaluation of the elliptic integral of the third kind. Math. Comp.19, 494–496 (1965)
Gautschi, W.: Computational methods in special functions — a survey, (R. Askey, ed.) Theory and Application of Special Functions. New York: Academic Press 1975
Heuman, C.: Tables of complete elliptic integrals. J. Math. and Phys.20, 127–206 (1941)
Midy, P.: An improved calculation of the general elliptic integral of the second kind in the neighbourhood ofx=0. Numer. Math.25, 99–101 (1975)
Milne-Thomson, L.M.: The Calculus of Finite Differences. London: Macmillan 1933
Todd, J.: The lemniscate constants. Commun. ACM18, 14–19, 462 (1969)
Van de Vel, H.: On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp.23, 61–69 (1969)
Zill, D.G., Carlson, B.C.: Symmetric elliptic integrals of the third kind. Math. Comp.24, 199–214 (1970)
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Carlson, B.C. Computing elliptic integrals by duplication. Numer. Math. 33, 1–16 (1979). https://doi.org/10.1007/BF01396491
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DOI: https://doi.org/10.1007/BF01396491