Precise and fast computation of the general complete elliptic integral of the second kind

Abstract

We developed an efficient procedure to evaluate two auxiliary complete elliptic integrals of the second kind B(m) and D(m) by using their Taylor series expansions, the definition of Jacobi’s nome, and Legendre’s relation. The developed procedure is more precise than the existing ones in the sense that the maximum relative errors are 1-3 machine epsilons, and it runs drastically faster; around 5 times faster than Bulirsch’s cel2 and 16 times faster than Carlson’s RF and RD.