## 4 Numerical Results  
Now, the new family Equation (4) without memory and the corresponding family Equation (17) with memory are employed to solve some nonlinear equations and compared with several known iterative methods. All algorithms are implemented using Symbolic Math Toolbox of MATLAB 7.0. For demonstration, we have selected three methods displayed below. [PERSON]'s methods without memory ( KM4, see [9] ):  
\[\left\{ \begin{aligned} y_{n}=& x_{n}-\frac{f(x_{n})}{f^{ \prime}(x_{n})},\\ x_{n+1}=& y_{n}-\frac{f(x_{n})+\beta f(y_{n})}{f(x_ {n})+(\beta-2)f(y_{n})}\frac{f(y_{n})}{f^{\prime}(x_{n})}\end{aligned} \right. \tag{39}\]  
where \(\beta\in R\). Bi-Wu-Ren method without memory ( BRM8, see [10] ):  
\[\left\{ \begin{aligned} y_{n}=& x_{n}-\frac{f(x_{n})}{f^{ \prime}(x_{n})}\\ z_{n}=& y_{n}-h(\frac{f(y_{n})}{f(x_{n})})\frac{f(y_ {n})}{f^{\prime}(x_{n})}\\ x_{n+1}=& z_{n}-\frac{f(x_{n})+(\gamma+2)f(z_{n}) }{f(x_{n})+\gamma f(z_{n})}\frac{f(z_{n})}{f[z_{n},y_{n}]+f[x_{n},x_{n},z_{n} ](z_{n}-y_{n})}\end{aligned} \right. \tag{40}\]  
where \(h(t)\)is a real-valued function satisfying the conditions\(h(0)=1,h^{\prime}(0)=2,h^{\prime\prime}(0)=10\), \(|h^{\prime\prime\prime}(0)|<\infty\) and \(\gamma\in R\). Petkovic-Ilic-Dzunic method with memory ( PD, see [12] )  
\[\left\{ \begin{aligned} y_{n}=& x_{n}-\frac{f(x_{n})}{f [x_{n},z_{n}]},\\ x_{n+1}=& y_{n}-\frac{f(y_{n})}{f[x_{n},z_{n}]}\left(1 +\frac{f(y_{n})}{f(x_{n})}+\frac{f(y_{n})}{f(z_{n})}\right)\end{aligned} \right. \tag{41}\]  
where \(z_{n}=x_{n}-\gamma_{n}f(x_{n})\). The parameter\(\gamma_{n}\) can be calculated by the following three formulas:  
\[\gamma_{n}=(x_{n}-y_{n-1})/(f(x_{n})-f(y_{n-1})) \tag{42}\]  
\[\gamma_{n}=1/(f[x_{n},y_{n-1}]+f[x_{n},x_{n-1}]-f[x_{n-1},y_{n-1}]) \tag{43}\]  
The absolute errors\(|x_{k}-a|\)in the first four iterations are given in Tables 1-4, where \(a\) is the exact root computed with 2400 significant digits. The computational order of convergence \(\rho\) is defined by [19]:  
\[\rho\approx\frac{\ln(\left|x_{n+1}-x_{n}\right|/\left|x_{n}-x_{n-1}\right|)}{ \ln(\left|x_{n}-x_{n-1}\right|/\left|x_{n-1}-x_{n-2}\right|)} \tag{44}\]  
The iterative processes of the Equations (4) and (17) are given in Figure 1, where Equation (4) (n=1) is one-point method. The parameters of the Equations (4) and (17) are \(\lambda=\lambda_{0}=1.0\). The initial value is \(x_{0}=-1.3\). The stopping criterium is \(|f(x)|<10^{-150}\). We will call \(f(x)\) the nonlinear residual or residual. The Figure 1 is a semilog plot of residual history, the norm of the nonlinear residual against the iteration number. Following test functions are used:  
\[f_{1}(x)=xe^{x^{2}}-\sin^{2}(x)+3\cos(x)+5,\ a\approx-1.2076478271309189,\,x_{0}=-1.3.\]  
\[f_{2}(x)=x^{5}+x^{4}+4x^{2}-15,\ a\approx 1.3474280989683050,\,x_{0}=1.6.\] \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Methods** & \(|x_{1}-a|\) & \(|x_{2}-a|\) & \(|x_{3}-a|\) & \(\rho\) \\ \hline Equation (4) \(n=2,\lambda=0.5\) & 0.32719E\(-\)4 & 0.57076E\(-\)18 & 0.52848E\(-\)73 & 4.0000005 \\ Equation (4) \(n=2,\lambda=1\) & 0.58111E\(-\)4 & 0.71445E\(-\)17 & 0.16328E\(-\)68 & 3.9999938 \\ \(KM4,\beta=2\) & 0.24269E\(-\)3 & 0.13078E\(-\)13 & 0.11033E\(-\)54 & 3.9999864 \\ \(BRM8,h(t)=1+2t+5t^{2},\gamma=1\) & 0.40513E\(-\)6 & 0.32351E\(-\)48 & 0.53484E\(-\)385 & 8.0000001 \\ Equation (4) \(n=3,\lambda=1\) & 0.22673E\(-\)8 & 0.83510E\(-\)70 & 0.28282E\(-\)561 & 8.0000000 \\ Equation (4) \(n=3,\lambda=1.5\) & 0.18012E\(-\)9 & 0.75259E\(-\)83 & 0.69916E\(-\)670 & 8.0000000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical results for \(f_{1}(x)\) by the methods without memory.