Comparing the geometry of the basins of attraction, the speed and the efficiency of several numerical methods

TL;DR

This work investigates how basins of attraction organize on the complex plane for a broad set of iterative root-finding methods across two root-count scenarios, using $f_1(z)=z^3-1$ and $f_2(z)=z^9-1$. It evaluates sixteen methods with orders $2$–$16$ on dense complex grids, quantifying convergence via the most probable iteration count $N^{*}$ and fractality measures $S_b$ and $S_{bb}$, including tail behavior via Laplace fits. Key findings show that while global basin geometries are similar across methods, fractal boundaries are pervasive; Halley’s method consistently yields the smoothest basins (lowest $S_b$), whereas Maheshwari’s method often produces highly fractal boundaries; increasing the number of roots from $3$ to $9$ amplifies complexity, ill-behaved initial conditions, and iteration counts. The results provide a quantitative framework for comparing method speed, efficiency, and robustness in complex-root problems, informing method selection beyond order alone.

Abstract

We use simple equations in order to compare the basins of attraction on the complex plane, corresponding to a large collection of numerical methods, of several order. Two cases are considered, regarding the total number of the roots, which act as numerical attractors. For both cases we use the iterative schemes for performing a thorough and systematic classification of the nodes on the complex plane. The distributions of the required iterations as well as the probability and their correlations with the corresponding basins of convergence are also discussed. Our numerical calculations suggest that most of the iterative schemes provide relatively similar convergence structures on the complex plane. In addition, several aspects of the numerical methods are compared in an attempt to obtain general conclusions regarding their speed and efficiency. Moreover, we try to determine how the complexity of the each case influences the main characteristics of the numerical methods.

Figures (12)

• Figure 1: The position of the roots on the complex plane for the equation (a-left): $f_1(z) = z^3 - 1 = 0$ and (b-right): $f_2(z) = z^9 - 1 = 0$. In both cases, all the roots lie on an circle with radius $R = 1$, which is indicated by red dashed line. (Color figure online).
• Figure 2: The basins of attraction on the complex plane for the first case, where three roots exist. The positions of the three roots are indicated by black dots. The color code is as follows: $R_1$ root (blue); $R_2$ root (green); $R_3$ root (red); initial conditions for which the iterative schemes lead to extremely large numbers (yellow); initial conditions for which the iterative schemes immediately abort (orange). The numbers of the panels correspond to the numerical formulae, as they have been listed at the beginning of Section \ref{['num']}. (Color figure online).
• Figure 3: The corresponding distributions of the number $N$ of the required iterations for obtaining the basins of attraction shown in Fig. \ref{['c1']}. All ill-behaved initial conditions are shown in white. (Color figure online).
• Figure 4: The corresponding probability distributions of the required iterations for obtaining the basins of attraction shown in Fig. \ref{['c1']}. The vertical, dashed, red lines indicate, in each case, the most probable number $N^{*}$ of iterations. (Color figure online).
• Figure 5: The basins of attraction on the complex plane for the first case, where nine roots exist. The positions of the three roots are indicated by black dots. The color code is as follows: $R_1$ root (blue); $R_2$ root (purple); $R_3$ root (olive); $R_4$ root (green); $R_5$ root (cyan); $R_6$ root (magenta); $R_7$ root (red); $R_8$ root (teal); $R_9$ root (brown); initial conditions for which the iterative schemes lead to extremely large numbers (yellow); initial conditions for which the iterative schemes immediately abort (orange). The numbers of the panels correspond to the numerical formulae, as they have been listed at the beginning of Section \ref{['num']}. (Color figure online).