On an application of Herzberger’s matrix method to multipoint families of root-solvers

Abstract

An application of Herzberger's matrix method, very rarely used in the topic of multipoint methods for solving nonlinear equations, is presented. It is shown that the area of application of Herzberger's matrix method is wider than it is presented in [J. Herzberger,Über Matrixdarstellungen für Iterationverfahren bei nichtlinearen Gleichungen, Computing, 12 (1974) 215-222]. This method is applied for the determination of the order of convergence of multipoint families of methods, Steffensen's type and Newton's type, with and without memory. The advantage and the elegance of this method arise from ease in handling matrices.

On the choice of the best members of the Kim family and the improvement of its convergence

“…There is an extensive literature of methods with memory that include derivatives in their iterative expressions (see, for example, Petković et al 2 and the references therein) and other ones that are derivative free, such as the papers of Petković and Dzunić [6][7][8] or by other authors, [9][10][11][12][13] all of them by using similar techniques.…”