On general families of multipoint iterations by inverse interpolation and their applications

In this paper, a general family of derivative-free n + 1-point iterative methods with n+1 parameters is constructed by inverse interpolation for solving nonlinear equations. A general family of n-point iterative methods with the first derivative and n parameters is also constructed by inverse interpolation. They satisfy the conjecture of Kung and Traub (1974) that an iterative method based on n + 1 evaluations per iteration without memory would have optimal order 2n. Furthermore, the two families are accelerated by divided difference expressions for the parameters with one memory f(xk-1,n) per iteration to achieve higher orders of convergence 2n + 2n-1 and 2n + 2n-2, respectively. Finally, the proposed families are verified by solving nonlinear equations and applied to solve nonlinear ODEs.(c) 2023 Elsevier B.V. All rights reserved.