Dynamics of the King and Jarratt iterations

Abstract

The purpose of this article is to present results that amount to a description of the conjugacy classes of two fourth-order root-finding iterative methods, namely King's family of iterative methods and Jarratt's iterative method, for complex polynomials of degrees two, three and four. For degrees two and three, a full description of the conjugacy classes is accomplished, in each case, by a one-parameter family of polynomials. This is done in such a way that, when one applies one of these two root-finding iterative methods to the elements of these parametrized families, a family of iterative methods is obtained, and its dynamics represents, up to conjugacy, the dynamics of the corresponding iterative root-finding method applied to any complex polynomial having the same degree. For degree four, partial results analogous to the ones just described are presented.

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

“…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] .…”

“…6. 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 2 − 1) 5 . Fig.…”

“…We conclude that 5YD is the best method overall. We have tried to find connection Table 8 Number of points requiring 40 iterations for each example (1)(2)(3)(4)(5)(6).…”

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.…”