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.