Keywords: nonlinear systems, Newton's method, iterative methods, efficiency index, computational efficiency, semilocal convergence, order of convergence, mildly nonlinear elliptic boundary value problem, molecular interaction equation.

The application of Newton's method (order of convergence two) is very widespread for solving nonlinear systems of equations. We propose the application of a new family of iterative methods with order of convergence four for solving such systems, which is constructed from the modification of a third-order iterative method of Chebyshev-type [1]. The proposal is based on the fact that the new iterative methods are more efficient than Newton's method. We can deduce the last by measuring the efficiency of the methods from two well-known indices [2]: the efficiency index and the computational efficiency, which are defined from the order of convergence of the iterative methods, the number of evaluations of functions that are carried out when the iterative methods are applied and the computational cost required to apply a step of the iterative methods. In particular, we construct iterative methods with a higher order of convergence than Newton's method, similar number of computations of functions and lower computational cost per iteration.

We study the semilocal convergence of the new iterative methods under the usual Kantorovich convergence conditions considered for studying Newton's method.

All the above-mentioned is illustrated by the solution of mildly nonlinear elliptic boundary value problems. The solution of a particular problem (the molecular interaction equation [3] in a square) is approximated by the most competitive iterative method of the family. To do this, we first follow a process of discretization using finite difference approximations, which yields a finite system of equations.

1

J.A. Ezquerro, J.M. Gutiérrez, M.A. Hernández, M.A. Salanova, "Chebyshev-like methods and quadratic equations", Rev. Anal. Numér. Théor. Approx., 28(1), 23-35, 1999.

2

J.F. Traub, "Iterative Methods for the Solution of Equations", Prentice Hall, Englewood Cliffs, N.J., 1964.

3

S.T. Pohozaev, "The Dirichlet problem for the equation Delta u=u²", Soviet Math. Dokl., 1, 1143-1146, 1960.

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