Algorithm 577: Algorithms for Incomplete Elliptic Integrals [S21]
ACM Transactions on Mathematical Software (TOMS)
The Mathematica Book
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Numerical Methods for Special Functions
Numerical Methods for Special Functions
Numerical Recipes 3rd Edition: The Art of Scientific Computing
Numerical Recipes 3rd Edition: The Art of Scientific Computing
NIST Handbook of Mathematical Functions
NIST Handbook of Mathematical Functions
IEEE Transactions on Signal Processing
Precise and fast computation of Lambert W-functions without transcendental function evaluations
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
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We developed the numerical procedures to evaluate the inverse functions of the complete elliptic integrals of the first and the 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).