Modified Newton’s Methods for Systems of Nonlinear Equations

Abstract

The main goal of this chapter is to obtain new iterative formulas in order to solve systems of nonlinear equations. They are proved to be modifications on classical Newton’s method which accelerate the convergence of the iterative process.

In previous works, the authors have obtained variants on Newton’s method based on quadrature formulas whose truncation error was up to O(h ⁵) (see [CoTo06] and [CoTo07]). Nevertheless, the approach used in this paper to solve a nonlinear system is different: by using Adomian polynomials, we obtain a family of multipoint iterative formulas, which include Newton and Traub (see [Tr82]) methods in the simplest cases.

The decomposition method using Adomian polynomials is used to solve different problems in applied mathematics in [Ad88]. Indeed, Babolian et al. (see [BaBiVa04]) apply this general method to a concrete nonlinear system. Nevertheless, with a different system, it is necessary to reconstruct the entire process.