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

Abstract

This is a continuation of our works to compute the incomplete elliptic integrals of the first and second kind (Fukushima (2010, 2011) [21,23]). We developed a method to compute an associate incomplete elliptic integral of the third kind, J ( φ , n | m ) ≡ [ Π ( φ , n | m ) − F ( φ | m ) ] / n , by the half argument formulas of the sine and cosine amplitude functions and the double argument transformation of the integral. The relative errors of J ( φ , n | m ) computed by the new method are sufficiently small as less than 20 machine epsilons. Meanwhile, the simplicity of the adopted algorithm makes the new method run 1.5 to 3.7 times faster than Carlson’s duplication method. The combination of the new method and that to compute simultaneously two other associate incomplete elliptic integrals of the second kind, B ( φ | m ) ≡ [ E ( φ | m ) − ( 1 − m ) F ( φ | m ) ] / m and D ( φ | m ) ≡ [ F ( φ | m ) − E ( φ | m ) ] / m , which we established recently [23], enables a precise and fast computation of arbitrary linear combination of Legendre’s incomplete elliptic integrals of all three kinds, F ( φ | m ) , E ( φ | m ) , and Π ( φ , n | m ) . These new procedures share the same device, the half argument transformations, while the double argument transformation of J ( φ , n | m ) includes those of B ( φ | m ) and D ( φ | m ) as its sub component. As a result, the simultaneous computation of the three associate integrals is significantly faster than computing them separately. In fact, our combined procedure is 2.7 to 5.9 times faster than the combination of Carlson’s duplication method to compute R D and R J .