Efficient n-point iterative methods with memory for solving nonlinear equations

Abstract

In this paper, we proposed a family of n-point iterative methods with and without memory for solving nonlinear equations. The convergence order of the new n-point iterative methods without memory is 2n requiring n+1 functional evaluations in per full iteration, which implies that the new n-point iterative methods without memory are optimal according to Kung-Traub conjecture. Based on the n-point methods without memory, the n-point iterative methods with memory are obtained by using n+1self-accelerating parameters. The maximal convergence order of the new n-point iterative methods with memory is (2n+1?1+22(n+1)+1)/2$(2^{n+1}-1+\sqrt {2^{2(n+1)}+1} )/2$, which is higher than any existing iterative methods with memory. Numerical examples are demonstrated to confirm theoretical results.