# A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve
## 4 Numerical Results  
Here, we check the effectiveness of our newly proposed iterative methods. We employ some elements of our schemes: specifically scheme (18) for \(\alpha=1\), \(\alpha=\frac{1}{2}\), \(\alpha=\frac{1}{10}\) and (19) for \(\alpha=1\), \(\alpha=\frac{1}{2}\), \(\alpha=\frac{1}{4}\), denoted by \(MHS1\), \(MHS2\), \(MHS3\), \(MHS1\), \(MHS2\) and \(MSHS3\), respectively, to solve nonlinear equations mentioned in Examples 1-5. We contrast our methods with existing schemes of same order: (3), (4), (5), (6) and (7), denoted by \(CS\), \(HS\), \(OS\), \(NS\) and \(CN\), respectively. In Tables 1-3, we display the values of absolute residual errors \(|f(x_{\sigma+1})|\), number of iterations in order to attain the desired accuracy, and the absolute errors \(|x_{\sigma+1}-x_{\sigma}|\). Further, it is known that formula [20],  
\[\rho\approx\frac{\ln|(x_{\sigma+1}-\xi)/(x_{\sigma}-\xi)|}{\ln|(x_{\sigma}- \xi)/(x_{\sigma-1}-\xi)|}.\]  
displays the computational order of convergence, COC. In this estimation, we need to know the exact zero \(\xi\) of the nonlinear function to be solved. But, there are several practical situations where the exact solution is not accessible. Therefore, [PERSON] and [PERSON] in [21], suggested the following estimation of the order of convergence, known as ACOC  
\[\rho\approx\frac{\ln|\xi_{\sigma+1}/\xi_{\sigma}|}{\ln|\xi_{\sigma}/\xi_{ \sigma-1}|}, \tag{27}\]  
where \(\xi_{\sigma}=x_{\sigma}-x_{\sigma-1}\) and exact zero is not needed. In tables, \({}^{\star}\), \(Div\) and \(F\) stand for converge to undesired root, case of divergence and case of failure, respectively. Moreover, _due_ stands for does not exist. During the current numerical experiments with programming language Mathematica (Version-9), all computations have been done with 1000 digits of mantissa, which minimize round-off errors. In tables, \((b_{1}\pm b_{2})\) denotes \((b_{1}\times 10^{\pm b_{2}})\). The configuration of the computer used is given below:  
Processor: Intel(R) Core(TM) i7-4790 CPU @ 3.60 GHz  
RAM: 8:00 GB  
System type: 64-bit-Operating System, x64-based processor. Now, we explore the performance of our proposed methods, in comparison with known ones, on some chemical and engineering problems, that can be found in [22; 23; 24; 25], among others. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline
**Methods** & **LG. ** & \(\left[f(\mathbf{x})\right]\) & **CS** & **_HS_** & **OS** & **ONS** & **_CN_ & **_MISS_** & **_MISS_** & **_MISS_** & **_MISS_** & **_MSHSS_** & **_MSHSS_** & **_MSHSS_** \\  & & \(\left[\mathbf{x}_{\
u+1}-\mathbf{x}_{\
u}\right]\) & & & & & & & & & & & & & \\ \hline Ex. (1) & 1.73 & \(\left[f(\mathbf{x})\right]\) & \(F\) & \(F\) & \(F\) & \(F\) & \(F\) & \(1.3(-15)\) & \(30.10\) & \(2.0(-6)\) & \(7.7(-10)\) & \(2.0(-67)\) & \(2.2(-45)\) \\  & \(\left[\mathbf{y}_{\
u}-\mathbf{x}\right]\) & \(F\) & \(F\) & \(F\) & \(F\) & \(F\) & \(21(-7)\) & \(10(-4)\) & \(8.9(-3)\) & \(16. (-50)\) & \(2.6(-33)\) & \(2.7(-22)\) \\ \hline \multirow{2}{*}{Ex. (1)} & \multirow{2}{*}{log5} & \multirow{2}{*}{log5} & \(\left[f(\mathbf{x})\right]\) & \(F\) & \(F\) & \(F\) & \(F\) & \(3.2(-97)\) & \(2.0(-226)\) & \(3.8(-79)\) & \(2.6(-12)\) & \(2.7(-40)\) & \(2.9(-924)\) \\  & & & \(\left[\mathbf{y}_{\
u}-\mathbf{x}\right]\) & \(F\) & \(F\) & \(F\) & \(F\) & \(F\) & \(3.3(-8)\) & \(6.5(-76)\) & \(1.7(-9)\) & \(1.5(-40)\) & \(1.5(-34)\) & \(7.4(-308)\) \\ \cline{2-13}  & & \(\left[\mathbf{y}_{\
u}-\mathbf{x}\right]\) & \(Dip\) & \(98. (-2)\) & \(7.2(-464)\) & \(Dip\) & \(Dip\) & \(28(-97)\) & \(1.4(-28)\) & \(1.2(-17)\) & \(2.1(-12)\) & \(4.7(-40)\) & \(1.0(-924)\) \\  & & \(\left[\mathbf{y}_{\
u}-\mathbf{x}\right]\) & \(Dip\) & \(9.3(-1)\) & \(2.2(-154)\) & \(Dip\) & \(Dip\) & \(3.4(-2)\) & \(57(-76)\) & \(12(-99)\) & \(1.4(-40)\) & \(1.9(-134)\) \\ \hline \multirow{2}{*}{Ex.