# Extended High Order Algorithms for Equations under the Same Set of Conditions
## 3 Comparison of Attraction Basins  
Comparison of the dynamical qualities of SM1 and SM2 are provided in this section by employing the tool attraction basin. Suppose \(\mathcal{M}(z)\) is the notation for a second or higher degree complex polynomial. Then, the set \(\{z_{0}\in\mathbb{C}:z_{j}\to z_{*}\) as \(j\to\infty\}\) represents the attraction basin corresponding to a zero \(z_{*}\) of \(\mathcal{M}\), where \(\{z_{j}\}_{j=0}^{\infty}\) is formed by an iterative algorithm with a starting choice \(z_{0}\in\mathbb{C}\). Let us select a region \(\mathcal{E}=[-4,4]\times[-4,4]\) on \(\mathbb{C}\) with a grid of \(400\times 400\) points. To prepare attraction basins, we apply SM1 and SM2 on variety of complex polynomials by selecting every point \(z_{0}\in\mathcal{E}\) as a starter. The point \(z_{0}\) remains in the basin of a zero \(z_{*}\) of a considered polynomial if \(\lim\limits_{j\to\infty}z_{j}=z_{*}\). Then, we display \(z_{0}\) with a fixed color corresponding to \(z_{*}\). As per the number of iterations, we employ the light to dark colors to each \(z_{0}\). Black color is the sign of non-convergence zones. The terminating condition of the iteration is \(||z_{j}-z_{*}||<10^{-6}\) with the maximum limit of 300 iterations. We used MATLAB 2019a to design the fractal pictures. This numerical experiment begins with polynomials \(\mathcal{M}_{1}(z)=z^{2}-1\) and \(\mathcal{M}_{2}(z)=z^{2}-z-1\) of degree two. These polynomials are used to compare the attraction basins for SM1 and SM2. The results of comparison are displayed in Figures 1 and 2. In Figure 1, green and pink areas indicate the attraction basins corresponding to the zeros \(-1\) and \(1\), respectively, of \(\mathcal{M}_{1}(z)\). The basins of the solutions \(\frac{1+\sqrt{5}}{2}\) and \(\frac{1-\sqrt{5}}{2}\) of \(\mathcal{M}_{2}(z)=0\) are shown in Figure 2 by applying pink and green colors, respectively. Figures 3 and 4 offer the attraction basins for SM1 and SM2 associated to the zeros of \(\mathcal{M}_{3}(z)=z^{3}-1\) and \(\mathcal{M}_{4}(z)=z^{3}+(-0.7250+1.6500i)z-0.275-1.65i\). In Figure 3, the basins of the solutions \(-\frac{1}{2}-0.866025i\), \(1\) and \(-\frac{1}{2}+0.866025i\) of \(\mathcal{M}_{3}(z)=0\) are painted in blue, green and pink, respectively. The basins for SM1 and SM2 associated to the zeros \(1\), \(-1.401440+0.915201i\) and \(0.4014403-0.915201i\) of \(\mathcal{M}_{4}(z)\) are given in Figure 4 by means of green, pink and blue regions, respectively. Next, we use polynomials \(\mathcal{M}_{5}(z)=z^{4}-1\) and \(\mathcal{M}_{6}(z)=z^{4}-10z^{2}+9\) of degree four to compare the attraction basins for SM1 and SM2. Figure 5 provides the comparison of basins for these algorithms associated to the solutions \(1\), \(-i\), \(i\) and \(-1\) of \(\mathcal{M}_{5}(z)=0\), which are denoted in yellow, green, pink and blue regions. The basins for SM1 and SM2 corresponding to the zeros \(-1\), \(3\), \(-3\) and \(1\) of \(\mathcal{M}_{6}(z)\) are demonstrated in Figure 6 using yellow, pink, green and blue colors, respectively. Moreover, we select polynomials \(\mathcal{M}_{7}(z)=z^{5}-5z^{3}+4z\) and \(\mathcal{M}_{8}(z)=z^{5}+z\) of degree five to design and compare the attraction basins for SM1 and SM2. Figure 7 gives the basins of zeros \(0\), \(2\), \(-1\), \(-2\) and \(1\) of \(\mathcal{M}_{7}(z)\) in yellow, magenta, red, green and cyan colors, respectively. In Figure 8, green, cyan, red, pink and yellow regions illustrate the attraction basins of the solutions \(-0.707106-0.707106i\), \(-0.707106+0.707106i\), \(0.707106i\), \(0.707106i\) and \(0.707106i\) and \(0\), respectively, of \(\mathcal{M}_{8}(z)=0\). Lastly, sixth degree complex polynomials \(\mathcal{M}_{9}(z)=z^{6}-0.5z^{5}+\frac{11}{4}(1+i)z^{4}-\frac{1}{4}(19+3i)z^{3} +\frac{1}{4}(1+i)z^{2}-\frac{1}{4}(19+3i)z+\frac{3}{2}-3i\) and \(\mathcal{M}_{10}(z)=z^{6}+z-1\) are considered.