Precise and fast computation of a general incomplete elliptic integral of second kind by half and double argument transformations

We developed a method to compute simultaneously two associate incomplete elliptic integrals of the second kind, B ( φ | m ) and D ( φ | m ) , by the half argument formulas of Jacobian elliptic functions and the double argument transformations of the integrals. The relative errors of the new method are sufficiently small as 5–10 machine epsilons. Meanwhile, the new method runs 3–6 times faster than that using Carlson’s R D . As a result, it enables a precise and fast computation of arbitrary linear combination of the incomplete elliptic integrals of the first and the second kind, F ( φ | m ) and E ( φ | m ) .