# Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations
## 4 Numerical Results  
Here, we perform several numerical tests in order to check the theoretical convergence and stability results of CMT(\(\alpha\)) family obtained in previous sections. To do this, we use some stable and unstable methods of (29). These methods are applied on five nonlinear test functions, whose expressions and corresponding roots are  
\[f_{1}(x) =\sin\left(x\right)-x^{2}+1,\quad\xi\approx-0.6367326508,\] \[f_{2}(x) =\cos\left(x\right)-x\exp\left(x\right)+x^{2},\quad\xi\approx 0.639 1540963,\] \[f_{3}(x) =x^{3}+4x^{2}-10,\quad\xi\approx 1.3652300134,\] \[f_{4}(x) =\sqrt{x^{2}+2x+5}-2\sin\left(x\right)-x^{2}+3,\quad\xi\approx 2.3319676559,\] \[f_{5}(x) =\sqrt{x^{4}}+\sin\left(\frac{\pi}{x^{2}}\right)-\frac{3}{16}, \quad\xi\approx-0.9059869793.\]  
Thus, we performed two experiments. In a first experiment, we carried out a efficiency analysis of CMT(\(\alpha\)) family through a comparative study between one of its stable methods and five different methods given in the literature: Newton of order 2, Ostrowski of order 4, and three other methods of order 6 proposed by [PERSON] et al. in [10] (ABA), [PERSON] in [11] (CH), and [PERSON] et al. in [12] (AHR). In a second experiment, we carried out a stability analysis of CMT (\(\alpha\)) family using six of its methods obtained with three good and three bad values of parameter \(\alpha\), in terms of stability.  
In the development of the numerical tests we start the iterations with different initial estimates: close (\(x_{0}\approx\xi\)), far (\(x_{0}\approx 10\xi\)), and very far (\(x_{0}\approx 100\xi\)) to the root \(\xi\), respectively. This allows us to measure, to some extent, how demanding the methods are relative to the initial estimation for finding a solution.  
The calculations are developed in Matlab R2020a programming package using variable precision arithmetics with 200 digits of mantissa. For each method, we analyze the number of iterations (iter) required to converge to the solution, so that the stopping criteria \(|x_{k+1}-x_{k}|<10^{-100}\) or \(|f(x_{k+1})|<10^{-100}\) are satisfied. Note that \(|x_{k+1}-x_{k}|\) represents the error estimation between two consecutive iterations and \(|f(x_{k+1})|\) is the residual error of the nonlinear test function. This stopping criterium does not need the exact solution, on the contrary of absolute error, and differs from recent ones as CESTAC (see in [13]) in the absence of additional calculations or functional evaluations, as \(f(x_{k+1})\) is needed for the following iteration and its absolute values is an efficient control element of the proximity to the exact root, where \(f\) is zero. Indeed, although a precision of one hundred exact digits is not usually necessary in the applications, we employ this value in the stopping criterium as it is useful to check the robustness and effectiveness of the numerical methods.  
To check the theoretical order of convergence (\(p\)), we calculate the approximate computational order of convergence (ACOC) given by [PERSON] and [PERSON] in [14]. In the numerical results presented below, if the ACOC vector inputs do not stabilize their values throughout the iterative process, it is marked as \"-\"; and, if any of the methods used does not reach convergence in a maximum of 50 iterations, it is marked as \"nc\".  
To illustrate the computational efficiency of each used method, the processing time (tcpu) in seconds required by the iterative scheme to converge to the solution is measured. This value is determined as the arithmetic mean of 10 runs of the method.