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

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:253

Article

Fukushima, Toshio

关键词: Complete elliptic integral;    Double argument transformation;    Half argument transformation;   

DOI  :  10.1016/j.cam.2013.04.015

来源: Elsevier

【 摘 要 】

We developed a novel method to calculate an associate complete elliptic integral of the third kind, J(n vertical bar m) equivalent to [Pi(n vertical bar m) - K(m)]/n. The key idea is the double argument formula of J(n vertical bar 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, Pi(1)(a vertical bar m), with respect to a, which is a real or pure imaginary argument connected with n and m as n = m sn(2)(a vertical bar m). Repeatedly using the half argument transformation (Fukushima 2010) [28] of a new variable, y equivalent to n/m, or its complement, x equivalent to (m - n)/m, we reduce vertical bar y vertical bar 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) equivalent to [E(m) - (1 - m)K(m)]/m and D(m) equivalent to [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 vertical bar 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 vertical bar m)), the new method runs 1.2-1.5 and 4.7-5.5 times faster than Bulirsch's ce1 and Carlson's R-J, 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 ce1 and Carlson's R-D and R-J, respectively. (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

Free