Keywords: modified Newton method, iterative methods, nonlinear equations, systems of equations, ill-conditioning, local convergence.

Iterative methods for solving nonlinear equations have been widely proposed, devised, analyzed and applied in the literature [1,2,3]. In fact, it is one of the classic topics in numerical analysis. From the bibliography it can be deduced that, in order for a method to be effective, some theoretical and computational issues must be achieved. Theoretical convergence, order, computational low cost must be accomplished for an iterative method [4]. Some of these issues are still being studied, because mathematicians are looking for general answers summing up different frameworks, [5,6].

Computational efficiency of iterative methods to solve nonlinear equations of the type F(x)=0 is handicapped in case of ill conditioned problems. A well known result states that high order methods, such as Newton, decay to first order in the presence of multiple roots. However, even simple roots may be problematic.

Most methods are analyzed in ideal conditions, where the equation is regular enough as not to introduce any additional difficulty. From a local point of view (that is, from a neighbourhood of the simple root), there is always a region where the method can be considered in these ideal conditions. The main problem arises when that regular region can be hardly reached, because it is difficult to distinguish different roots or the function is extremely flat around the root. This problem gets worse when solving nonlinear systems, arising from the fact that roots interact and are difficult to separate.

In this work, we deal with these ill conditioned cases. We propose a general strategy in order to reduce the harmful effects of the above mentioned (strategy which, in the other side, may be used even in the most positive conditions), based on the regularization of the iteration. When applied, this regularization provides different methods, not all of them new. In general, these methods depend on some parameters which have to be chosen. Criteria for these choices are also provided, following the properties of the equation, in such a way that the methods are adapted to the problem instead of the usual techniques which try to adapt the problem to the methods.

In particular, we study regularization on Newton's method, giving the perturbed Newton method depending on a conditioning matrix C. We will analize this method, and will illustrate through examples its properties.

1

L.V. Kantorovich, G.P. Akilov, " Functional Analysis", Pergamon Press, Oxford, 1982.

2

J.M. Ortega, W.C. Rheinbold, "Iterative Solution of Nonlinear Equations in Several Variables", Academic Press, Reading, MA, 1970.

3

W.C. Rheinbold, "Methods for Solving Systems of Nonlinear Equations", SIAM, Philadelphia, 1966.

4

L.B. Rall, "Computational Solution of Nonlinear Operator Equations", R.E. Krieger, New York, 1979.

5

V.F. Candela, A. Marquina, "Recurrence relations for rational cubic methods I: The Halley method", Computing, 44, 169-184, 1990. doi:10.1007/BF02241866

6

V.F. Candela, A. Marquina, "Recurrence relations for rational cubic methods II: The Chebyshev method", Computing, 45, 355-367, 1990. doi:10.1007/BF02238803

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