A Family of Combined Iterative Methods for Solving Nonlinear Equations

Abstract

In this article we construct some higher-order modifications of Newton's method for solving nonlinear equations, which is based on the undetermined coefficients. This construction can be applied to any iteration formula. It can be found that per iteration the resulting methods add only one additional function evaluation, their order of convergence can be increased by two or three units. Higher order convergence of our methods is proved and corresponding asymptotic error constants are expressed. Numerical examples, obtained using Matlab with high precision arithmetic, are shown to demonstrate the convergence and efficiency of the combined iterative methods. It is found that the combined iterative methods produce very good results on tested examples, compared to the results produced by the existing higher order schemes in the related literature.

Ball convergence for combined three-step methods under generalized conditions in Banach space

“…The radius of convergence q used in [9] is smaller than the radius r DS given by Dennis and Schabel [4]…”

“…Therefore, we can choose w 0 (t) = w(t) = 1 8 3 2 t 1 2 + t and by Remark 2.2(a) v(t) = 1 + w 0 (t). The results in [16,9] can not be used to solve this problem, since F is not Lipschitz. However, our results can apply.…”

“…Approximating x * is very important, since numerous problems can be reduced to equation (1.1) using mathematical modeling [4,7,12,9,16,21,23,24]. However, it is not always possible to find the solution x * in a closed form.…”

“…. It was shown in [9] using Taylor series expansions when X = Y = R that method (1.2) is of order at least 2m, if m < 3 and of order at least m + 3, if m ≥ 3 provided that F is eighth times differentiable. The hypotheses on the derivatives of F restrict the applicability of method (1.2).…”

Ball Convergence for Combined Three-Step Methods Under Generalized Conditions in Banach Space

“…. If X = Y = R, then it was shown in [1]. The proof uses Taylor series expansions and the conditions on function Θ is up to the seventh differentiable.…”