# Local Convergence Study for an Iterative Scheme with a High Order of Convergence
## 4 Numerical Results  
In this section, several numerical examples are provided to validate the efficiency of our analysis for local convergence (Examples 1-3). Figure 1: Computational Efficiency Index \((CEI)\). **Example 1**. : _The van der Waals equation of state (for a vapor) is (see [19; 20]):_  
\[\Big{(}P+\frac{a}{V^{2}}\Big{)}(V-b)=RT,\]  
_This equation can be rearranged to the form:_  
\[PV^{3}-(Pb+RT)V^{2}+aV-ab=0\]  
_in \(V\), where every constant possesses physical significance, with values that can be found in [19]. Let \(X=\mathbb{R}\) and \(D=[0,100]\). Choose \(P=10,000\) kPa and \(T=800\) K. Then, the solution \(V\) from the resulting equation is \(36.9167\). Applying the conditions given in Theorems 1 and 2, we obtain \(L_{0}=0.1151\), \(L=0.1215\) and \(M=1.12\). The radii are:_  
\[(C_{1})-(C_{3}): \alpha=0.10,\quad r=0.1575<r_{1}=3.0365\] \[(C_{4}): \alpha=0,\quad\overline{r}=0.6397<\overline{r_{1}}=3.5000\]  
_The uniqueness radius is:_  
\[R<\frac{2}{L_{0}}=\frac{2}{0.1151}=17.38\]  
**Example 2** (Beam Design Problem). : _Here, we examine a beam positioning problem (see [21]), where a beam of length \(l\) units is positioned against the edge of a cubical box with one end placed on the floor and the other end positioned against the wall. The box, with sides of \(1\) units each, is positioned on the floor braced against the wall and one end rests on the floor, as shown in Figure 2._  
_The triangles formed by the floor, box and beam (lower) and by the box, wall, and beam (upper) are similar._  
_In this case, the measurement from the base of the wall to where the beam touches the floor is \(l=4\). Suppose \(y\) represents the length measured from the floor to the edge of the box along the length of the beam, and let \(x\) be the distance measured from the bottom of the box to the lower edge of the beam. Let \(X=\mathbb{R}\) and \(D=|0,3[\). Then, we have:_  
\[x^{4}+2x^{3}-14x^{2}+2x+1=0.\]  
_One positive solution of the equation is the zero \(x=2.7609\), and the first derivative is \(4x^{3}+6x^{2}-28x+2=0\). Applying the conditions given in Theorems 1 and 2, we obtain \(L_{0}=1.87\), \(L=1.9723\) and \(M=1.1634\). The radii are:_  
\[(C_{1})-(C_{3}): \alpha=0,\quad r=0.0100<r_{1}=0.1916\] \[(C_{4}): \alpha=0,\quad\overline{r}=0.0208<\overline{r_{1}}=0.2143\]  
\[\Big{(}P+\frac{a}{V^{2}}\Big{)}(V-b)=RT,\]0 Beam problem. _The uniqueness radius is:_  
\[\Big{(}P+\frac{a}{V^{2}}\Big{)}(V-b)=RT,\]1  
In Examples 3 and 4, we choose two members of the described iterative method and perform numerical tests to compare them with iterative schemes [13; 14; 15]. Numerical calculations were executed on a computer with 16 MB RAM and Intel Core i7 processor using MATLAB R2024a software, using variable precision arithmetics with 2000 digits of mantissa. Since we are comparing schemes of high order of convergence, the error tolerance is set at \(\epsilon=10^{-100}\), to reduce the risk of significant error accumulation, and therefore, ensure more accurate results. For each method, the error estimate between two consecutive iterations is \(\|x_{k+1}-x_{k}\|\) and the residual error is \(\|\mathcal{T}(x_{k+1})\|\). The stopping criterium is set at \(\|x_{k+1}-x_{k}\|+\|\mathcal{T}(x_{k+1})\|<\epsilon\). The necessary number of iterations (iter) to converge to the solution is provided. To validate the theoretical order of convergence \(p\), the ACOC is computed. The execution time, denoted as e-time, required to achieve convergence is displayed (in seconds), based on the average of 10 successive runs for each scheme. **Example 3**. : _Let us examine a system of differential equations that describe the motion of an object given by (see [22]):_  
\[\Big{(}P+\frac{a}{V^{2}}\Big{)}(V-b)=RT,\]2  
_with initial conditions \(\mathcal{T}_{1}(0)=\mathcal{T}_{2}(0)=\mathcal{T}_{3}(0)=0\). Let us assume \(\mathcal{T}=(\mathcal{T}_{1},\mathcal{T}_{2},\mathcal{T}_{3})\). Let \(X=Y=R^{3}\), \(D=\tilde{U}(0,1)\) and \(x^{*}=(0,0,0)^{T}\).