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

Tom 34 / 2007

Ioannis K. Argyros Applicationes Mathematicae 34 (2007), 193-204 MSC: 65H10, 65G99, 47H17, 49M15. DOI: 10.4064/am34-2-4


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.


  • Ioannis K. ArgyrosDepartment of Mathematical Sciences
    Cameron University
    Lawton, OK 73505, U.S.A.

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