An analysis of the convergence of Newton iterations for solving elliptic Kepler’s equation

Abstract

In this note a study of the convergence properties of some starters E0= E0(e, M) in the eccentricity–mean anomaly variables for solving the elliptic Kepler’s equation (KE) by Newton’s method is presented. By using a Wang Xinghua’s theorem (Xinghua in Math Comput 68(225):169–186, 1999) on best possible error bounds in the solution of nonlinear equations by Newton’s method, we obtain for each starter E0(e, M) a set of values (e, M) ∈ [0 , 1) × [0 , π] that lead to the q-convergence in the sense that Newton’s sequence (En)n≥0 generated from E0= E0(e, M) is well defined, converges to the exact solution E∗= E∗(e, M) of KE and further |En-E∗|≤q2n-1|E0-E∗| holds for all n≥ 0. This study completes in some sense the results derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014) by using Smale’s α-test with q= 1 / 2. Also since in KE the convergence rate of Newton’s method tends to zero as e→ 0 , we show that the error estimates given in the Wang Xinghua’s theorem for KE can also be used to determine sets of q-convergence with q=ekq~ for all e∈ [0 , 1) and a fixed q~ ≤ 1. Some remarks on the use of this theorem to derive a priori estimates of the error | En- E∗| after n Kepler’s iterations are given. Finally, a posteriori bounds of this error that can be used to a dynamical estimation of the error are also obtained.