On improved three-step schemes with high efficiency index and their dynamics

Abstract

This paper presents an improvement of the sixth-order method of Chun and Neta as a class of three-step iterations with optimal efficiency index, in the sense of Kung-Traub conjecture. Each member of the presented class reaches the highest possible order using four functional evaluations. Error analysis will be studied and numerical examples are also made to support the theoretical results. We then present results which describe the dynamics of the presented optimal methods for complex polynomials. The basins of attraction of the existing optimal methods and our methods are presented and compared to illustrate their performances.

An Efficient Family of Optimal Eighth-Order Iterative Methods for Solving Nonlinear Equations and Its Dynamics

“…But if we put w n = x n + f (x n ) 3 then its error equation (under the same conditions on weight functions as of theorem (2.1)). Thus the method preserves its order of convergence for w n = x n + f (x n ) 3 . In fact if we put w n = x n +α.…”

Computing multiple zeros by using a parameter in Newton–Secant method

“…The Julia set of a nonlinear map G(z), denoted by J(G), is the closure of the set of its repelling periodic points. The complement of J(G) is the Fatou set F (G), where the basin of attraction of the different roots lie [1].…”

New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations

“…C be a rational map on the complex plane. The orbit of a point z 2 C is defined as [23]. For the dynamical point of view, in fact, we take a 256 256 grid of the square OE 3; 3 OE 3; 3 2 C and assign a color to each point z 0 2 D according to the simple root to which the corresponding orbit of the iterative method starting from z 0 converges, and we mark the point as black if the orbit does not converge to a root, in the sense that after at most 100 iterations it has a distance to any of the roots, which is larger than 10 3 .…”