# Expanding the Applicability of Some High Order Householder-Like Methods
## 1 Introduction  
Let \(B_{1}\) and \(B_{2}\) be Banach spaces and \(D\) be a non-empty open convex subset of \(B_{1}\). Let also \(\mathcal{L}(B_{1},B_{2})\) stand for the space of bounded linear operators from \(B_{1}\) into \(B_{2}\). In this study, we are concerned with the problem of approximating a locally unique solution \(x^{*}\) of the nonlinear equation:  
\[F(x)=0, \tag{1}\]  
where \(F:D\subset B_{1}\to B_{2}\) is a nonlinear Frechet-differentiable operator. Beginning from \(t_{0}\), we can consider different third-order iterative methods to solve the scalar equation \(F\left(t\right)=0\), with \(F:X\to Y\) and \(X=Y=\mathbb{R}\), [1, 2]. According to [PERSON]'s classification [3], these third-order methods are divided into two broad classes. On the one hand, the one-point methods, which require the evaluation of \(F\), \(F^{\prime}\) and \(F^{\prime\prime}\) at the current point only. For instance,  
1. _The Halley method_ \[t_{n+1}=t_{n}-\left(\frac{1}{1+\frac{1}{2}L_{F}(t_{n})}\right)\frac{F(t_{n})} {F^{\prime}(t_{n})},\] where \(L_{F}(t)=\frac{F(t)F^{\prime\prime}(t)}{F^{\prime}(t)^{2}}\). 2. The Chebyshev method: \[t_{n+1}=t_{n}-\left(1+\frac{1}{2}L_{F}(t_{n})\right)\frac{F(t_{n})}{F^{\prime} (t_{n})},\]The family of Newton-like methods:  
\[\left\{ \begin{array}{l}t_{n+1}=G(t_{n})=t_{n}-H(L_{F}(t_{n}))\frac{F(t_{n})}{F^{ \prime}(t_{n})},\quad n\geq 0,\\ H(y)=\sum_{j\geq 0}\tilde{A}_{j}y^{j};\quad\tilde{A}_{0}=1,\,\tilde{A}_{1}=1/2,\, \tilde{A}_{j}\in R^{+}\cup\{0\},\forall j\geq 2,\end{array} \right. \tag{2}\]  
which includes the well-known iterative methods of the third-order Chebyshev, Halley, Super-Halley, Ostrowski and Euler methods. Notice that the family (2) has the inconvenience of reaching order greater than three only for certain equations. For instance, it is known that the family (2) has fourth order of convergence for quadratic equations, if \(\tilde{A}_{2}=1/2\). On the other hand, the second class is the multipoint methods, which cannot require \(F^{\prime\prime}\). For instance:  
_Two-step method_  
\[ \begin{array}{rcl}s_{n}&=&t_{n}-\frac{F(t_{n})}{F^{\prime}(t_{n})},\\ t_{n+1}=TS(t_{n})&=&s_{n}-\frac{F(s_{n})}{F^{\prime}(t_{n})}.\end{array}  \tag{3}\]  
Notice that this iterative method cannot be included in the family (2). 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:  
\[ \begin{array}{rcl}s_{n}&=&t_{n}-\frac{F(t_{n})}{F^{\prime}(t_{n})},\\ t_{n+1}&=&H(t_{n})=s_{n}-\frac{A_{0}(F;t_{n})}{F^{\prime}(t_{n})}-\left(\sum_ {k=2}^{\infty}\frac{A_{k}(F;t_{n})}{F^{\prime}(t_{n})}\left[\frac{F(t_{n})}{F^ {\prime}(t_{n})}\right]^{k-1}\right)\frac{F(t_{n})}{F^{\prime}(t_{n})}.\end{array}  \tag{4}\]  
Notice that (4) is well defined if \(|\frac{A_{k}(F;t)}{F(t)}|\leq k^{p}\), for all \(p\) and \(k\geq 2\) and \(|\frac{F(t)}{F^{\prime}(t)}|<1\). Thus, for instance, for \(A_{0}(F;t)=0\) and \(A_{k}(F;t)=\tilde{A}_{k-1}\frac{F^{\prime\prime}(t)^{k-1}}{F^{\prime}(t)^{k-2}}\), we obtain the previous family (2). On the other hand, if we take \(A_{k}(F;t_{n})=0\), for all \(k\geq 1\), and:  
\[A_{0}(F;t_{n})=F(s_{n}),\]  
then we get the two-step method (3). In general, the methods (2) have higher operational cost than the methods (3) when solving a non-linear system. Another measure that takes into account the operational cost that requires an iterative process \(\Phi\) is the computational efficiency given by \(CE(\phi,F)=o^{\frac{1}{\theta(F)}}\) where, again, \(o\) is the order of convergence of \(\Phi\) and \(\theta(F)\) is the operational cost to apply a step with the iterative process \(\Phi\), that is the number of products needed per iteration and the number of products that appear in the evaluation of the corresponding operator.