New fourth- and sixth-order classes of iterative methods for solving systems of nonlinear equations and their stability
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New fourth- and sixth-order classes of iterative methods for solving systems of nonlinear equations and their stability analysis Munish Kansal1 · Alicia Cordero2 · Sonia Bhalla3 · Juan R. Torregrosa2 Received: 4 October 2019 / Accepted: 4 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, a two-step class of fourth-order iterative methods for solving systems of nonlinear equations is presented. We further extend the two-step class to establish a new sixth-order family which requires only one additional functional evaluation. The convergence analysis of the proposed classes is provided under several mild conditions. A complete dynamical analysis is made, by using real multidimensional discrete dynamics, in order to select the most stable elements of both families of fourth- and sixth-order of convergence. To get this aim, a novel tool based on the existence of critical points has been used, the parameter line. The analytical discussion of the work is upheld by performing numerical experiments on some application-oriented problems. We provide an implementation of the proposed scheme on nonlinear optimization problem and zero-residual nonlinear least-squares problems taken from the constrained and unconstrained testing environment test set. Finally, based on numerical results, it has been concluded that our methods are comparable with the existing ones of similar nature in terms of order, efficiency, and computational time and also that the stability results provide the most efficient member of each class of iterative schemes. Keywords Systems of nonlinear equations · Order of convergence · Multipoint iterative methods · Stability analysis Mathematics Subject Classification 2010 65H10 · 65Y20
This research was partially supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE), Generalitat Valenciana PROMETEO/2016/089. Juan R. Torregrosa
[email protected]
Extended author information available on the last page of the article.
Numerical Algorithms
1 Introduction Systems of nonlinear equations are of immense importance for applications in many areas of science and engineering. For a given nonlinear system, G(X) = 0, where G : D ⊆ Rn → Rn , we are interested to find a vector X∗ = (x1∗ , x2∗ , · · · , xn∗ )T such that G(X∗ ) = 0 , where G(X) = (g1 (X), g2 (X), . . . , gn (X))T is a Fr´echet differentiable function and X = (x1 , x2 , . . . , xn )T ∈ Rn . The classical Newton’s method is the most basic procedure to solve systems of nonlinear equations. It is given by: X(k+1) = X(k) − {G (X(k) )}−1 G(X(k) ), k = 0, 1, . . .
(1)
where {G (X(k) )}−1 is the inverse of first-order Fr´echet derivative of the function G evaluated in (X(k) ). Assuming that the function G is continuously differentiable and the initial approximation is close enough to the solution, then this method converges quadratically. In literature, there are a variety of higher-order methods which improve the order of convergence of Newton’s scheme. For example, several authors have developed thir
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