Complex dynamics of derivative-free methods for nonlinear equations

Comparison of several families of optimal eighth order methods

“…This idea was started by Stewart [16] and continued in the work of Amat et al [17][18][19][20], Scott et al [21], Chun et al [22], Chun and Neta [23], Chicharro et al [24], Cordero et al [25], Neta et al [26,27], Argyros and Magreñan [28], Magreñan [29], Geum et al [30] and Chun et al [31]. The only papers comparing basins of attraction for methods to obtain multiple roots are due to Neta et al [32], Neta and Chun [33,34], Chun and Neta [35][36][37].…”

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 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.…”