# Convergence and Stability of a New Parametric Class of Iterative Processes for Nonlinear Systems
## 2 Family ACTV for Nonlinear Systems
\begin{split}\text{F}\Big{(}x^{(k)}\Big{)}&=F^{\prime }(\xi)\Big{[}e^{(k)}+C_{2}e^{(k)^{2}}+C_{3}e^{(k)^{3}}+C_{4}e^{(k)^{4}}+C_{5}e^ {(k)^{5}}+C_{6}e^{(k)^{6}}+C_{7}e^{(k)^{7}}\Big{]}\\ &\quad+\text{O}\Big{(}e^{(k)^{8}}\Big{)},\end{split}  \tag{12}\]  
\[ \begin{split}\text{F}^{\prime}\Big{(}x^{(k)}\Big{)}&=F^{ \prime}(\xi)\Big{[}I+2C_{2}e^{(k)}+3C_{3}e^{(k)^{2}}+4C_{4}e^{(k)^{3}}+5C_{5}e^ {(k)^{4}}+6C_{6}e^{(k)^{5}}+7C_{7}e^{(k)^{6}}\Big{]}\\ &\quad+O\Big{(}e^{(k)^{7}}\Big{)}.\end{split}  \tag{13}\]  
We suppose that the Jacobian matrix \(F^{\prime}(\xi)\) is nonsingular and calculate the Taylor expansion of \(\Big{[}F^{\prime}\Big{(}x^{(k)}\Big{)}\Big{]}^{-1}\) as follows:  
\[ \begin{split}\Big{[}F^{\prime}\Big{(}x^{(k)}\Big{)}\Big{]}^{-1}& =\Big{[}I+X_{2}e^{(k)}+X_{3}e^{(k)^{2}}+X_{4}e^{(k)^{3}}+X_{5}e^{( k)^{4}}+X_{6}e^{(k)^{5}}+X_{7}e^{(k)^{6}}\Big{]}\big{[}F^{\prime}(\xi)\big{]}^{-1} \\ &+\text{O}\Big{(}\Big{(}e^{k}\Big{)}^{7}\Big{)},\end{split}  \tag{14}\]  
where \(X_{2}\), \(X_{3}\), \(X_{4}\), \(X_{5}\), \(X_{6}\), \(X_{7}\) are unknowns such that  
\[\Big{[}F^{\prime}\Big{(}x^{(k)}\Big{)}\Big{]}^{-1}F^{\prime}\Big{(}x^{(k)} \Big{)}=I.\]  
Then, it can be proven that  
\[ \begin{split}&\Big{[}F^{\prime}\Big{(}x^{(k)}\Big{)}\Big{]}^{-1}= \Big{[}I-2C_{2}e^{(k)}+\Big{(}4C_{2}^{2}-3C_{3}\Big{)}e^{(k)^{2}}+\Big{(}6C_{3 }C_{2}+6C_{2}C_{3}-4C_{4}-8C_{2}^{3}\Big{)}e^{(k)^{3}}+\\ &\Big{(}16C_{2}^{4}+9C_{3}^{2}-12C_{3}C_{2}^{2}-12C_{2}C_{3}C_{2 }-12C_{2}^{2}C_{3}+8C_{4}C_{2}+8C_{2}C_{4}-5C_{5}\Big{)}e^{(k)^{4}}+\\ &\Big{(}-32C_{2}^{5}-18C_{3}^{2}C_{2}+24C_{3}C_{2}^{3}+24C_{2}C_{ 3}C_{2}^{2}+24C_{2}^{2}C_{3}C_{2}-16C_{4}C_{2}^{2}-16C_{2}C_{4}C_{2}+10C_{5}C _{2}\\ &\quad-18C_{3}C_{2}C_{3}-18C_{2}C_{3}^{2}+12C_{4}C_{3}+24C_{3}^{ 2}C_{3}-16C_{2}^{2}C_{4}+12C_{3}C_{4}+10C_{2}C_{5}-6C_{6}\Big{)}e^{(k)^{5}}\\ &\quad+\Big{(}64C_{2}^{6}+36C_{3}^{2}C_{2}^{2}-48C_{3}C_{2}^{4}-4 