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In this paper, from Traub's method and by applying weight function technique, a bi-parametric family of predictor-corrector iterative schemes with optimal fourth-order of convergence, for solving nonlinear equations, is presented. By using some algebraic manipulations and a divided difference operator, we extend this family to the multidimensional case, preserving its order of convergence. Some numerical test are made in order to confirm the theoretical results and to compare the new methods with other known ones.

In this paper, a three-step iterative method with sixth-order local convergence for approximating the solution of a nonlinear system is presented. From Ostrowski's scheme adding one step of Newton with 'frozen' derivative and by using a divided difference operator we construct an iterative scheme of order six for solving nonlinear systems. The computational efficiency of the new method is compared with some known ones, obtaining good conclusions. Numerical comparisons are made with other existing methods, on standard nonlinear systems and the classical 1D-Bratu problem by transforming it in a nonlinear system by using finite differences. From this numerical examples, we confirm the theoretical results and show the performance of the presented scheme.

A secant-type method is designed for approximating the inverse and some generalized inverses of a complex matrix A. For a nonsingular matrix, the proposed method gives us an approximation of the inverse and, when the matrix is singular, an approximation of the Moore–Penrose inverse and Drazin inverse are obtained. The convergence and the order of convergence is presented in each case. Some numerical tests allowed us to confirm the theoretical results and to compare the performance of our method with other known ones. With these results, the iterative methods with memory appear for the first time for estimating the solution of a nonlinear matrix equations.

In this paper we design, by using the weight function technique, two families of iterative schemes with order of convergence eight. These weight functions depend on one, two and three variables and they are used in the second and third step of the iterative expression. Dynamics on polynomial and non-polynomial functions is analyzed and they are applied on the problem of preliminary orbit determination by using a modified Gauss method. Finally, some standard test functions are to check the reliability of the proposed schemes and allow us to compare them with other known methods.