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

The incomplete elliptic integral of the second kind, EðsinðTÞ;mÞ R T 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m sinðT Þ q dT 0 where m 2 1⁄20;1 is the elliptic modulus, can be inverted with respect to angle T by solving the transcendental equation EðsinðTÞ;mÞ z 1⁄4 0. We show that Newton’s iteration, T 1⁄4 T EðsinðTÞ;mÞ z f g ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m sinðTÞ q , always converges to Tðz;mÞ 1⁄4 E ðz;mÞ within a relative error of less than 10 10 in three iterations or less from the first guess Tðz;mÞ 1⁄4 p=2þ ffiffi r p ðh p=2Þ where, defining f 1 z=Eð1;mÞ; r 1⁄4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 mÞ þ f q and h 1⁄4 atanðð1 mÞ=fÞ. We briefly discuss three alternative initialization strategies: ‘‘homotopy’’ initialization [Tðz;mÞ ð1 mÞðz;0Þ þmTðz;mÞðm;1Þ], 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’’. 2011 Elsevier Inc. All rights reserved.