A new approach for finding weaker conditions for the convergence of Newton's method

Ioannis K. Argyros¹

¹ Department of Mathematical Sciences Cameron University Lawton, OK 73505, U.S.A.

Applicationes Mathematicae, Tome 32 (2005) no. 4, pp. 465-475

Résumé

The Newton–Kantorovich hypothesis (15) has been used for a long time as a sufficient condition for convergence of Newton's method to a locally unique solution of a nonlinear equation in a Banach space setting. Recently in [3], [4] we showed that this hypothesis can always be replaced by a condition weaker in general (see (18), (19) or (20)) whose verification requires the same computational cost. Moreover, finer error bounds and at least as precise information on the location of the solution can be obtained this way. Here we show that we can further weaken conditions (18)–(20) and still improve on the error bounds given in [3], [4] (see Remark 1(c)).

Détail

DOI : 10.4064/am32-4-7

Keywords: newton kantorovich hypothesis has long time sufficient condition convergence newtons method locally unique solution nonlinear equation banach space setting recently showed hypothesis always replaced condition weaker general see whose verification requires computational cost moreover finer error bounds least precise information location solution obtained here further weaken conditions still improve error bounds given see remark