Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations

Abstract

A number of higher order iterative methods with derivative evaluations are developed in literature for computing multiple zeros. However, higher order methods without derivative for multiple zeros are difficult to obtain and hence such methods are rare in literature. Motivated by this fact, we present a family of eighth order derivative-free methods for computing multiple zeros. Per iteration the methods require only four function evaluations, therefore, these are optimal in the sense of Kung-Traub conjecture. Stability of the proposed class is demonstrated by means of using a graphical tool, namely, basins of attraction. Boundaries of the basins are fractal like shapes through which basins are symmetric. Applicability of the methods is demonstrated on different nonlinear functions which illustrates the efficient convergence behavior. Comparison of the numerical results shows that the new derivative-free methods are good competitors to the existing optimal eighth-order techniques which require derivative evaluations.

Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms

“…where, R: universal gas constant, T: temperature, P: pressure, V: volume, n: number of moles, a 1 , a 2 : variables with values depending on the gas. To calculate the volume V, we can write (33) as…”

“…Derivative-free iterative approaches are crucial for optimizing complex systems and resolving difficult engineering problems [31]. In the literature, researchers [32][33][34][35][36][37][38][39] have also developed the derivative-free multiple root iterative algorithms that are based on second-order modified Traub-Steffensen iteration [11]. The modified Traub-Steffensen method is given by…”

An optimal eighth order derivative free multiple root finding numerical method and applications to chemistry

“…In this section, we check the convergence behavior and performance of Case 1 to Case 4 of our proposed eighth order scheme denoted respectively by FZ1, FZ2, FZ3 and FZ4 by carrying out some nonlinear equations from real life applications of chemical engineering. We compare the methods with the recent derivative free methods of seventh order (see [15], Case I(a), Case II(c)) denoted by SH1, SH2 and eighth order (see [16], M-4 and [14]) denoted as SH3 and SH4. The recent seventh order methods are defined by, in case of SH1…”

“…Sharma et al [15] proposed seventh order convergent iterative scheme for multiple roots. Sharma et al in [14,16] presented eight order scheme for computing multiple root of nonlinear equations. nevertheless, other derivative-free methods for multiple roots have been generated by using different approaches, such as [4].…”

“…3, we discuss some particular cases of weight functions and choose four of them for further investigation. We compare our methods with two of the recent derivative-free schemes of seventh [15] and eighth order [14,16] using physical applications from chemical engineering and dynamical behaviour in Sect. 4.…”