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Elliptic integrals of the first and second kind – comparison of Bulirsch’s and Carlson’s algorithms for numerical calculation. (English) Zbl 1097.33512
Dunkl, Charles (ed.) et al., Special functions. Proceedings of the international workshop on special functions – asymptotics, harmonic analysis and mathematical physics, Hong Kong, China, June 21–25, 1999. Singapore: World Scientific (ISBN 981-02-4393-6/hbk). 293-308 (2000).

Summary: Numerical calculations of elliptic integrals of the first and second kinds are usually done using algorithms of R. Bulirsch [Numer. Math. 7, 353–354 (1965; Zbl 0128.37204)] or B. C. Carlson [Numer. Math. 33, No. 1, 1–16 (1979; Zbl 0438.65029)]. These algorithms are based on the descending Landen transformation and the duplication theorem respectively. The algorithms are compared as to the computing time and keeping the prescribed tolerance. The comparison is done in single precision with a given tolerance and leads to a rating concerning the range of the arguments. Bulirsch’s algorithm for calculating the elliptic integral of the second kind,

0 x 1+k c 2 ξ 2 (1+ξ 2 )(1+ξ 2 )(1+k c 2 ξ 2 )dξ,

produces cancellation errors in the range {(x,k c );0<|x|<1,|k c |>1}. A numerically stable alternative in this range is presented. The duplication theorem is interpreted both in the sense of Euler’s addition formula for the elliptic integral of the first kind or the Jacobian elliptic function sinus amplitudinis and in the sense of Weierstrass.

33F10Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
33E05Elliptic functions and integrals
65D30Numerical integration