Approximating solutions of equations using Newton's method with a modified Newton's method iterate as a starting point

Abstract

In this study we are concerned with the problem of approximating a locally unique solution of an equation in a Banach space setting using Newton's and modified Newton's methods. We provide weaker convergence conditions for both methods than before [6]-[8]. Then, we combine Newton's with the modified Newton's method to approximate locally unique solutions of operator equations in a Banach space setting. Finer error estimates, a larger convergence domain, and a more precise information on the location of the solution are obtained under the same or weaker hypotheses than before [6]-[8]. Numerical examples are also provided.

Keywords

• Banach space
• Newton-Kantorovich method
• radius of convergence
• Fréchet-derivative
• Banach lemma on invertible operators