Weighted Gaussian correction of Newton-type methods for solving nonlinear systems

Abstract

A new technique to design predictor-corrector methods for solving non- linear equations or nonlinear systems is presented. With Newton's scheme as a predictor and any Gaussian quadrature as a corrector we construct, by using weight function procedure, iterative schemes of order four, with inde- pendence of both the number of nodes used in the quadrature and the or- thogonal polynomials employed. These methods are obtained by assuming some conditions on the weight function related to the weights and nodes of the corresponding Gaussian quadrature. These methods are optimal, in the sense of Kung-Traub conjecture, in one-dimensional case. Some numerical tests allow us to con rm the theoretical results and show that the proposed methods need less computational time than well-known procedures, such as Newton' and Jarratt's schemes.