# Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations
## 2 Convergence of the New Family
\begin{split} f(x_{k})&=f(\xi+e_{k})\\ &=f(\xi)+f^{\prime}(\xi)e_{k}+\dfrac{1}{2!}f^{\prime\prime}(\xi) e_{k}^{2}+\cdots+\dfrac{1}{6!}f^{(vi)}(\xi)e_{k}^{6}+\mathcal{O}(e_{k}^{7})\\ &=f^{\prime}(\xi)\Bigg{[}e_{k}+\dfrac{1}{2!}\dfrac{f^{\prime \prime}(\xi)}{f^{\prime}(\xi)}e_{k}^{2}+\cdots+\dfrac{1}{6!}\dfrac{f^{(vi)}( \xi)}{f^{\prime}(\xi)}e_{k}^{6}+\mathcal{O}(e_{k}^{7})\Bigg{]}\\ &=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}+\mathcal{O}(e_{k}^{7})\Big{]},\end{split}  \tag{15}\]  
and  
\[ \begin{split} f^{\prime}(x_{k})&=f^{\prime}(\xi+e _{k})\\ &=f^{\prime}(\xi)+f^{\prime\prime}(\xi)e_{k}+\dfrac{1}{2!}f^{ \prime\prime\prime}(\xi)e_{k}^{2}+\cdots+\dfrac{1}{5!}f^{(vi)}(\xi)e_{k}^{5}+ \mathcal{O}(e_{k}^{6})\\ &=f^{\prime}(\xi)\Bigg{[}1+\dfrac{f^{\prime\prime}(\xi)}{f^{ \prime}(\xi)}e_{k}+\dfrac{1}{2!}\dfrac{f^{\prime\prime\prime}(\xi)}{f^{\prime}( \xi)}e_{k}^{2}+\cdots+\dfrac{1}{5!}\dfrac{f^{(vi)}(\xi)}{f^{\prime}(\xi)}e_{k}^ {5}+\mathcal{O}(e_{k}^{6})\Bigg{]}\\ &=f^{\prime}(\xi)\Big{[}1+2C_{2}e_{k}+3C_{3}e_{k}^{2}+4C_{4}e_{k }^{2}+5C_{5}e_{k}^{4}+6C_{6}e_{k}^{5}+\mathcal{O}(e_{k}^{6})\Big{]},\end{split}  \tag{16}\]  
where \( C_{q}=\dfrac{1}{q! }\dfrac{f^{(q)}(\xi)}{f^{\prime}(\xi)}\), \(q=2,3,...\)  
Dividing (15) by (16), we get  
\[ \begin{split}\dfrac{f(x_{k})}{f^{\prime}(x_{k})}&=e _{k}-C_{2}e_{k}^{2}+2\Big{(}C_{2}^{2}-C_{3}\Big{)}e_{k}^{3}-\Big{(}4C_{2}^{3}-7C _{2}C_{3}+3C_{4}\Big{)}e_{k}^{4}+\\ &\Big{(}8C_{2}^{4}-20C_{2}^{2}C_{3}+6C_{3}^{2}+10C_{2}C_{4}-4C_{ 5}\Big{)}e_{k}^{5}-\Big{(}16C_{2}^{5}-52C_{2}^{3}C_{3}+\\ & 28C_{2}^{2}C_{4}-17C_{3}C_{4}+C_{2}\Big{(}33C_{3}^{2}-13C_{5} \Big{)}+5C_{6}\Big{)}e_{k}^{6}+\mathcal{O}\Big{(}e_{k}^{7}\Big{)}.\end{split}  \tag{17}\]  
Replacing (17) in the first step of family (1), we have  
