# Expanding the Applicability of Some High Order Householder-Like Methods
## 1 Introduction
Therefore, in order to compare the efficiency of some iterative methods of the new family, we consider the computational efficiency \(CE(\Phi,F)\). Note that the methods selected in the family (4) as the most efficient have a lower computational efficiency as more terms in the series are considered. Notice that, for quadratic equations, the Chebyshev method and the two-step method (method in (7) with \(A_{0}(F;x_{n})=F(y_{n})\)) have the same algorithm:  
\[ \begin{array}{rcl}y_{n}&=&x_{n}-F^{{}^{\prime}}(x_{n})^{-1}F(x_{n}),\\ x_{n+1}&=&y_{n}-F^{{}^{\prime}}(x_{n})^{-1}F(y_{n}).\end{array}  \tag{5}\]On the other hand, we note that the family of iterative processes (7), with \(A_{2}=1/2\), has the property of having four-order convergence when we consider quadratic equations. Therefore, the iterative process that is more efficient in this family, for quadratic equations, is the known Chebyshev-like method (\(A_{2}=1/2\) and \(A_{k}=0\) for \(k\geq 3\) in (7)). Then, we compare this iterative process with the Chebyshev method. This fourth-order method for quadratic equations can be written as:  
\[ \begin{array}{lll}y_{n}&=&x_{n}-F^{{}^{\prime}}(x_{n})^{-1}F(x_{n}),\\ z_{n}&=&y_{n}-F^{{}^{\prime}}(x_{n})^{-1}F(y_{n}),\\ x_{n+1}&=&z_{n}-F^{{}^{\prime}}(x_{n})^{-1}(2F(z_{n})-C(z_{n}-y_{n})(z_{u}-x_{n })),\end{array}  \tag{6}\]  
For the quadratic equation, the computational efficiency, \(CE\), for the Chebyshev method, (5) and (6) methods in the family (7) is \(CE=3^{3/(m^{3}+9m^{2}+17m)}\), \(CE=4^{3/(m^{3}+12m^{2}+24m)}\), respectively. 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 [PERSON] et al. [4], a semilocal convergence analysis was given in a Banach space setting:  
\[ \begin{array}{lll}y_{n}&=&x_{n}-\Gamma_{n}F(x_{n}),\\ x_{n+1}&=&y_{n}-\Gamma_{n}A_{0}(F;x_{n})-\left(\sum_{k=2}^{\infty}\Gamma_{n}A _{k}(F;x_{n})\left[\Gamma_{n}F(x_{n})\right]^{k-1}\right)\Gamma_{n}F(x_{n}), \end{array}  \tag{7}\]  
where \(\Gamma_{n}=[F^{\prime}(x_{n})]^{-1}\) and the operators \(A_{k}(F;-):D\rightarrow\mathcal{L}(\overbrace{D\times\ldots\times D}^{(k},B_ {2})\) are some particular \(k\)-linear operators. 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. We are able to find the best method in the family (different of the two-step) for this particular problem. Indeed, the founded method has order four. The denoising model that we propose in this paper permits a  
Figure 1: Computational efficiency for the Chebyshev method, (6) and (5) methods in the family (7). good reconstruction of the edges that are the most important visual parts of an image. See also [5; 6] for more applications. On the other hand, the semilocal convergence analysis of method (7) was based on \(\gamma\)-type conditions [7] that constitute an alternative to the classical Kantorovich-type conditions [8]. 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 \(\gamma\)-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. The rest of the paper is organized as follows: the semilocal convergence analysis of the method (7) is given in Section 2, whereas the numerical examples are presented in the concluding Section 3.