# On the Kung-Traub Conjecture for Iterative Methods for Solving Quadratic Equations
## 3 Convergence Analysis  
**Theorem 3**. : _Let a sufficiently smooth function \(f:D\subset\mathbb{R}\to\mathbb{R}\) has a simple root \(x^{*}\) in the open interval \(D\). Then the six members of 2-point \((r+2)\)thBQIM's family in Equation (7) (\(r=1,2,3,4,5,6\)) are of local 3 rd to 8 th order convergence, respectively._  
Proof. : We will prove the 3 rd order convergence of the 2-point 3 rdBQIM I.F. and 8 th order convergence of the 2-point 8 thBQIM I.F. The proofs for the 2-point 4 th to 7 th order I.F.s follow on similar lines. It is easy to see that for a quadratic function,  
\[f(x)=f^{\prime}(x^{*})\Bigg{[}e^{(k)}+c_{2}(e^{(k)})^{2}\Bigg{]}\]  
and  
\[f^{\prime}(x)=f^{\prime}(x^{*})\Bigg{[}1+2c_{2}\,e^{(k)}\Bigg{]}\]  
By Taylor expansion and using computer algebra software as Maple  
\[ \begin{split} u(x)&=e^{(k)}-c_{2}\big{(}e^{(k)} \big{)}^{2}+2\,c_{2}{}^{2}\big{(}e^{(k)}\big{)}^{3}-4\,c_{2}{}^{3}\big{(}e^{(k )}\big{)}^{4}+8\,c_{2}{}^{4}\big{(}e^{(k)}\big{)}^{5}-16\,c_{2}{}^{5}\big{(}e ^{(k)}\big{)}^{6}\\ &+32\,c_{2}{}^{6}\big{(}e^{(k)}\big{)}^{7}-64\,c_{2}{}^{7}\big{(} e^{(k)}\big{)}^{8}+128\,c_{2}{}^{8}\big{(}e^{(k)}\big{)}^{9}+...\end{split}  \tag{8}\]  
so that  
\[ \begin{split}\tau=1&-\frac{4}{3}\,c_{2}\,e^{(k)}+4 \,c_{2}{}^{2}\big{(}e^{(k)}\big{)}^{2}-\frac{32}{3}\,c_{2}{}^{3}\big{(}e^{(k)} \big{)}^{3}+\frac{80}{3}\,c_{2}{}^{4}\big{(}e^{(k)}\big{)}^{4}-64\,c_{2}{}^{5} \big{(}e^{(k)}\big{)}^{5}+\frac{448}{3}\,c_{2}{}^{6}\big{(}e^{(k)}\big{)}^{6} \\ &-\frac{1024}{3}\,c_{2}{}^{7}\big{(}e^{(k)}\big{)}^{7}+768\,c_{2}{ }^{8}\big{(}e^{(k)}\big{)}^{8}+...\end{split}  \tag{9}\]  
Now,  
\[H(\tau,1)=1+c_{2}\,e^{(k)}-3\,c_{2}{}^{2}\big{(}e^{(k)}\big{)}^{2}+8\,c_{2}{}^ {3}\big{(}e^{(k)}\big{)}^{3}+... \tag{10}\]  
Using Equations (8) and (10), we have  
\[u(x)\ H(\tau,1)=e^{(k)}-2\,c_{2}{}^{2}\big{(}e^{(k)}\big{)}^{3}+O\left(\big{(} e^{(k)}\big{)}^{4}\right)\]  
which leads to the error equation for the 2-point 3 rdBQIM I.F. Similarly,  
\[H(\tau,6)=1+c_{2}\,e^{(k)}-c_{2}{}^{2}\big{(}e^{(k)}\big{)}^{2}+c_{2}{}^{3} \big{(}e^{(k)}\big{)}^{3}-c_{2}{}^{4}\big{(}e^{(k)}\big{)}^{4}+c_{2}{}^{5} \big{(}e^{(k)}\big{)}^{5}-c_{2}{}^{6}\big{(}e^{(k)}\big{)}^{6}-428c_{2}{}^{7} \big{(}e^{(k)}\big{)}^{7}+... \tag{11}\]  
Using Equations (8) and (11), we have  
\[u(x)\ H(\tau,6)=e^{(k)}-429\,c_{2}{}^{7}\big{(}e^{(k)}\big{)}^{8}+O\left(\big{(} e^{(k)}\big{)}^{9}\right)\]which leads to the error equation for the 2-point \(8 thBQIM\) I.F. We next prove the local convergence of the 2-point \((r+2)thBQIM\)'s family for any \(r\). **Theorem 4**. : _Let a sufficiently smooth function \(f:D\subset\mathbb{R}\rightarrow\mathbb{R}\) has a simple root \(x^{*}\) in the open interval \(D\). Then the members of 2-point \((r+2)\)thBQIM's family in Equation (7) are of local \((r+2)th\) order convergence._  
Proof. : We prove this result by induction. The case \(r=1\) corresponds to the \(3 rdBQIM\) I.F. Assume the 2-point \((r+2)thBQIM\) family has order of convergence of \((r+2)\). Then it satisfies the error equation  
\[\psi_{(r+2)^{th}BQIM}(x)-x^{*}=C_{r}\;c_{2}{}^{r+1}(e^{(k)})^{r+2}+O\left((e^{( k)})^{r+3}\right) \tag{12}\]  
where \(C_{r}\) is the asymptotic error constant. Assume that Equation (12) holds for \(r=m\).