# On the Kung-Traub Conjecture for Iterative Methods for Solving Quadratic Equations
## 2 Developments of the Methods  
Let \(x^{(k+1)}=\psi(x^{(k)})\) define an Iterative Function (I.F.). **Definition 1**. : _[_7_]_ _If the sequence \(\{x^{(k)}\}\) tends to a limit \(x^{*}\) in such a way that_  
\[\lim_{n\rightarrow\infty}\frac{x^{(k+1)}-x^{*}}{(x^{(k)}-x^{*})^{p}}=C\]  
_for \(p\geq 1\), then the order of convergence of the sequence is said to be \(p\), and \(C\) is known as the asymptotic error constant. If \(p=1\), \(p=2\) or \(p=3\), the convergence is said to be linear, quadratic or cubic, respectively. Let \(e^{(k)}=x^{(k)}-x^{*}\), then the relation_  
\[e^{(k+1)}=C\ (e^{(k)})^{p}+O\Big{(}(e^{(k)})^{p+1}\Big{)}=O\Big{(}(e^{(k)})^{p} \Big{)} \tag{1}\]  
_is called the error equation. The value of \(p\) is called the order of convergence of the method._  
**Definition 2**. : _[_8_]_ _The Efficiency Index is given by_  
\[EI=p^{\frac{1}{d}} \tag{2}\]  
_where \(d\) is the total number of new function evaluations (the values of \(f\) and its derivatives) per iteration._  
Let \(x^{(k+1)}\) be determined by new information at \(x^{(k)},\phi_{1}(x^{(k)}),...,\phi_{1}(x^{(k)}),i\geq 1\)  
No old information is reused. Thus,  
\[x^{(k+1)}=\psi(x^{(k)},\phi_{1}(x^{(k)}),...,\phi_{1}(x^{(k)})) \tag{3}\]  
Then \(\psi\) is called a multipoint I.F without memory. **[PERSON]'s Conjecture**[9]  
Let \(\psi\) be an I.F. without memory with \(d\) evaluations. Then  
\[p(\psi)\leq p_{Opt}=2^{d-1} \tag{4}\]  
where \(p_{opt}\) is the maximum order. The second order Newton I.F. (\(2^{nd}NR\)) is given by  
\[\psi_{2^{nd}NR}(x)=x-u(x),\,u(x)=\frac{f(x)}{f^{\prime}(x)} \tag{5}\]  
The \(2^{nd}NR\) I.F. is a 1-point I.F. with 2 functions evaluations and it satisfies the [PERSON] conjecture with \(d=2\). Thus, \(EI_{2^{nd}NR}=1.414\). The 2-point fourth order Jarratt I.F. (\(4 th\)JM) [10] is given by  
\[\psi_{4^{th}JM}(x)=x-u(x)\left(\frac{3\tau+1}{6\tau-2}\right)\quad\tau=\frac{ f^{\prime}\left[x-\frac{2}{3}u(x)\right]}{f^{\prime}(x)} \tag{6}\]  
The \(4 th\)JM I.F. with 3 function evaluations satisfies the [PERSON] conjecture with \(d=3\). According to [PERSON]'s conjecture, it is not possible to obtain an I.F. with three function evaluations reaching an order greater than four. We show that this conjecture fails for quadratic functions. We consider the quadratic function \(f(x)=\kappa_{2}x^{2}+\kappa_{1}x+\kappa_{0}\), where \(\kappa_{2}\
eq 0\), \(\kappa_{1}\), \(\kappa_{0}\) are constants. Consider the following I.F. for quadratic function:  
\[\psi_{(r+2)^{th}BQIM}(x)=x-u(x)\ H(\tau,r) \tag{7}\]  
where  
\[H(\tau,r)=1+\sum_{i=1}^{r}\ a_{i}(\tau-1)^{i}\]  
where \(a_{i}\)'s are constants. The error equation of the I.F. defined by Equation (7) for \(r=6\) is given by  
\[\psi(x)-x^{*}=\left(\left(\frac{4}{3}\,a_{1}+1\right)c_{2}\right) \left(e^{(k)}\right)^{2}+\left(\left(-\frac{16}{3}\,a_{1}-\frac{16}{9}\,a_{2}- 2\right)c_{2}{}^{2}\right)\left(e^{(k)}\right)^{3}\] \[e^{(k+1)}=C\ (e^{(k)})^{p}+O\Big{(}(e^{(k)})^{p+1}\Big{)}=O\Big{(}(e^{(k)})^{p} \Big{)} \tag{1}\]0 \[e^{(k+1)}=C\ (e^{(k)})^{p}+O\Big{(}(e^{(k)})^{p+1}\Big{)}=O\Big{(}(e^{(k)})^{p} \Big{)} \tag{1}\]1 \[e^{(k+1)}=C\ (e^{(k)})^{p}+O\Big{(}(e^{(k)})^{p+1}\Big{)}=O\Big{(}(e^{(k)})^{p} \Big{)} \tag{1}\]2 \[e^{(k+1)}=C\ (e^{(k)})^{p}+O\Big{(}(e^{(k)})^{p+1}\Big{)}=O\Big{(}(e^{(k)})^{p} \Big{)} \tag{1}\]3 \[e^{(k+1)}=C\ (e^{(k)})^{p}+O\Big{(}(e^{(k)})^{p+1}\Big{)}=O\Big{(}(e^{(k)})^{p} \Big{)} \tag{1}\]4  
where \(c_{2}=\frac{f^{\prime\prime}(x^{*})}{f^{\prime}(x^{*})}\), \(f^{\prime}(x^{*})\
eq 0\). Eliminating the terms in \(\left(e^{(k)}\right)^{j}\), \(j=2,3,4,5,6,7\) we obtain a system of 6 linear equations with 6 unknowns:  
\[e^{(k+1)}=C\ (e^{(k)})^{p}+O\Big{(}(e^{(k)})^{p+1}\Big{)}=O\Big{(}(e^{(k)})^{p} \Big{)} \tag{1}\]5  
where  
\[\mathbf{A}=\left(