8C_{2}C_{3}C_{2}^{3}-48C_{2}^{2}C_{3}C_{2}^{2}+32C_{4}C_{2}^{3}+32C_{2}C_{4}C_ {2}^{2}-20C_{5}C_{2}^{2}\\ &\quad+36C_{3}C_{2}C_{3}C_{2}+36C_{2}C_{3}^{2}C_{2}-24C_{4}C_{3} C_{2}-48C_{3}^{2}C_{3}C_{2}+32C_{2}^{2}C_{4}C_{2}-24C_{3}C_{4}C_{2}-20C_{2}C_{5}C_{2} \\ &\quad+12C_{6}C_{2}-48C_{2}^{4}C_{3}-27C_{3}^{3}+36C_{3}C_{2}^{2 }C_{3}+36C_{2}C_{3}C_{2}C_{3}+36C_{2}^{2}C_{3}^{2}-24C_{4}C_{2}C_{3}-24C_{2}C_ {4}C_{3}\\ &\quad+15C_{5}C_{3}-24C_{3}C_{2}C_{4}-24C_{2}C_{3}C_{4}+16C_{4}^ {2}+32C_{2}^{3}C_{4}-20C_{2}^{2}C_{5}+15C_{3}C_{5}+12C_{2}C_{6}-7C_{7}\Big{)} \\ &\quad e^{(k)^{6}}\Big{]}\big{[}F^{\prime}(\xi)\big{]}^{-1}+O\Big{(}e ^{(k)^{7}}\Big{)},\end{split}  \tag{15}\]  
and multiplying expressions (12) and (15), we obtain  
\[ \begin{split}&\Big{[}F^{\prime}\Big{(}x^{(k)}\Big{)}\Big{]}^{-1}F \Big{(}x^{(k)}\Big{)}=e^{(k)}-C_{2}e^{(k)^{2}}+2\Big{(}C_{2}^{2}-C_{3}\Big{)}e^{( k)^{3}}+\Big{(}3C_{3}C_{2}+4C_{2}C_{3}-3C_{4}-4C_{2}^{3}\Big{)}\\ &\quad e^{(k)^{4}}+\Big{(}-6C_{3}C_{2}^{2}-8C_{2}^{2}C_{3}+6C_{3}^ {2}-6C_{2}C_{3}C_{2}+6C_{2}C_{4}+4C_{4}C_{2}+8C_{2}^{4}-4C_{5}\Big{)}e^{(k)^{5}}+ \\ &\quad\Big{(}-16C_{2}^{5}-9C_{3}^{2}C_{2}+12C_{3}C_{2}^{3}+12C_{2 }C_{3}C_{2}^{2}+12C_{2}^{2}C_{3}C_{2}-8C_{4}C_{2}^{2}-8C_{2}C_{4}C_{2}+5C_{5}C _{2}\\ &\quad-12C_{3}C_{2}C_{3}-12C_{2}C_{3}^{2}+8C_{4}C_{3}+16C_{3}^{2 }C_{3}-12C_{2}^{2}C_{4}+9C_{3}C_{4}+8C_{2}C_{5}-5C_{6}\Big{)}e^{(k)^{6}}. \end{split} \]Taking into account \(e^{(k)}=x^{(k)}-\xi\), the expansion of the error at the first step of family (1) is  
\[ \begin{split}& y^{(k)}-\xi=C_{2}e^{(k)^{2}}-2\Big{(}C_{2}^{2}-C_{3} \Big{)}e^{(k)^{3}}-\Big{(}3C_{3}C_{2}+4C_{2}C_{3}-3C_{4}-4C_{2}^{3}\Big{)}e^{(k )^{4}}\\ &-\Big{(}-6C_{3}C_{2}^{2}-8C_{2}^{2}C_{3}+6C_{3}^{2}-6C_{2}C_{3} C_{2}+6C_{2}C_{4}+4C_{4}C_{2}+8C_{2}^{4}-4C_{5}\Big{)}e^{(k)^{5}}\\ &-\Big{(}-16C_{2}^{5}-9C_{3}^{2}C_{2}+12C_{3}C_{2}^{3}+12C_{2}C_{3 }C_{2}^{2}+12C_{2}^{2}C_{3}C_{2}-8C_{4}C_{2}^{2}-8C_{2}C_{4}C_{2}+5C_{5}C_{2}\\ &-12C_{3}C_{2}C_{3}-12C_{2}C_{3}^{2}+8C_{4}C_{3}+16C_{2}^{3}C_{3 }-12C_{2}^{2}C_{4}+9C_{3}C_{4}+8C_{2}C_{5}-5C_{6}\Big{)}e^{(k)^{6}}.\end{split}