A convergence analysis of Newton's method under the gamma-condition in Banach spaces

Ioannis K. Argyros¹

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

Applicationes Mathematicae, Tome 36 (2009) no. 2, pp. 225-239

Résumé

We provide a local as well as a semilocal convergence analysis for Newton's method to approximate a locally unique solution of an equation in a Banach space setting. Using a combination of center-gamma with a gamma-condition, we obtain an upper bound on the inverses of the operators involved which can be more precise than those given in the elegant works by Smale, Wang, and Zhao and Wang. This observation leads (under the same or less computational cost) to a convergence analysis with the following advantages: local case: larger radius of convergence and finer error estimates on the distances involved; semilocal case: larger domain of convergence, finer error bounds on the distances involved, and at least as precise information on the location of the solution.

Détail

DOI : 10.4064/am36-2-9

Keywords: provide local semilocal convergence analysis newtons method approximate locally unique solution equation banach space setting using combination center gamma gamma condition obtain upper bound inverses operators involved which precise those given elegant works smale wang zhao wang observation leads under computational cost convergence analysis following advantages local larger radius convergence finer error estimates distances involved semilocal larger domain convergence finer error bounds distances involved least precise information location solution