_Algorithms_**2015**, \(8\), 669-679; doi:10.3390/a8030669  
_algorithms_  
**ISSN 1999-4893**  
www.mdpi.com/journal/algorithms  
_Article_  
**Expanding the Applicability of a Third Order Newton-Type Method Free of Bilinear Operators**  
**[PERSON] 1, [PERSON] 1, [PERSON] 1 and [PERSON] 2,*  
In this paper we study the following two-step method [4,5] that improves the order of [PERSON]'s method up to third order, but without evaluating any second Frechet derivative:  
\[y_{n} = x_{n}+F^{{}^{\prime}}(x_{n})^{-1}F(x_{n})\] \[x_{n+1} = y_{n}-F^{{}^{\prime}}(x_{n})^{-1}F(y_{n}) \tag{1}\]  
The basic advantage of this method is that, as the matrix that appears at each iteration is the same, only one \(LU\) decomposition is computed. In most real problems, the computational cost of solving a linear system is more expensive than some extra evaluations of the operator. Moreover, from a dynamical point of view [6] the method seem better than the classical two-point [PERSON] method, that is  
\[y_{n} = x_{n}-F^{{}^{\prime}}(x_{n})^{-1}F(x_{n})\] \[x_{n+1} = y_{n}-F^{{}^{\prime}}(x_{n})^{-1}F(y_{n}) \tag{2}\]  
since the regions of non convergence are reduced. The main assumption for obtaining convergence of third order iterative methods [2,7,8], is a Lipschitz condition in the second Frechet derivatives,  
\[\|F^{\prime\prime}(x)-F^{\prime\prime}(y)\|\leq c\,\|x-y\| \tag{3}\]  
With this hypothesis, and by choosing an initial guess such that \(F^{{}^{\prime}}(x_{0})^{-1}\) exists and \(||F(x_{0})||\) is sufficiently close to zero, cubic convergence is obtained. The Lipschitz condition (3) can be relaxed to a \(p\)-Holder condition  
\[\|F^{\prime\prime}(x)-F^{\prime\prime}(y)\|\leq c\,\|x-y\|^{p} \tag{4}\]  
or even to some \(\omega\)-condition  
\[\|F^{\prime\prime}(x)-F^{\prime\prime}(y)\|\leq\omega(||x-y||) \tag{5}\]  
where \(\,\omega:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}\,\) is a nondecreasing continuous function. By means of this type of conditions we can ensure the convergence of the scheme (1). Alternatively, since the scheme (1) only uses first derivatives, we could also consider the possibility of obtaining convergence by assuming the main condition just in the first derivatives, instead of in the second derivatives. There are many theories on the local and semilocal convergence of [PERSON]'s type methods, see for instance [9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Following [19, 20, 21, 22], in [5] the semilocal convergence of (1) under \(\omega\)-conditioned divided differences was analyzed. Remember that a continuous bounded linear operator \([x,y;F]\) associated to a nonlinear operator \(F:B\subset X\to Y\), is called a _divided difference of first order_ for the operator \(F\) on the points \(x\) and \(y\) if \([x,y;F](x-y)=F(x)-F(y)\). If \(F\) is Frechet differentiable, then \(F^{{}^{\prime}}(x)=[x,x;F]\) for all \(x\in B\). Moreover, a divided difference satisfies an \(\omega\)-condition if  
\[||[x,y;F]-[v,w;F]||\leq\omega(||x-v||,||y-w||),\qquad x,y,v,w\in B \tag{6}\]  
where \(\,\omega:\mathbb{R}_{+}\times\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}\,\) is a continuous function, which is nondecreasing in both variables. In this paper, following [23], we expand the applicability of (1) relaxing the hypotheses of convergence. We also analyze a modification of scheme (1) using divided differences:  
\[y_{n} = x_{n}+[x_{n}-\alpha_{n}F(x_{n}),x_{n}+\alpha_{n}F(x_{n});F]^{-1} F(x_{n}),\] \[x_{n+1} = y_{n}-[x_{n}-\alpha_{n}F(x_{n}),x_{n}+\alpha_{n}F(x_{n});F]^{-1} F(y_{n}). \tag{7}\]Here, the parameters \(\alpha_{n}\) can be considered as a control of the good approximation to the first Frechet derivative. For this method we are able to obtain convergence without assuming any Frechet differentiability in the operator \(F\). Other higher order methods, in some cases with a better behavior in the real case, have been proposed during the last few years [24; 25; 26; 27].