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

• We developed a new method to compute the associate complete elliptic integral of the third kind, J(n|m)J(n|m).
• The method utilizes the double argument formula of J(n|m)J(n|m) with respect to nn.
• The errors of the new method is sufficiently small as around 10 machine epsilons.
• The new method is 20%–50% faster than Bulirsch’s cel.
• The new method is around 5 times faster than Carlson’s RJRJ.

We developed a novel method to calculate an associate complete elliptic integral of the third kind, J(n|m)≡[Π(n|m)−K(m)]/nJ(n|m)≡[Π(n|m)−K(m)]/n. The key idea is the double argument formula of J(n|m)J(n|m) with respect to nn. We derived it from the not-so-popular addition theorem of Jacobi’s complete elliptic integral of the third kind, Π1(a|m)Π1(a|m), with respect to aa, which is a real or pure imaginary argument connected with nn and mm as n=msn2(a|m). Repeatedly using the half argument transformation (Fukushima 2010) [28] of a new variable, y≡n/my≡n/m, or its complement, x≡(m−n)/mx≡(m−n)/m, we reduce |y||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)]/mB(m)≡[E(m)−(1−m)K(m)]/m and D(m)≡[K(m)−E(m)]/mD(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 nn by successively applying the double argument formula of J(n|m)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)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 RJRJ, 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 RDRD and RJRJ, respectively.