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

Elipe, A. (Universidad de Zaragoza) ; Montijano, J.I. (Universidad de Zaragoza) ; Rández, L. (Universidad de Zaragoza) ; Calvo, M. (Universidad de Zaragoza)

Resumen: In this note a study of the convergence properties of some starters (Formula presented.) 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 (Formula presented.) a set of values (Formula presented.) that lead to the q-convergence in the sense that Newton’s sequence (Formula presented.) generated from (Formula presented.) is well defined, converges to the exact solution (Formula presented.) of KE and further (Formula presented.) holds for all (Formula presented.). 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 (Formula presented.)-test with (Formula presented.). Also since in KE the convergence rate of Newton’s method tends to zero as (Formula presented.), 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 (Formula presented.) for all (Formula presented.) and a fixed (Formula presented.). Some remarks on the use of this theorem to derive a priori estimates of the error (Formula presented.) 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.

Idioma: Inglés

DOI: 10.1007/s10569-017-9785-5

Año: 2017

Publicado en: Celestial Mechanics and Dynamical Astronomy 129, 4 (2017), 415-432
ISSN: 0923-2958

Factor impacto JCR: 2.121 (2017)

Categ. JCR: MATHEMATICS, INTERDISCIPLINARY APPLICATIONS rank: 30 / 103 = 0.291 (2017) - Q2 - T1
Categ. JCR: ASTRONOMY & ASTROPHYSICS rank: 34 / 66 = 0.515 (2017) - Q3 - T2

Factor impacto SCIMAGO: 1.092 - Applied Mathematics (Q1) - Computational Mathematics (Q1) - Modeling and Simulation (Q1) - Mathematical Physics (Q1) - Space and Planetary Science (Q2) - Astronomy and Astrophysics (Q2)

Financiación: info:eu-repo/grantAgreement/ES/MICINN/MTM2013-47318-C2-1-P
Financiación: info:eu-repo/grantAgreement/ES/MINECO/2013-44217-R

Tipo y forma: Artículo (PostPrint)

Área (Departamento): Área Matemática Aplicada (Dpto. Matemática Aplicada)