Four Ways to Compute the Inverse of the Complete Elliptic Integral of the First Kind John P. Boyd Department of Atmospheric, Oceanic & Space Science University of Michigan, 2455 Hayward Avenue, Ann Arbor MI 48109 jpboyd@umich.edu Abstract The complete elliptic integral of the first kind arises in many applications. This article furnishes four different ways to compute the inverse of the elliptic inte- gral. One motive for this study is simply that the author needed to compute the inverse integral for an application. Another is to develop a case study com- paring different options for solving transcendental equations like those in the author’s book [11]. A third is to develop analytical approximations, more use- ful to theorists than mere numbers. A fourth is to provide robust “black box” software for computing this function. The first solution strategy is “polynomial- ization” which replaces the elliptic integral by an exponentially convergent series of Chebyshev polynomials. The transcendental equation becomes a polynomial equation which is easily solved by finding the eigenvalues of the Chebyshev companion matrix. (The numerically ill-conditioned step of converting from the Chebyshev to monomial basis is never necessary). The second approximation is a regular perturbation series, accurate where the modulus is small. The third is a power-and-exponential series that converges over the entire range param- eter range, albeit only sub-exponentially in the limit of zero modulus. Lastly, Newton’s iteration is promoted from a local iteration to a global method by a Never-Failing Newton’s Iteration (NFNI) in the form of the exponential of the ratio of a linear function divided by another linear polynomial. A short Matlab implementation is provided, easily translatable into other languages. The Mat- lab/Newton code is recommended for numerical purposes. The other methods are presented because (i) all are broadly applicable strategies useful for other rootfinding and inversion problems (ii) series and substitutions are often much more useful to theorists than numerical software and (iii) the Never-Failing Newton’s Iteration was discovered only after a great deal of messing about with power series, inverse power series and so on. Keywords: pseudospectral; Chebyshev polynomials ; elliptic integral; inverse elliptic integral Preprint submitted to Elsevier February 2, 2015 © 2015. This manuscript version is made available under the Elsevier user license http://www.elsevier.com/open-access/userlicense/1.0/