Stable high-order iterative methods for solving nonlinear models

Abstract

There are several problems of pure and applied science which can be studied in the unified framework of the scalar and vectorial nonlinear equations. In this paper, we propose a sixth-order family of Jarratt type methods for solving nonlinear equations. Further, we extend this family to the multidimensional case preserving the order of convergence. Their theoretical and computational properties are fully investigated along with two main theorems describing the order of convergence. We use complex dynamics techniques in order to select, among the elements of this class of iterative methods, those more stable. This process is made by analyzing the conjugacy class, calculating the fixed and critical points and getting conclusions from parameter and dynamical planes. For the implementation of the proposed schemes for system of nonlinear equations, we consider some applied science problems namely, Van der Pol problem, kinematic syntheses, etc. Further, we compare them with existing sixth-order methods to check the validity of the theoretical results. From the numerical experiments, we find that our proposed schemes perform better than the existing ones. Further, we also consider a variety of nonlinear equations to check the performance of the proposed methods for scalar equations.

On an iterative method without inverses of derivatives for solving equations

“…(b) Notice that L 0 ≤ L, L 1 ≤ L, hold in general and L L 0 , L L 1 can be arbitrarily large [1,2,3]. In the literature with the exception of ours L 0 = L = L 1 is used for the study of Secant-type methods [4], [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. However, the latter choice is leading to less precise error estimates and stronger sufficient convergence conditions (see also the numerical examples).…”

On a Class of Efficient Higher Order Newton-Like Methods

“…This method converges if the initial approximation x 0 is closer to solution x * and F (x) −1 exists in the neighborhood Ω of x * . In order to attain the higher order of convergence, a number of modified Newton's or Newton-like methods have been proposed in literature, see, for example [3,4,5,6,7,8,9,10,11,14] and references therein.…”

“…versus M 6,2 case: The ratio 5.2 is given by R 5,1;6,2 = 7m 2 +4 3 m log(5) 8m 2 + 8 3 m log(6) .It is easy to prove that R 5,1;6,2 > 1 for m 8. Thus, we conclude that E 5,1 > E 6,2 for m 8.…”

Local convergence analysis of Jarratt-type schemes for solving equations

“…is very important since problems from many disciplines can be reduced to (1.1); see, e.g., [1,2,3,4,5,8,13,21] and the references therein. A locally unique solution x * is desirable in a closed form but this can be achieved only in special cases.…”

“…The most widely used schemes are the so called Newton-type. Recently, there are many results, based on the Lipschitz-type conditions on the higher order derivatives, on local and semi-local convergence analysis of Newton-type schemes; see [1,2,3,4,5,10,11,12,14,15,16,17,18,19,22] and the references therein. The convergence domain in these studies is small in general.…”