New optimal class of higher-order methods for multiple roots, permitting f′(xn)=0

Comparing the basins of attraction for Kanwar–Bhatia–Kansal family to the best fourth order method

“…We have used the basin of attraction idea to recommend the best optimal fourth order method. Here we compare the family of methods developed by Kanwar et al (2013)-to the best known method. We will also point out some mistake in deriving that family.…”

“…Here we concentrate on the optimal fourth-order family of methods [23]. We correct the errors in the paper and compare two members of the family to the best known fourth-order methods found in the literature.…”

“…Remark: Actually Kanwar et al [23] have suggested a linear polynomial in the denominator, but this will not be different from the first case. We have decided here to try something close to what they have, namely a quadratic in the denominator.…”

“…a method of order 1.5 [5], a method of order 2 [6], third order methods [7][8][9][10][11][12][13][14][15][16][17], and [18]. The fourth order methods [19][20][21][22][23] and [24]. Some of these methods are considered optimal in the sense of Kung and Traub [25], i.e.…”

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] .…”