Dynamics of a family of third-order iterative methods that do not require using second derivatives

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

“…The idea of using basins of attraction was initiated by Stewart [23] and followed by the works of Amat et al [1][2][3][4], Scott et al [22] and Chun et al [5]. The only papers comparing basins of attraction for methods to obtain multiple roots is due to Neta et al [18] and Neta and Chun [16,15].…”

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

“…Based on Table 6 we find that 5YD uses the least number of iterations per point (6.06 on average) followed closely by 5XF (7.02). The method requiring the highest number of iterations per point is 5XB (10.26) which is slightly less than the best sixth order method GKNPA in our previous paper 4. The top row for 5XA , 5XB , 5XD in order from left to right, the second row for 5XE , 5XF , and 5XG , and the bottom row for 5YD , and 5YF , for the roots of the polynomial (z 5 − 1) 3 .…”

“…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] .…”

“…5. The top row for 5XB , 5XE , 5XF , in order from left to right, and the bottom row for 5XG , and 5YD , for the roots of the polynomial (z 3 − z) 4 . Fig.…”

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.…”