Abstract
In this paper, a one-step Steffensen-type method with super-cubic convergence for solving nonlinear equations is suggested. The convergence order 3.383 is proved theoretically and demonstrated numerically. This super-cubic convergence is obtained by self-accelerating second-order Steffensen’s method twice with memory, but without any new function evaluations. The proposed method is very efficient and convenient, since it is still a derivative-free two-point method. Its theoretical results and high computational efficiency is confirmed by Numerical examples.
Highlights
Introduction(2014) Steffensen-Type Method of Super Third-Order Convergence for Solving Nonlinear Equations
Finding the root of a nonlinear equation f (x) = 0 (1)is a classical problem
Traub conjectured that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2m−1, Traub proposed a self-accelerating two-point method of order 2.414 with memory:
Summary
(2014) Steffensen-Type Method of Super Third-Order Convergence for Solving Nonlinear Equations. Zhang unavailable or is expensive to be obtained, the derivative-free method is necessary. Besides H.T. Kung and J.F. Traub conjectured that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2m−1 (see [2]), Traub proposed a self-accelerating two-point method of order 2.414 with memory (see [3]): xn +=1. A lot of self-accelerating Steffensen-type methods were derived in the literature (see [1]-[7]). Džunića and M.S. Petkovića proposed a cubically convergent Steffensen-like method (see [7]):.
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