# Dynamics and Stability on a Family of Optimal Fourth-Order Iterative Methods
## 6 Numerical Results  
We use the functions proposed in Table 1 to perform some numerical tests in order to analyze the results obtained with different known methods in comparison with the one proposed in the iterative family (11).  
We call the methods used for the numerical tests M1, M2, and M3. M3 is the method that we worked with in this manuscript, and M1 and M2 are two known fourth-order  
\begin{table}
\begin{tabular}{l l} \hline \hline \(f(x)\) & \(\xi\) \\ \hline \(f_{1}(x)=e^{-x}+\cos x\) & \(1.7461395304080124176507030889537802\) \\ \(f_{2}(x)=x^{4}+\sin\left(\frac{x}{x^{2}}\right)-5\) & \(1.4142135623730950488016887242096980\) \\ \(f_{3}(x)=e^{x}-1.5-\tan^{-1}x\) & \(0.7676532662012788981900298911398052\) \\ \(f_{4}(x)=10 xe^{-x^{2}}-1\) & \(1.6796306104284499406749203388379704\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Test functions.  
methods that we use to compare the numerical results; these are the scheme of [PERSON] and the BCMT family, which can be found in [16; 17].  
\[\text{M1}=\left\{\begin{array}{l}y_{n}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})} \\ x_{n+1}=y_{n}-\frac{f(\frac{f(x_{n})}{x_{n}}+2f(y_{n})}{f(x_{n})}\frac{f(y_{n})}{ f^{\prime}(x_{n})},\end{array}\right.\]  
where \(n=0,1,2,\ldots\).  
\[\text{M2}=\left\{\begin{array}{l}y_{n}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n} )}\\ x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}\left[\frac{\beta+2}{3a^{2}} \Big{(}\frac{f(x_{n})}{f(x_{n})+af(y_{n})}\Big{)}^{3}-\frac{2(1+a)}{a^{2}} \Big{(}\frac{f(x_{n})}{f(x_{n})+af(y_{n})}\Big{)}+\frac{3a^{2}+5a+4}{3a^{2}} \right]\end{array}\right.\]  
where \(\alpha=1\) and \(n=0,1,2,\ldots\).  
\[\text{M3}=\left\{\begin{array}{l}y_{n}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n })},\\ x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}\left[\frac{\beta}{6}\Big{(} \frac{f(y_{n})}{f(x_{n})}\Big{)}^{3}+2\Big{(}\frac{f(y_{n})}{f(x_{n})}\Big{)}^ {2}+\Big{(}\frac{f(y_{n})}{f(x_{n})}\Big{)}+1\right].\end{array}\right.\]  
where \(\beta=10\) and \(n=0,1,2,\ldots\).  
The following expression can be used to estimate the theoretical order of convergence:  
\[p\approx COC=\frac{\ln\lvert(x_{n+1}-\xi)/(x_{n}-\xi)\rvert}{\ln\lvert(x_{n} -\xi)/(x_{n-1}-\xi)\rvert},\]  
as presented by [PERSON] and [PERSON] in [18]. However, the principal inconvenience of this COC is that it requires knowledge of the precise root \(\xi\). In order to solve such a problem, in [19], [PERSON] and [PERSON] overhauled the definition of COC so that it would not need the root to be exactly known at all, as shown in the following:  
\[p\approx ACOC=\frac{\ln\lvert x_{n+1}-x_{n}\rvert/\lvert x_{n}-x_{n-1}\rvert} {\ln\lvert x_{n}-x_{n-1}\rvert/\lvert x_{n-1}-x_{n-2}\rvert}.\]  
Calculations were performed by using the Matlab programming software with variable-precision arithmetic, with 200 digits of mantissa and \(\epsilon=10^{-8}\) as the error tolerance. For the computer programs, the following stopping criteria were used:  
(i) \(\lvert x_{n+1}-x_{n}\rvert<\epsilon\) or (ii) \(\lvert f(x_{n+1})\rvert<\epsilon\).  
Table 2 displays the best outcomes for each experiment. We can see how the following method is highly efficient compared to schemes such as [PERSON]'s and the BCMT, which we used for comparison.  
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline \(f(x)\) & \multicolumn{4}{c}{**Iterations**} & \multicolumn{4}{c}{**Computational Time in (sec)**} & \multicolumn{4}{c}{**ACOC**} \\ \hline  & **M1** & **M2** & **M3** & **M1** & **M2** & **M3** & **M1** & **M2** & **M3** \\ \hline \(f_{1}(x)\), 2 & 3 & 3 & 3 & 0.050211 & 0.044647 & 0.053968 & 4.1497 & 3.4747 & 4.1213 \\ \(f_{2}(x)\), 2 & 5 & 4 & 4 & 0.185660 & 0.103332 & 0.096203 & 3.9928 & 3.9584 & 4.0173 \\ \(f_{3}(x)\), 1 & 3 & 3 & 3 & 0.054889 & 0.058569 & 0.061012 & 3.7652 & 3.6640 & 3.8134 \\ \(f_{4}(x)\), 1.9 & 4 & 5 & 4 & 0.074889 & 0.123747 & 0.098324 & 3.9630 & 3.9885 & 3.9945 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Benchmark comparison of fourth-order methods in relation to the convergence order and computational time on the function \(f(x)\).