Fast computation of a general complete elliptic integral of third kind by half and double argument transformations

Abstract

We developed a novel method to calculate an associate complete elliptic integral of the third kind, J(n|m)≡[Π(n|m)−K(m)]/n. The key idea is the double argument formula of J(n|m) with respect to n. We derived it from the not-so-popular addition theorem of Jacobi’s complete elliptic integral of the third kind, Π1(a|m), with respect to a, which is a real or pure imaginary argument connected with n and m as n=msn2(a|m). Repeatedly using the half argument transformation (Fukushima 2010) [28] of a new variable, y≡n/m, or its complement, x≡(m−n)/m, we reduce |y| sufficiently small, say less than 0.3 or so. Then, we evaluate the integral for the reduced variable by its Maclaurin series expansion. The coefficients of the series expansion are recursively computed from two other associate complete elliptic integrals, B(m)≡[E(m)−(1−m)K(m)]/m and D(m)≡[K(m)−E(m)]/m. The precise and fast computation of these two integrals is found in our previous work (Fukushima 2011) [17]. Finally, we recover the integral value for the original n by successively applying the double argument formula of J(n|m). The new method is sufficiently precise in the sense that the maximum errors are less than around 10 machine epsilons. For the sole computation of J(n|m), the new method runs 1.2–1.5 and 4.7–5.5 times faster than Bulirsch’s cel and Carlson’s RJ, respectively. In the simultaneous computation of three associate complete integrals, the new method runs 1.6–1.7 and 5.3–8.0 times faster than cel and Carlson’s RD and RJ, respectively.