Optimal Order of One-Point and Multipoint Iteration

Abstract

The problem is to calculate a simple zero of a non-linear function f n-1 by iteration. We exhibit a family of iterations of order 2 which use n evaluations of f and no derivative evaluations, as well as a second family of iterations of order 2 n ^ based on n-1 evaluations of f and one of f f . In particular, with four evaluations we construct an iteration of eighth order.The best previous result for four evaluations was fifth order.We prove that the optimal order of one general class of multipoint n 1 iterations is 2 and that an upper bound on the order of a multipoint iteration based on n evaluations of f (no derivatives) is 2 n .CONJECTURE. A multipoint iteration without memory based on n evaluations has optimal order 2 n

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

“…They have not considered the methods with such weight functions. In Section 4 we present the results of several numerical examples with six different combinations of the parameters a and g and the methods (7)- (9). We close with concluding remarks.…”

“…[17,[6][7][8]10,11,13,14,21,24,25,27]. Some of these methods are considered optimal in the sense of Kung and Traub [9], i.e. they have a maximal order of 2 n when using n + 1 function-(and derivative-) evaluation per iteration step.…”

“…In the next 2 sections, we will analyze the basins of attraction to compare several cases of the pair of parameters a and g with (7)- (9). 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].…”

A Variant of Sharma-Arora's Optimal Eighth-Order Family of Methods for Finding A Simple Root of Nonlinear Equation

“…Remark: Some expressions such as (8), (9) and (12) are intentionally omitted to display in full form for the sake of simplicity. All the displayed results can be easily verified using Wolfram's Mathematica software for symbolic computations.…”

“…In [9] Kung and Traub conjectured that the iterative method which requires n + 1 function evaluations per iteration can reach at most 2 n convergence order in general. The methods that satisfy KungTraub conjecture are known as optimal methods (see [10], [11], [12], [13], [14], [15], [16], [14], [18], [19]).…”