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 [f(xn),f′(xn),f′(yn)] per iteration, where yn 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 H to generalize the quadrature methods, of which (xn+1−xn)f′(xn)/H signifies an approximate area under the curve f′(x) between xn and xn+1. Then, a generalization of the midpoint Newton method is obtained by using another weight function, which is based on three evaluations of functions on [f(xn),f′(xn),f′((xn+yn)/2)] 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.
Communicated by N. C. Wong
