A new family of generalized quadrature methods for solving nonlinear equations

Abstract

Weerakoon and Fernando [A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87–93] were resorted on a trapezoidal quadrature rule to derive an arithmetic mean Newton method with third-order convergence of the iterative scheme to solve nonlinear equations. Different quadrature methods have been developed, which form a special class of third-order iterative schemes requiring three evaluations of functions on [Formula: see text] per iteration, where [Formula: see text] is generated from the first Newton step. As an extension of these methods, we derive a new family of iterative schemes by using a new weight function [Formula: see text] to generalize the quadrature methods, of which [Formula: see text] signifies an approximate area under the curve [Formula: see text] between [Formula: see text] and [Formula: see text]. Then, a generalization of the midpoint Newton method is obtained by using another weight function, which is based on three evaluations of functions on [Formula: see text] per iteration. The sufficient conditions of these two weight functions are derived, which guarantee that the convergence order of the proposed iterative schemes is three.