Modified Potra-Pták Multi-step Schemes with Accelerated Order of Convergence for Solving Systems of Nonlinear Equations

Himani Arora,Juan Torregrosa,Alicia Cordero

doi:10.3390/mca24010003

Himani Arora, Juan Torregrosa + Show 1 more

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https://doi.org/10.3390/mca24010003

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Abstract

In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of third order Potra-Pták method and last is weighted-Newton step. Furthermore, we generalize our work to derive a family of multi-step iterative methods with order of convergence 3 r + 6 , r = 0 , 1 , 2 , … . The sixth order method is the special case of this multi-step scheme for r = 0 . The family gives a four-step ninth order method for r = 1 . As much higher order methods are not used in practice, so we study sixth and ninth order methods in detail. Numerical examples are included to confirm theoretical results and to compare the methods with some existing ones. Different numerical tests, containing academical functions and systems resulting from the discretization of boundary problems, are introduced to show the efficiency and reliability of the proposed methods.

Highlights

Many applied problems in Science and Engineering [1,2,3] are reduced to solve nonlinear systems F(x) = 0 numerically, that is, for a given nonlinear function F(x) : D ⊂ Rm −→ Rm, where F(x) = ( f1(x), f2(x), ..., fm(x))T and x = (x1, x2, ..., xm)T, to find a vector α = (α1, α2, ..., αm)T such that F(α) = 0
The third order method by Potra–Pták [11] for systems of nonlinear equations is given by y(k) = x(k) − F (x(k))−1F(x(k)), x(k+1) = y(k) − F (x(k))−1F(y(k)) k = 0, 1
We propose a three-step iterative method with accelerated sixth order of convergence; of the three steps, the first two are those of Potra–Pták method whereas the third is a weighted Newton-step

Summary

Introduction

The most widely used method for this purpose is the classical Newton’s method [3,4], which converges quadratically under the conditions that the function F is continuously differentiable and a good initial approximation x(0) is given. It is quite clear that this scheme requires the evaluation of two functions, one derivative and one matrix inversion per iteration, that is usually avoided by solving a linear system. This algorithm is illustrious for its simplicity and for its efficient character

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