A class of optimal eighth-order derivative-free methods for solving the Danchick–Gauss problem

Abstract

A derivative-free optimal eighth-order family of iterative methods for solving nonlinear equations is constructed using weight functions approach with divided first order differences. Its performance, along with several other derivativefree methods, is studied on the specific problem of Danchick's reformulation of Gauss' method of preliminary orbit determination. Numerical experiments show that such derivative-free, high-order methods offer significant advantages over both, the classical and Danchick's Newton approach.

On developing a higher-order family of double-Newton methods with a bivariate weighting function

“…The dynamical idea behind basins of attraction was initiated by Stewart [28] and followed by works of Chun et al [6][7][8], Cordero et al [10] and Neta et al [21]. More recent results on basins of attraction can be found in [2,18,19].…”

A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics

“…The dynamics behind basins of attraction was initiated by Stewart [41] and followed by works of Amat et al [2][3][4][5], Scott et al [38], Chun et al [10], Chun and Neta [11], Chicharro et al [8], Cordero et al [16], Neta et al [30], [33], Argyros and Magreñan [7], Magreñan [29], Magreñan et al [28], Andreu et al [6] and Chun et al [12]. The only papers comparing basins of attraction for methods to obtain multiple roots are due to Neta et al [31], Neta and Chun [32,34], and Chun and Neta [13,14].…”

A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points

“…The dynamics underlying basins of attraction was initiated by Stewart [41] and followed by works of Amat et al [2][3][4][5] , Scott et al [38] , Chun et al [10] , Chun-Neta [11] , Chicharro et al [8] , Cordero et al [15] , Neta et al [28,33] , Argyros-Magreñan [7] , Magreñan [27] , Magreñan et al [26] , Andreu et al [6] and Chun et al [12] . The only papers comparing basins of attraction for methods to obtain multiple roots are due to Neta et al [29] , Neta-Chun [30,34] , Chun-Neta [13,14] and Geum-Kim-Neta [19] .…”

“…We conclude that 5YD is the best method overall. We have tried to find connection Table 8 Number of points requiring 40 iterations for each example (1)(2)(3)(4)(5)(6).…”

“…where 6) and ρ j (1 ≤ j ≤ 6) is a bivariate polynomial in b and c . As a result, we can obtain the extraneous fixed points ξ = ζ 1 / 2 by finding the zeros ζ of F 1 or F 2 .…”