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

作者：Boyd John P^*

来源：Applied Mathematics and Computation, 2012, 218(13): 7005-7013.

DOI：10.1016/j.amc.2011.12.021

• Elliptic integral
• Inverse elliptic integral
• Chebyshev polynomial approximation
• Homotopy continuation

The incomplete elliptic integral of the second kind, epsilon(sin(T), m) equivalent to integral(T)(0) root 1-m sin(2)(T') dT' where m is an element of vertical bar 0, 1 vertical bar is the elliptic modulus, can be inverted with respect to angle T by solving the transcendental equation epsilon(sin(T), m) - z = 0. We show that Newton's iteration, Tn+1 - T-n - {epsilon(sin(T), m) - z}/{root 1 - m sin(2)(T)}, always converges to T(z;m) - epsilon(-1)(z; m) within a relative error of less than 10(-10) in three iterations or less from the first guess T-0(z, m) = pi/2 root r(theta - pi/2) where, defining zeta equivalent to 1 - z/epsilon(1;m); r = root(1 - m)(2) + zeta(2) and theta = atan((1 - m)/zeta). We briefly discuss three alternative initialization strategies: "homotopy" initialization [T-0(z, m) equivalent to (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".

• 出版日期2012-3-1