Iterative Methods with Memory for Solving Systems of Nonlinear Equations Using a Second Order Approximation

Alicia Cordero,Juan R Torregrosa,Javier G Maimó,María P Vassileva

doi:10.3390/math7111069

Alicia Cordero, Juan R Torregrosa + Show 2 more

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https://doi.org/10.3390/math7111069

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Abstract

Iterative methods for solving nonlinear equations are said to have memory when the calculation of the next iterate requires the use of more than one previous iteration. Methods with memory usually have a very stable behavior in the sense of the wideness of the set of convergent initial estimations. With the right choice of parameters, iterative methods without memory can increase their order of convergence significantly, becoming schemes with memory. In this work, starting from a simple method without memory, we increase its order of convergence without adding new functional evaluations by approximating the accelerating parameter with Newton interpolation polynomials of degree one and two. Using this technique in the multidimensional case, we extend the proposed method to systems of nonlinear equations. Numerical tests are presented to verify the theoretical results and a study of the dynamics of the method is applied to different problems to show its stability.

Highlights

This paper deals with iterative methods for approximating the solutions of a nonlinear system of n equations and n unknowns, F ( x ) = 0, where F : D ⊆ Rn −→ Rn is a nonlinear vectorial function defined in a convex set D
Several authors have constructed iterative methods with memory for solving nonlinear systems, by approximating the accelerating parameters by Newton’s polynomial interpolation of first degree, see for example the results presented by Petković and Sharma in [5] and by Narang et al in [6]
It is constructed by defining a mesh of points, each of which is taken as a initial estimation of the iterative method

Summary

Introduction

Several authors have constructed iterative methods with memory for solving nonlinear systems, by approximating the accelerating parameters by Newton’s polynomial interpolation of first degree, see for example the results presented by Petković and Sharma in [5] and by Narang et al in [6]. We illustrate the technique of the accelerating parameters, introduced by Traub in [2], starting from a very simple method with a real parameter α x k +1 = x k − α f ( x k ), k = 1, 2, This method has first order of convergence, with error equation ek+1 = (1 − α f 0 (ξ ))ek + O[e2k ],.

Modified Secant Method

Dynamical and Numerical Study

Findings

Conclusions