\[ \begin{split} y_{k}&=\xi+C_{2}e_{k}^{2}-2\Big{(}C_{2 }^{2}-C_{3}\Big{)}e_{k}^{3}+\Big{(}4C_{2}^{3}-7C_{2}C_{3}+3C_{4}\Big{)}e_{k}^{4}- \\ &\Big{(}8C_{2}^{4}-20C_{2}^{2}C_{3}+6C_{3}^{2}+10C_{2}C_{4}-4C_{ 5}\Big{)}e_{k}^{5}+\Big{(}16C_{2}^{5}-52C_{2}^{3}C_{3}+\\ & 28C_{2}^{2}C_{4}-17C_{3}C_{4}+C_{2}\Big{(}33C_{3}^{2}-13C_{5} \Big{)}+5C_{6}\Big{)}e_{k}^{6}+\mathcal{O}\Big{(}e_{k}^{7}\Big{)}.\end{split}  \tag{18}\]Using Taylor expansion again, similar to (15), to expand \(f(y_{k})\) around \(\xi\), we get  
\[ \begin{split} f(y_{k})&=f^{\prime}(\xi)\Big{[}C_{2}e_{ k}^{2}-2\Big{(}C_{2}^{2}-C_{3}\Big{)}e_{k}^{3}+\Big{(}5C_{2}^{3}-7C_{2}C_{3}+3C_{4} \Big{)}e_{k}^{4}-\\ &\quad 2\Big{(}6C_{2}^{4}-12C_{2}^{2}C_{3}+3C_{3}^{2}+5C_{2}C_{4}-2C_{ 5}\Big{)}e_{k}^{5}+\Big{(}28C_{2}^{5}-73C_{2}^{3}C_{3}+\\ &\quad 34C_{2}^{2}C_{4}-17C_{3}C_{4}+C_{2}\Big{(}37C_{3}^{2}-13C_{5} \Big{)}+5C_{6}\Big{)}e_{k}^{6}+\mathcal{O}\Big{(}e_{k}^{7}\Big{)}\Big{]}.\end{split}  \tag{19}\]  
With (15), (18), and (19), we calculate the divided difference operator defined in (2), obtaining  
\[ \begin{split} f[x_{k},y_{k}]&=f^{\prime}(\xi)\Big{[} 1+C_{2}e_{k}+\Big{(}C_{2}^{2}+C_{3}\Big{)}e_{k}^{2}-\Big{(}2C_{2}^{3}-3C_{2}C_ {3}-C_{4}\Big{)}e_{k}^{3}+\\ &\quad\Big{(}4C_{2}^{4}-8C_{2}^{2}C_{3}+2C_{3}^{2}+4C_{2}C_{4}+C_ {5}\Big{)}e_{k}^{4}+\Big{(}-8C_{2}^{5}+20C_{2}^{3}C_{3}-\\ &\quad 11C_{2}^{2}C_{4}+5C_{3}C_{4}+C_{2}\Big{(}-9C_{3}^{2}+5C_{5} \Big{)}+C_{6}\Big{)}e_{k}^{5}+\mathcal{O}\Big{(}e_{k}^{6}\Big{)}\Big{]}.\end{split}  \tag{20}\]  
Then, substituting (15), (16), (18), and (20) in the second step of family (1), we have  
\[ \begin{split} z_{k}&=\xi+\Big{(}C_{2}^{3}-C_{2}C_{3 }\Big{)}e_{k}^{4}-2\Big{(}2C_{2}^{4}-4C_{2}^{2}C_{3}+C_{3}^{2}+C_{2}C_{4}\Big{)} e_{k}^{5}+\\ &\quad\Big{(}10C_{2}^{5}-30C_{2}^{3}C_{3}+12C_{2}^{2}C_{4}-7C_{3 }C_{4}+3C_{2}\Big{(}6C_{3}^{2}-C_{5}\Big{)}\Big{)}e_{k}^{6}+\mathcal{O}\Big{(} e_{k}^{7}\Big{)}.\end{split}  \tag{21}\]  
Using [PERSON] series once again, similar to (15), to expand \(f(z_{k})\) around \(\xi\), we get  
\[