Basins of attraction for several methods to find simple roots of nonlinear equations

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

“…As we can see in the next section, the new methods have both attractive and repulsive extraneous fixed points. We should also mention here that we have found previously [39] that if the extraneous fixed points are on the imaginary axis the method performs better.…”

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] .…”

Comparative study of eighth-order methods for finding simple roots of nonlinear equations

“…In order to find the extraneous fixed points, we substitute the quadratic polynomial z 2 − 1 for f (z) and then find the zeros of H f . For example, Super Halley method has extraneous fixed points which are the solution of L f = 2,which are (see [12] …”

“…For HK8, we have previously used β = 3 − 2 √ 2 which is the optimal parameter for King's method (see [12]). Here, we have taken β = 0 (denoted HK8d).…”

“…In recent work, one can find a visual comparison, by plotting the basins of attraction for the methods. The idea of using basins of attraction appeared first in Stewart [3] and followed by the works of Amat et al [4][5][6] and [7], Scott et al [8], Chun et al [9], Chicharro et al [10], Cordero et al [11], Neta et al [12,13], Chun et al [14][15][16], Argyros and Magreñan [17], and Magreñan [18]. In later works ( [15,16,[19][20][21]), we have introduced a more quantitative comparison, by listing the average number of iterations per point, the CPU time, and the number of points requiring 40 iterations.…”