Keywords: nonlinear system, Newton's method, quadratic convergence, efficiency index.

In this paper we consider the problem of finding a real solution of a nonlinear system F(x)=0 with n equations and n unknowns. The most used iterative method is the classical Newton's method. It is well known that this method requires the Jacobian matrix of F to be nonsingular in a neighborhood of the solution in order to obtain quadratic convergence.

A known acceleration technique consists of the composition of two iterative methods of order p and q, respectively, to obtain a method of order pq [1]. Usually, new evaluations of the Jacobian matrix and the nonlinear functions are required in order to increase the order of convergence. However, some modifications of Newton's method can be made in order to limit the number of functional evaluations and increase the convergence order.

In this work we present two new iterative methods with a high convergence order for solving nonlinear systems. These new methods are obtained by composing known iterative methods of order 3 and 4 with a modification of Newton's method that introduces just one evaluation of the function, increasing the convergence order in two units.

Firstly, we consider the third order method introduced by Frontini and Sormani [2], and combine it with a modified Newton's method obtaining a three-step method with fifth order convergence.

This idea is also applied to the fourth order method introduced by Cordero et al. [3] in order to obtain a sixth-order method when we combine it with the modified Newton's method.

In order to compare different methods, it is very common to use the efficiency index [4] which involves the order of convergence and the number of functional evaluations per iteration required by the method. However, in the n-dimensional case, it is also important to take into account the number of arithmetic operations performed, since for each iteration a number of linear systems must be solved. For this reason we define the computational efficiency index, which also takes into account the number of products and quotients per iteration. We also use this new index to compare the different methods.

The efficiency index and the computational efficiency index of the new methods are checked. We also perform different numerical tests that confirm the theoretical results and allow us to compare these new methods with the ones from which they have been derived and with the classical Newton's method.

1

J.F. Traub, "Iterative methods for the solution of equations", Chelsea Publishing Company, New York, 1982.

2

M. Frontini, E. Sormani, "Third-order methods from quadrature formulae for solving systems of nonlinear equations", Applied Mathematics and Computation, 149, 771-782, 2004. doi:10.1016/S0096-3003(03)00178-4

3

A. Cordero, E. Martínez, J. R. Torregrosa "Iterative methods of order four and five for systems of nonlinear equations", Journal of Computational and Applied Mathematics, 231, 541-551, 2009. doi:10.1016/j.cam.2009.04.015

4

A.M. Ostrowski, "Solutions of equations and systems of equations", Academic Press, New York-London, 1966.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £125 +P&P)