A stable family with high order of convergence for solving nonlinear equations

Abstract

have published a new family of iterative methods, without memory, with order of convergence five or six, which are not optimal in the sense of Kung and Traub's conjecture. Therefore, we attempt to modify this suggested family in such a way that it becomes optimal. To this end, we consider the same two first steps of the mentioned family, and furthermore, we introduce a better approximation for f ′ (z) in the third step based on interpolation idea as opposed to the Taylor's series used in the work of Li et al. Theoretical, dynamical and numerical aspects of the new family are described and investigated in details.

Efficacy of Optimal Methods for Nonlinear Equations with Chemical Engineering Applications

“…Several optimal fourth-order iterative methods were constructed; see, for example, [9][10][11]. The optimal eighth-order of convergence was reached by many authors as presented in [12][13][14]. Also, many sixteenth-order iterative methods were proposed; for instance, see [15][16][17].…”

Local convergence for a Chebyshev-type method free from second derivative in Banach spaces

“…To avoid the second derivative, some scholars have proposed some variants of Chebyshev-Halley type methods free from second derivative [16,17]. Cordero et al [18] proposed a three-step form of modified Chebyshev-Halley type method:…”

On the local convergence of Kung-Traub's two-point method and its dynamics

“…This algorithm is presently known as the Newton-Raphson method, or more commonly as Newton's method [41]. In the last decade, many modified iterative methods have been developed to improve these classical methods, see [2], [1], [6], [9], [11], [13], [14], [24], [23], [25], [26], [30], [33], [35], [37], [36], [43], [48], [49] and the references therein. DOI: 10.21136/AM.2020.0322 -18 In 1974 Kung and Traub proposed an optimal fourth-order method [34], [42].…”

“…Some of the important problems in the study of iterative procedures are to find the estimates of the radii of convergence balls or to discuss of the local convergence analysis. There are many studies which deal with the local and semilocal convergence analyses of Newton-like methods such as [8], [24], [27], [28], [29], [33], [38]. Recently, Veiseh et al [51] studied the local convergence and dynamical behavior of derivativefree Kung-Traub's method.…”