# On the Kung-Traub Conjecture for Iterative Methods for Solving Quadratic Equations
## 2 Developments of the Methods
\begin{array}{cccccc}\frac{4}{3}&0&0&0&0&0\\ -\frac{16}{3}&-\frac{16}{9}&0&0&0&0\\ \frac{52}{3}&\frac{112}{9}&\frac{64}{27}&0&0&0\\ -\frac{152}{3}&-\frac{176}{3}&-\frac{640}{27}&-\frac{256}{81}&0&0\\ \frac{416}{3}&\frac{688}{3}&\frac{3968}{27}&\frac{3328}{81}&\frac{1024}{243}&0 \\ -\frac{1088}{3}&-800&-\frac{19456}{27}&-\frac{25600}{81}&-\frac{16384}{243}&- \frac{4096}{729}\end{array} \right)\]  
\[\mathbf{X}=\left( \begin{array}{c}a_{1}\\ a_{2}\\ a_{3}\\ a_{4}\\ a_{5}\\ a_{6}\end{array} \right),\ \mathbf{B}=\left( \begin{array}{c}-1\\ 2\\ -4\\ 8\\ -16\\ 32\end{array} \right)\]whose solutions are given by  
\[\mathbf{X}=\left( \begin{array}{c}a_{1}\\ a_{2}\\ a_{3}\\ a_{4}\\ a_{5}\\ a_{6}\end{array} \right)=\mathbf{A}^{-1}\mathbf{B}=\left( \begin{array}{c}-\frac {3}{4}\\ \frac{9}{8}\\ -\frac{135}{64}\\ \frac{567}{128}\\ -\frac{5103}{512}\\ \frac{24057}{1024}\end{array} \right)\]  
We note that \(\mathbf{A}\) is a lower triangular matrix and the solutions are easily obtained once the first solution is obtained from the first equation. In this way, we obtain a family of higher order I.F.s which we term as higher order 2-point Babajee's Quadratic Iterative Methods for solving quadratic equations (\((r+2)thBQIM\)). The first six members of \((r+2)thBQIM\)'s family in Equation (7) with their error equation are  
1. \(r=1\): 2-point \(3\pi BQIM\) I.F. \[H(\tau,1)=1-\frac{3}{4}(\tau-1)\] \[\psi_{3^{th}BQIM}(x)-x^{*}=\left(2\,c_{2}{}^{2}\right)\left(e^{(k)}\right)^{ 3}+O\Big{(}\left(e^{(k)}\right)^{4}\Big{)}\]
2. \(r=2\): 2-point \(4 thBQIM\) I.F. \[H(\tau,2)=1-\frac{3}{4}(\tau-1)+\frac{9}{8}(\tau-1)^{2}\] \[\psi_{4^{th}BQIM}(x)-x^{*}=\left(5\,c_{2}{}^{3}\right)\left(e^{(k)}\right)^{ 4}+O\Big{(}\left(e^{(k)}\right)^{5}\Big{)}\]
3. \(r=3\): 2-point \(5 thBQIM\) I.F. \[H(\tau,3)=1-\frac{3}{4}(\tau-1)+\frac{9}{8}(\tau-1)^{2}-\frac{135}{64}(\tau-1 )^{3}\] \[\psi_{5^{th}BQIM}(x)-x^{*}=\left(14\,c_{2}{}^{4}\right)\left(e^{(k)}\right) ^{5}+O\Big{(}\left(e^{(k)}\right)^{6}\Big{)}\]
4. \(r=4\): 2-point \(6 thBQIM\) I.F. \[H(\tau,4)=1-\frac{3}{4}(\tau-1)+\frac{9}{8}(\tau-1)^{2}-\frac{135}{64}(\tau- 1)^{3}+\frac{567}{128}(\tau-1)^{4}\] \[\psi_{6^{th}BQIM}(x)-x^{*}=\left(42\,c_{2}{}^{5}\right)\left(e^{(k)}\right) ^{6}+O\Big{(}\left(e^{(k)}\right)^{7}\Big{)}\]
5. \(r=5\): 2-point \(7 thBQIM\) I.F. \[H(\tau,5)=1-\frac{3}{4}(\tau-1)+\frac{9}{8}(\tau-1)^{2}-\frac{135}{64}(\tau- 1)^{3}+\frac{567}{128}(\tau-1)^{4}-\frac{5103}{512}(\tau-1)^{5}\] \[\psi_{7^{th}BQIM}(x)-x^{*}=\left(132\,c_{2}{}^{6}\right)\left(e^{(k)} \right)^{7}+O\Big{(}\left(e^{(k)}\right)^{8}\Big{)}\]
6. \(r=6\): 2-point \(8 thBQIM\) I.F. \[\psi_{3^{th}BQIM}(x)-x^{*}=\left(2\,c_{2}{}^{2}\right)\left(e^{(k)}\right)^{ 3}+O\Big{(}\left(e^{(k)}\right)^{4}\Big{)}\]0\[\psi_{3^{th}BQIM}(x)-x^{*}=\left(2\,c_{2}{}^{2}\right)\left(e^{(k)}\right)^{ 3}+O\Big{(}\left(e^{(k)}\right)^{4}\Big{)}\]1  
We note that the maximum order reached by optimal methods with four function evaluations is eight. We have obtained an eighth order 2-point method with only three function evaluations for solving quadratic equations. This implies that the Kung-Traub conjecture fails for quadratic equations.