# Convergence and Stability of a New Parametric Class of Iterative Processes for Nonlinear Systems
## 2 Family ACTV for Nonlinear Systems
\]\[F\Big{(}y^{(k)}\Big{)}= F^{\prime}(\xi)\Big{[}\Big{(}y^{(k)}-\xi\Big{)}+C_{2}\Big{(}y^{(k)} -\xi\Big{)}^{2}\Big{]}+O\Big{(}\Big{(}y^{(k)}-\xi\Big{)}^{3}\Big{)} \tag{18}\] \[= F^{\prime}(\xi)\Big{[}C_{2}e^{(k)^{2}}+2\Big{(}C_{3}-C_{2}^{2} \Big{)}e^{(k)^{3}}+\Big{(}3C_{4}+5C_{2}^{3}-3C_{3}C_{2}-4C_{2}C_{3}\Big{)}e^{( k)^{4}}\] \[+\Big{(}4C_{5}-6C_{2}C_{4}+10C_{2}^{2}C_{3}-6C_{3}^{2}-4C_{4}C_{2 }+8C_{2}C_{3}C_{2}-12C_{2}^{4}+6C_{3}C_{2}^{2}\Big{)}e^{(k)^{5}}\] \[+\Big{(}28C_{2}^{5}-27C_{3}^{2}C_{3}+16C_{2}C_{3}^{2}+15C_{2}^{2} C_{4}-9C_{3}C_{4}-8C_{2}C_{5}+5C_{6}-16C_{2}^{2}C_{3}C_{2}\] \[+9C_{3}^{2}C_{2}+11C_{2}C_{4}C_{2}-5C_{5}C_{2}-18C_{2}C_{3}C_{2}^{ 2}+8C_{4}C_{2}^{2}-12C_{3}C_{2}^{3}-8C_{4}C_{3}\] \[+12C_{3}C_{2}C_{3})e^{(k)^{6}}\Big{]}+O\Big{(}e^{(k)^{7}}\Big{)}.\]  
Considering the results obtained in (16)-(18), the second step has as the error equation  
\[z^{(k)}-\xi=\Big{(}C_{2}^{3}-C_{3}C_{2}\Big{)}e^{(k)^{4}}+\Big{(} -2C_{4}C_{2}+2C_{2}^{2}C_{3}+2C_{2}C_{3}C_{2}+4C_{3}C_{2}^{2}-2C_{3}^{2}-4C_{2 }^{4}\Big{)}e^{(k)^{5}}\] \[+\Big{(}10C_{2}^{5}-5C_{2}^{3}C_{3}-8C_{2}^{2}C_{3}C_{2}-8C_{2}C_ {3}C_{2}^{2}-9C_{3}C_{2}^{3}+4C_{2}C_{3}^{2}+6C_{3}^{2}C_{2}+8C_{3}C_{2}C_{3}+ 3C_{2}^{2}C_{4}\] \[+3C_{2}C_{4}C_{2}+6C_{4}C_{2}^{2}-3C_{3}C_{4}-4C_{4}C_{3}-3C_{5}C _{2}\Big{)}e^{(k)^{6}}.\]  
To obtain the error equation of the third step, we need the calculation of \([x^{k},y^{k};F]^{-1}\) and \(F\Big{(}z^{(k)}\Big{)}\) since the other elements were previously obtained. Following a process similar to that seen in (17) and developing only to order two, we have that  
\[[x^{k},y^{k};F]^{-1}=I-C_{2}e^{(k)}-C_{3}e^{(k)^{2}}+O\Big{(}e^{(k)^{3}}\Big{)}.\]  
For the calculation of \(F\Big{(}z^{(k)}\Big{)}\), we are only interested in the terms up to order six, so applying what we see in formula (18), we obtain  
\[= F^{\prime}(\xi)\Big{[}C_{2}e^{(k)^{2}}+2\Big{(}C_{3}-C_{2}^{2} \Big{)}e^{(k)^{3}}+\Big{(}3C_{4}+5C_{2}^{3}-3C_{3}C_{2}-4C_{2}C_{3}\Big{)}e^{( k)^{4}}\]0  
The resulting error equation for the family of methods (9) is  
\[= F^{\prime}(\xi)\Big{[}C_{2}e^{(k)^{2}}+2\Big{(}C_{3}-C_{2}^{2} \Big{)}e^{(k)^{3}}+\Big{(}3C_{4}+5C_{2}^{3}-3C_{3}C_{2}-4C_{2}C_{3}\Big{)}e^{( k)^{4}}\]1  
\(\square\)  
Once the convergence order of the proposed class of the iterative method is proven, we undertake a complexity analysis, taking into account the cost of solving the linear systems and the rest of the computational effort, not only of the proposed class but also of [PERSON]'s and those schemes presented in the Introduction, with the same order six. In order to calculate it, let us remark that the computational cost (in terms of products/quotients) of solving a linear system of size \(n\times n\) is  
\[= F^{\prime}(\xi)\Big{[}C_{2}e^{(k)^{2}}+2\Big{(}C_{3}-C_{2}^{2} \Big{)}e^{(k)^{3}}+\Big{(}3C_{4}+5C_{2}^{3}-3C_{3}C_{2}-4C_{2}C_{3}\Big{)}e^{( k)^{4}}\]2  
but if another linear system is solved with the same coefficient matrix, then the cost increases only in \(n^{2}\) operations. Moreover, a matrix-vector product corresponds to \(n^{2}\) operations. From these bases, the computational effort of each scheme is presented in \[= F^{\prime}(\xi)\Big{[}C_{2}e^{(k)^{2}}+2\Big{(}C_{3}-C_{2}^{2} \Big{)}e^{(k)^{3}}+\Big{(}3C_{4}+5C_{2}^{3}-3C_{3}C_{2}-4C_{2}C_{3}\Big{)}e^{( k)^{4}}\]3 As the ACTV class depends on parameter \(\alpha\), we consider \(\alpha=1\), as this value eliminates one of the terms in the iterative expression, reducing the complexity. Observing the data in Table 1, there seems to be a great difference among [PERSON]'s and sixth-order methods, with \(PSH6_{2}\), \(B_{6}\) and \(C6_{2}\) being the most costly, in this order. Our proposed scheme ACTV for \(\alpha=1\), stays in the middle values of the table. However, this must be seen in contrast with the order of convergence \(p\) of each scheme. With this point of view, the comparison among the methods is more clear.