Convergence for variants of Chebyshev–Halley methods using restricted convergence domains

Ioannis K. Argyros¹ ; Santhosh George²

¹ Department of Mathematical Sciences Cameron University Lawton, OK 73505, U.S.A.
² Department of Mathematical and Computational Sciences NIT Karnataka Karnataka, India 575 025

Applicationes Mathematicae, Tome 46 (2019) no. 1, pp. 115-126

Résumé

We present a local convergence analysis for some variants of Chebyshev–Halley methods of approximating a locally unique solution of a nonlinear equation in a Banach space setting. We only use hypotheses reaching up to the second Fréchet derivative of the operator involved in contrast to earlier studies using Lipschitz hypotheses on the second Fréchet derivative and other more restrictive conditions. This way the applicability of these methods is expanded. We also show how to improve the semilocal convergence in the earlier studies under the same conditions using our new idea of restricted convergence domains leading to: weaker sufficient convergence criteria, tighter error bounds on the distances involved and an at least as precise information on the location of the solution. Numerical examples where earlier results cannot be applied but our results can, are also provided.

Détail

DOI : 10.4064/am2321-4-2017

Keywords: present local convergence analysis variants chebyshev halley methods approximating locally unique solution nonlinear equation banach space setting only hypotheses reaching second chet derivative operator involved contrast earlier studies using lipschitz hypotheses second chet derivative other restrictive conditions applicability these methods expanded improve semilocal convergence earlier studies under conditions using idea restricted convergence domains leading weaker sufficient convergence criteria tighter error bounds distances involved least precise information location solution numerical examples where earlier results cannot applied results provided