Keywords: nonlinear system, multipoint iterative methods, Newton's method, order of convergence, efficiency index, computational efficiency.

In this paper we consider the problem of finding a real solution of a nonlinear system 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 the nonlinear function to be nonsingular in the neighborhood of the solution in order to obtain quadratic convergence. A known acceleration technique consists of the composition of two iterative methods, to obtain a method whose order is the product of the respective order of convergence of the original methods [1]. Usually, new evaluations of the Jacobian matrix and the nonlinear functions are needed in order to increase the order of convergence. However, some modifications can be made in order to limit the number of functional evaluations and improve the convergence order of the original method.

In this work we present a family of new iterative methods obtained by composing known iterative methods of order of convergence four [2,3] with a modification of Newton's method that introduces just one evaluation of the nonlinear function, increasing the convergence order in one unit. Then, the new methods are multi-point iterative methods, with order of convergence five, which only need to evaluate one Jacobian matrix in each iteration.

In order to compare different methods, it is very common to use the efficiency index [4], where the order of convergence and the number of functional evaluations per iteration required by the method are taken into account. However, in the n-dimensional case, it is also important to take into account the number of operations performed, since for each iteration a number of linear systems must be solved. For this reason we define the computational efficiency index where the number of products/quotients per iteration is also considered. 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. Taking into account that these methods only need to evaluate one Jacobian matrix per iteration, their efficiency and computational indices are better than the ones of all known methods for nonlinear systems including: Newton's method, Jarratt's method, methods based on quadrature formulae, etc.

We check the effectiveness of the modified methods by applying them to several nonlinear systems. The numerical computations confirm the theoretical results. We compare the number of iterations needed, the computational order introduced in [2], the efficiency index and the computational efficiency index of the presented methods with a large family of known methods. For each example, we run each method starting from 25 points randomly chosen in the neighborhood of the solution. The numerical computations have been carried out using variable precision arithmetic that uses floating point representation of 200 decimal digits of mantissa in Matlab 7.1.

1

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

2

J.L. Hueso, E. Martínez, J.R. Torregrosa, "Third and fourth order iterative methods free from second derivative for nonlinear systems", Applied Mathematics and Computation, 211, 190-197, 2009. doi:10.1016/j.amc.2009.01.039

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)