Numerical, perturbative and Chebyshev inversion of the incomplete elliptic integral of the second kind

The incomplete elliptic integral of the second kind, E(sin(T),m)≡∫0T1-msin2(T′)dT′ where m∈[0,1]m∈[0,1] is the elliptic modulus, can be inverted with respect to angle T   by solving the transcendental equation E(sin(T);m)-z=0E(sin(T);m)-z=0. We show that Newton’s iteration, Tn+1=Tn-E(sin(T);m)-z1-msin2(T), always converges to T(z;m)=E-1(z;m)T(z;m)=E-1(z;m) within a relative error of less than 10−10 in three iterations or less from the first guess T0(z,m)=π/2+r(θ-π/2) where, defining ζ≡1-z/E(1;m),r=(1-m)2+ζ2 and θ=atan((1-m)/ζ)θ=atan((1-m)/ζ). We briefly discuss three alternative initialization strategies: “homotopy” initialization [T0(z,m)≡(1-m)(z;0)+mT(z,m)(m;1)T0(z,m)≡(1-m)(z;0)+mT(z,m)(m;1)], perturbation series (in powers of m), and inversion of the Chebyshev interpolant of the incomplete elliptic integral. Although all work well, and are general strategies applicable to a very wide range of problems, none of these three alternatives is as efficient as the empirical initialization, which is completely problem-specific. This illustrates Tai Tsun Wu’s maxim “usefulness is often inversely proportional to generality”.