A Three-Step Seventh-Order Iterative Method Using Divided Difference Approximations for Solving Nonlinear Equations

Abstract

Nonlinear equations are used in numerous modeling approaches, and yet the closed form solutions are hard to find. Thus, iterative algorithms become a popular choice when tackling such problems, although the search for faster and better performing schemes does not stop there. In this paper, we present a novel three step seventh order iterative scheme for finding a root of a nonlinear function in one variable. Our iterative approach is designed based on a Newton predictor, a correction step, and the final step, where a derivative free divided difference estimation is applied instead of calculating a complete derivative. The number of functional evaluations used in the new scheme is reduced to only five, thus providing a very high efficiency index of 715=1.47. We analytically prove the seventh order convergence by performing Taylor series expansions. Then, the new scheme called ZAH7 is tested against other known seventh order schemes by applying it to a range of nine different functions, among which there were polynomials, exponentials, logarithms, and trigonometric expressions. High precision numerical experiments performed with the new algorithm showed that it produces lower errors compared to others and guarantees seventh order computational order of convergence.