On the gap between the semilocal convergence domains of two Newton methods

Ioannis K. Argyros¹

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

Applicationes Mathematicae, Tome 34 (2007) no. 2, pp. 193-204

Résumé

We answer a question posed by Cianciaruso and De Pascale: What is the exact size of the gap between the semilocal convergence domains of the Newton and the modified Newton method? In particular, is it possible to close it? Our answer is yes in some cases. Using some ideas of ours and more precise error estimates we provide a semilocal convergence analysis for both methods with the following advantages over earlier approaches: weaker hypotheses; finer error bounds on the distances involved, and at least as precise information on the location of the solution; and a smaller gap between the two methods.

Détail

DOI : 10.4064/am34-2-4

Keywords: answer question posed cianciaruso pascale what exact size gap between semilocal convergence domains newton modified newton method particular possible close answer yes cases using ideas ours precise error estimates provide semilocal convergence analysis methods following advantages earlier approaches weaker hypotheses finer error bounds distances involved least precise information location solution smaller gap between methods