Some high-Order convergence modifications of the Householder method for Nonlinear Equations

Abstract

One major setback of iterative methods that require higher derivatives in their iterative procedures is that of computational cost. The Householder’s method is one of such methods that require second derivative evaluation in its procedures. To circumvent this setback, the second derivative is annihilated by estimation via the use of the interpolating polynomial and the divided difference techniques. Consequently, three new modifications of the Householder’s method that are of two and three steps were put forward in this article. To further improve the efficiency of the modified methods, a weight function is introduced to the iterative cycle to enhance the methods convergence order. From the convergence analysis conducted on the methods, revealed that they are of fifth, ninth and tenth order convergence respectively. To test the applicability of the methods, they were applied to locate the solutions of some nonlinear equations and modeled practical problems that are nonlinear equations. From the computational experience, it was observed that the methods performed better than the compared methods that are also modifications of the Householder methods.