A Study on Convergence of Newton's Method in Real and Interval Number

Abstract

In order to find the approximate numerical solution to a system of nonlinear equations as well as an integral and a differential operator equations, Newton's algorithm is widely used. L. V. Kantorovich [1948] and Moore [1977] studied the existence and uniqueness of solution to the system of nonlinear equations and their error bounds. M. Urabe [ 1965] also studied the existence and uniqueness of the solution to nonlinear operator equations (mainly differential operator equations). Kantorovich and Urabe's methods are two variants of Newton's method in some sense. We study the existence and uniqueness of solutions to the nonlinear systems and their error bounds. Our results will be stated in a theorem that ensures the best possible generalized error bound that is different from that given by Kantorovich and Moore. We also develop a technique that may be applied to find an approximate numerical solution to an algebraic as well as to a system of nonlinear equations both in real and interval number systems. Finally, we have treated the error estimation for the quasiperiodic solution to the Van der pol type differential operator equation based on Urabe's theorem.