# Convergence and Stability of a New Parametric Class of Iterative Processes for Nonlinear Systems
## 2 Family ACTV for Nonlinear Systems  
Taking into account the iterative expression of family (1), we can extend, in a natural way, this expression for solving nonlinear systems \(F(x)=0\). We change scalar \(f^{\prime}\) by vectorial \(F^{\prime}\) and \(f[x,y]\) by the divided difference operator \([x,y;F]\). The resulting expression is\[y^{(k)} =x^{(k)}-\Big{[}F^{\prime}\Big{(}x^{(k)}\Big{)}\Big{]}^{-1}F\Big{(}x^ {(k)}\Big{)}, \tag{9}\] \[z^{(k)} =y^{(k)}-\Big{(}2\Big{[}x^{(k)},y^{(k)};F\Big{]}-F^{\prime}\Big{(} x^{(k)}\Big{)}\Big{)}^{-1}F\Big{(}y^{(k)}\Big{)},\] \[x^{(k+1)} =z^{(k)}-\Big{[}\alpha F^{\prime}\Big{(}x^{(k)}\Big{)}^{-1}+(1- \alpha)\Big{[}x^{(k)},y^{(k)};F\Big{]}^{-1}\Big{]}F\Big{(}z^{(k)}\Big{)}\] \[\quad-(1+\alpha)\bigg{[}I_{n}-F^{\prime}\Big{(}x^{(k)}\Big{)}^{-1 }\Big{[}x^{(k)},y^{(k)};F\Big{]}\bigg{]}\cdot F^{\prime}\Big{(}x^{(k)}\Big{)}^ {-1}F\Big{(}z^{(k)}\Big{)},\]  
where \(I_{n}\) is the \(n\times n\) identity matrix. Mapping \([\cdot,\cdot;F]:\Omega\times\Omega\subset\mathbb{R}^{n}\times\mathbb{R}^{n} \longrightarrow\mathcal{L}(\mathbb{R}^{n})\) such that  
\[[x,y;F](x-y)=F(x)-F(y),\text{ for any }x,y\in\Omega,\]  
is the divided difference operator of \(F\) on \(\mathbb{R}^{n}\) (see [14]). The proof of the main result is based on the Genochi-Hermite formula (see [14]),  
\[[x,y;F]=\int_{0}^{1}F^{\prime}(x+t(x-y))dt,\text{ for all }(x,y)\in\Omega \times\Omega.\]  
By developing \(F^{\prime}(x+th)\) in Taylor series around \(x\), we obtain  
\[\int_{0}^{1}F^{\prime}(x+th)dt=F^{\prime}(x)+\frac{1}{2}F^{\prime\prime}(x)h+ \frac{1}{6}F^{\prime\prime\prime}(x)h^{2}+\mathcal{O}\Big{(}h^{3}\Big{)}. \tag{10}\]  
Denoting by \(e=x-\xi\), where \(\xi\) is a zero of \(F(x)\), and assuming that \(F^{\prime}(\xi)\) is invertible, we obtain  
\[F(x) =F^{\prime}(\xi)\Big{(}e+C_{2}e^{2}+C_{3}e^{3}+C_{4}e^{4}+C_{5}e^ {5}\Big{)}+\mathcal{O}\Big{(}e^{6}\Big{)},\] \[F^{\prime}(x) =F^{\prime}(\xi)\Big{(}I+2C_{2}e+3C_{3}e^{2}+4C_{4}e^{3}+5C_{5}e^ {4}\Big{)}+\mathcal{O}\Big{(}e^{5}\Big{)},\] \[F^{\prime\prime}(x) =F^{\prime}(\xi)\Big{(}2C_{2}+6C_{3}e+12C_{4}e^{2}+20C_{5}e^{3} \Big{)}+\mathcal{O}\Big{(}e^{4}\Big{)},\] \[z^{(k)} =y^{(k)}-\Big{(}2\Big{[}x^{(k)},y^{(k)};F\Big{]}-F^{\prime}\Big{(} x^{(k)}\Big{)}\Big{)}^{-1}F\Big{(}y^{(k)}\Big{)},\]0  
where \(C_{q}=\frac{1}{q! }[F^{\prime}(\xi)]^{-1}F^{(q)}(\xi),q\geq 2\). Replacing these expressions in the Genocchi-Hermite formula and denoting the second point of the divided difference by \(y=x+h\) and the error of \(y\) by \(e_{y}=y-\xi\), we obtain  
\[z^{(k)} =y^{(k)}-\Big{(}2\Big{[}x^{(k)},y^{(k)};F\Big{]}-F^{\prime}\Big{(} x^{(k)}\Big{)}\Big{)}^{-1}F\Big{(}y^{(k)}\Big{)},\]1  
Particularly, if \(y\) is the Newton approximation, i.e., \(h=x-y=[F^{\prime}(x)]^{-1}F(x)\), we obtain  
\[z^{(k)} =y^{(k)}-\Big{(}2\Big{[}x^{(k)},y^{(k)};F\Big{]}-F^{\prime}\Big{(} x^{(k)}\Big{)}\Big{)}^{-1}F\Big{(}y^{(k)}\Big{)},\]2  
_Convergence Analysis_  
**Theorem 3**. : _Being \(F:\Omega\subseteq\mathbb{R}^{n}\to\mathbb{R}^{n}\) differentiable enough in an open convex neighborhood \(\Omega\) of \(\xi\), root of \(F(x)\). Consider a seed \(x^{(0)}\) close enough to the solution \(\zeta\) and that \(F^{\prime}(x)\) is continuous and invertible in \(\zeta\). Then, (9) has a local convergence of order six, for all \(\alpha\in\mathbb{R}\), with the error equation_  
\[z^{(k)} =y^{(k)}-\Big{(}2\Big{[}x^{(k)},y^{(k)};F\Big{]}-F^{\prime}\Big{(} x^{(k)}\Big{)}\Big{)}^{-1}F\Big{(}y^{(k)}\Big{)},\]3  
_being \(C_{k}=\frac{1}{R}[F^{\prime}(\xi)]^{-1}F^{(k)}(\xi),\quad k=2,3,\ldots,\quad e ^{(k)}=x^{(k)}-\xi\)._Proof. : From  
\[z^{(k)} =y^{(k)}-\Big{(}2\Big{[}x^{(k)},y^{(k)};F\Big{]}-F^{\prime}\Big{(} x^{(k)}\Big{)}\Big{)}^{-1}F\Big{(}y^{(k)}\Big{)},\]4  
We perform the Taylor series of \(F\Big{(}x^{(k)}\Big{)}\) and \(F^{\prime}\Big{(}x^{(k)}\Big{)}\) around \(\xi\),  
\[