Alicia Cordero, Juan R. Torregrosa and María P. Vassileva: Weighted Gaussian correction of Newton-type methods for solving nonlinear systems, p.23-38


A new technique to design predictor-corrector methods for solving nonlinear 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 independence of both the number of nodes used in the quadrature and the orthogonal 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 confirm the theoretical results and show that the proposed methods need less computational time than well-known procedures, such as Newton' and Jarratt's schemes.

Key Words: Nonlinear system of equations, Gaussian quadrature, Pseudocomposition, Weight function procedure, Multipoint method, Optimal order, Efficiency.

2000 Mathematics Subject Classification: Primary: 65H10;
Secondary: 65H05, 65D30.