An optimal reconstruction of Chebyshev–Halley type methods for nonlinear equations having multiple zeros

Abstract

Establishment of a new optimal higher-order iterative scheme for multiple zeros with known multiplicity ( m ≥ 1 ) of univariate function is one of the hard and demanding task in the area of computational mathematics and numerical analysis. We write this paper with the aim to propose a new higher-order reconstruction of Chebyshev–Halley type iteration functions for multiple zeros in a simple way. Minimum order of convergence of our scheme is seven and maximum reaches to eight without using any extra functional evaluation. With regard to computational cost, each member of our scheme needs four functional evaluations at each step. So, the maximum efficiency index of our scheme is 1.6818 for β = 1 . In addition, we thoroughly study the convergence analysis which confirms the theoretical order of convergence. Moreover, we check the efficiency and compare them to the existing ones on four real life problems namely, fractional conversion, chemical engineering, continuous stirred tank reactor and eigen value problems and two well-known academic problems. Finally, we confirm on the basis of obtained results that our methods have better residual error, errors between two consecutive iterations and stable computational order of convergence as compared to the other existing ones.