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.

A Two-Dimensional Variant of Newton’s Method and a Three-Point Hermite Interpolation: Fourth- and Eighth-Order Optimal Iterative Schemes

“…Besides the Halley method, there are many two-point iterative schemes which are of third-order convergence. Liu and Lee [48] generalized many quadrature-type third-order iterative schemes to…”

“…Now, we propose some new FOIS by a constantly weighting combination of the thirdorder iterative schemes from Equations ( 17) and (24), as well as the following one. Before that, we cite the following result [48].…”

“…Inserting a and b in Equation (48) into Equation ( 17), we solve Equation (35) and the related data are tabulated in Table 5. We solve Equations ( 36)-( 39), and the related data are tabulated in Table 6.…”