On Convergence Domains of Newton's and Modified Newton Methods

ABSTRACT

We analyze convergence domains of Newton's and the modified Newton methods for solving operator equations in Banach spaces assuming first that the operator in question is ω-smooth in a ball centered at the starting point. It is shown that the gap between convergence domains of these two methods cannot be closed under ω-smoothness. Its exact size for Hölder smooth operators is computed. Then we proceed to investigate their convergence domains under regular smoothness. As our analysis reveals, both domains are the same and wider than their counterparts in the previous case.

Keywords:

• Modified Newton method
• Newton method
• Operator equations
• Regular smoothness
• ω-Smoothness

Mathematics Subject Classification:

• 65J15