Abstract
A generalized high-order class for approximating the solution of nonlinear systems of equations is introduced. First, from a fourth-order iterative family for solving nonlinear equations, we propose an extension to nonlinear systems of equations holding the same order of convergence but replacing the Jacobian by a divided difference in the weight functions for systems. The proposed GH family of methods is designed from this fourth-order family using both the composition and the weight functions technique. The resulting family has order of convergence 9. The performance of a particular iterative method of both families is analyzed for solving different test systems and also for the Fisher’s problem, showing the good performance of the new methods.
Highlights
During the last few decades, there has been a wide amount of interest in designing iterative schemes for estimating both equations and systems of equations presenting nonlinearities
We present a new family of iterative methods for solving nonlinear systems of equations with convergence order nine
In order to check the feasibility of the proposed schemes to solve nonlinear systems of equations, Section 3 shows the numerical results when the fourth-order scheme is used for solving Fisher’s partial differential equation and when the ninth-order family is applied to several test functions
Summary
During the last few decades, there has been a wide amount of interest in designing iterative schemes for estimating both equations and systems of equations presenting nonlinearities. We present a new family of iterative methods for solving nonlinear systems of equations with convergence order nine. This class, named GH family, has two weight functions and four steps on its iterative expression. A fourth-order family with only a weight function is proposed This family is the basis for designing the iterative schemes of the GH family with a composition-type technique. In order to check the feasibility of the proposed schemes to solve nonlinear systems of equations, Section 3 shows the numerical results when the fourth-order scheme is used for solving Fisher’s partial differential equation and when the ninth-order family is applied to several test functions.
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