Three novel fifth-order iterative schemes for solving nonlinear equations

Kung and Traub's conjecture indicates that a multipoint iterative scheme without memory and based on m evaluations of functions has an optimal convergence order p = 2m(-1). Consequently, a fifth-order iterative scheme requires at least four evaluations of functions. Herein, we derive three novel iterative schemes that have fifth-order convergence and involve four evaluations of functions, such that the efficiency index is E.I.=1.49535. On the basis of the analysis of error equations, we obtain our first iterative scheme from the constant weight combinations of three first- and second-class fourth-order iterative schemes. For the second iterative scheme, we devise a new weight function to derive another fifth-order iterative scheme. Finally, we derive our third iterative scheme from a combination of two second-class fourth-order iterative schemes. For testing the practical application of our schemes, we apply them to solve the van der Waals equation of state. (C) 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.