Some fourth-order nonlinear solvers with closed formulae for multiple roots

Basins of attraction for Zhou–Chen–Song fourth order family of methods for multiple roots

“…These two methods perform better than the other optimal fourth-order methods in [10,11]. Here we suggest a rational function φ having two parameters and examine the possibility of finding a better performer.…”

“…where 10 + 559872z 8 − 139968z 6 + 6912z 4 − 138z 2 + 1. The roots are as follows: ξ = ±0.132279278794500, ξ = ±0.144687458801151 ± 0.0387765609887301i, ξ = ±0.382013739639782 ± 0.254362184937334i.…”

“…[17,[6][7][8]10,11,13,14,21,24,25,27]. Some of these methods are considered optimal in the sense of Kung and Traub [9], i.e.…”

“…It was shown in [15] that the method is equivalent to an optimal fourth-order method given by (75) in [10]. These two methods perform better than the other optimal fourth-order methods in [10,11].…”

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

“…It is known that numerical scheme (1.1) is a second-order one-point optimal [23] method on the basis of Kung-Traub's conjecture [23] that any multipoint method [35] without memory can reach its convergence order of at most 2 r−1 for r functional evaluations. We can find other higher-order multiple-zero finders in a number of literatures [16][17][18]21,24,25,31,32,40,45] .…”