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 second kind, K(m) and E(m), with respect to the parameter m. The evaluation is executed by inverting eight sets of the truncated Taylor series expansions of the integrals in terms of m or of − 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).