Numerical computation of inverse complete elliptic integrals of first and second kinds

We developed the numerical procedures to evaluate the inverse functions of the complete elliptic integrals of the first and the second kind, K(m)K(m) and E(m)E(m), with respect to the parameter mm. The evaluation is executed by inverting eight sets of the truncated Taylor series expansions of the integrals in terms of mm or of −log(1−m)−log(1−m). The developed procedures are (1) so precise that the maximum absolute errors are less than 3–5 machine epsilons, and (2) 30%–40% faster than the evaluation of the integrals themselves by the fastest procedures (Fukushima 2009a, 2011).

► We developed the numerical procedures to compute the inverse of K(m)K(m) and E(m)E(m).
► The procedures consist of eight sets of the inverted Taylor series expansions.
► The inversion is conducted with respect to mm or −log(1−m)−log(1−m).
► The maximum absolute errors of the procedures are less than 3–5 machine epsilons.
► The procedures run 30%–40% faster than those to evaluate the integrals themselves.