On a family of high-order iterative methods under gamma conditions with applications in denoising

Expanding the Applicability of Some High Order Househölder-Like Methods

“…This way, to obtain iterative methods with higher order than three for any function F and to be able to represent some multipoint methods, as (3), we can consider a modification of the family (2) that allow getting this generalization [4]. Therefore, we consider the following family, which includes all of these methods:…”

“…As we can observe in Figure 1, the optimum computational efficiency is for the Chebyshev-like method given in (6), when the system has at least six equations. Recently, in the work by Amat et al [4], a semilocal convergence analysis was given in a Banach space setting:…”

“…An interesting application related to image denoising was presented also in [4]. The approximation, via our family, is of a new nonlinear mathematical model for the denoising of digital images.…”

“…The convergence domain for the method (7) is small in general. In the present study, we use our new idea of restricted convergence domains leading to smaller γ-parameters, which in turn lead to the following advantages over the work in [4] (and under the same computational cost): larger convergence domain; tighter error bounds on the distances involved and an at least as precise information on the location of the solution.…”

“…Then, the following semilocal convergence result for the method defined in (7) under γ-type conditions was given in [4], Theorem 1: Theorem 1. Assume that the operator F satisfies for each x ∈ D:…”