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 ) ≡ ∫ 0 T 1 - m sin 2 ( T ′ ) dT ′ where 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 = 0 . We show that Newton’s iteration, T n + 1 = T n - E ( sin ( T ) ; m ) - z 1 - m sin 2 ( T ) , always converges to 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 T 0 ( z , m ) = π / 2 + r ( θ - π / 2 ) where, defining ζ ≡ 1 - z / E ( 1 ; m ) , r = ( 1 - m ) 2 + ζ 2 and θ = atan ( ( 1 - m ) / ζ ) . We briefly discuss three alternative initialization strategies: “homotopy” initialization [ T 0 ( 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”.