Dynamics of four families of methods with the same weight function to solve nonlinear equations

1. Problem Statement

The paper addresses the need to analyze the complex dynamical behavior of iterative methods used to solve nonlinear equations $f(x) = 0$. Specifically, the authors investigate four distinct families of iterative methods derived from a specific weight function $H(t)$. These families are:

1. A one-point method (Eq. 1).
2. A convex combination of Newton's and Newton-Halley methods for multiple roots (Eq. 3).
3. A two-point method (Eq. 5).
4. Another two-point method (Eq. 7).

While these methods have different formulations for general functions, the authors focus on their behavior when applied to quadratic polynomials with two distinct roots (or multiple roots of equal multiplicity). The core problem is to determine the stability, convergence, and potential chaotic behavior of these methods by analyzing the rational maps they generate, identifying "strange" fixed points (points that are not roots of the equation), and mapping the parameter space to find optimal values for the free parameter $A$.

2. Methodology

The authors employ Complex Dynamics theory to study the iteration functions. The methodology follows these steps:

• Conjugacy and Reduction:

  • The authors utilize the Scaling Theorem (Theorem 2.1) to prove that the dynamics of the iteration function $R_f$ applied to a general quadratic polynomial $p(z) = (z-a)(z-b)$ are analytically conjugate to a simpler rational map $S(z)$.
  • Using the Möbius transformation $M(z) = \frac{z-a}{z-b}$, they reduce the study of the four families to a single universal rational map:
    $$S(z) = \frac{z^3((A-1)z + 2A - 1)}{(2A-1)z + A - 1}$$
  • This reduction allows the analysis of all four families simultaneously, as they share the same dynamical properties on quadratic polynomials.

• Fixed and Critical Point Analysis:

  • Fixed Points: The authors solve $S(z) = z$ to find fixed points. They identify the roots of the polynomial ($z=0, \infty$) and "strange" fixed points (those not corresponding to the roots of $f$).
  • Stability: The stability of these fixed points is determined by calculating the derivative $|S'(z)|$. A point is an attractor if $|S'(z)| < 1$, repulsive if $|S'(z)| > 1$, and parabolic if $|S'(z)| = 1$.
  • Critical Points: The authors solve $S'(z) = 0$ to find critical points. According to Fatou's theory, the behavior of critical points dictates the structure of the Fatou set (basins of attraction).

• Parameter Space and Dynamical Planes:

  • Parameter Space: By iterating the free critical point $z_{c1}$ over a grid of complex values for the parameter $A$, they generate parameter planes. These maps visualize regions where the method converges to roots (stable) versus regions where it converges to strange fixed points or diverges (unstable).
  • Dynamical Planes: For selected values of $A$, they plot the basins of attraction on the complex plane to visualize the convergence behavior.
  • Periodic Orbits: The authors search for and analyze periodic orbits of period two to further understand the chaotic regions.

3. Key Contributions

• Unified Dynamical Analysis: The paper establishes that four seemingly different families of iterative methods (one-point and two-point, single and multiple root variants) share the same conjugacy class when applied to quadratic polynomials. This unifies their dynamical study under a single rational map $S(z)$.
• Analytical Stability Conditions: The authors derive precise analytical conditions for the stability of the strange fixed point $z=1$ and the pair $z_{2,3}$.
  • For $z=1$, stability occurs when $|A - \frac{42}{55}| \leq \frac{2}{55}$.
  • For $z_{2,3}$, stability occurs when $|A - \frac{29}{35}| \leq \frac{1}{35}$.
  • They identify specific values where these points become superattractors (e.g., $A=3/4$ for $z=1$ and $A=5/6$ for $z_{2,3}$).
• Identification of Optimal Parameters: Through parameter space visualization, the authors identify specific regions of the parameter $A$ that yield stable methods (convergence to roots) and regions that lead to chaotic behavior or convergence to non-root fixed points.
• Discovery of Periodic Orbits: The study identifies and lists specific periodic orbits of period two for selected parameter values, providing a deeper understanding of the method's failure modes.

4. Results

• Special Cases:
  • Newton's Method ($A=1$): Reduces to $S(z)=z^2$. The strange fixed point $z=1$ is repulsive.
  • Halley's Method ($A=0$): Reduces to $S(z)=z^3$. Strange fixed points are repulsive.
  • Super Halley ($A=1/2$): Reduces to $S(z)=z^4$.
  • Case $A=2/3$: Reduces to $S(z)=-z^3$.
• Stability Regions:
  • The parameter space analysis reveals that for real values of $A$, the method is generally stable for specific intervals. However, as $A$ moves away from these intervals or becomes complex, the basins of attraction become fragmented, and strange fixed points may become attractive, causing the method to converge to wrong solutions.
  • Figure 4 & 5: Show the parameter plane where red indicates convergence to roots (good behavior) and blue/other colors indicate divergence or convergence to strange points (bad behavior).
• Dynamical Planes:
  • For stable parameters (e.g., $A=0, 1/2, 2/3, 1$), the dynamical planes show clear, separated basins of attraction for the two roots, bounded by the unit circle.
  • For unstable parameters (e.g., $A=2, 5, 10$), the basins become disconnected and complex, indicating chaotic behavior.
  • Complex values of $A$ (e.g., $A = -4+i$) cause significant distortion in the basins of attraction.
• Periodic Orbits: The paper explicitly calculates and tabulates six periodic orbits of period two, demonstrating that for certain parameters, the iteration cycles between two non-root values rather than converging.

5. Significance

This work is significant for the development of robust numerical algorithms for solving nonlinear equations.

1. Predictive Power: By mapping the parameter space, practitioners can select values for the weight function parameter $A$ that guarantee convergence to the roots and avoid regions where the method might get stuck in strange fixed points or periodic cycles.
2. Theoretical Unification: It demonstrates that diverse high-order iterative schemes can be analyzed through a single dynamical lens when applied to quadratic polynomials, simplifying the theoretical study of these families.
3. Safety in Application: The identification of "bad" parameter regions prevents the use of these methods in configurations where they are likely to fail or converge to incorrect solutions, which is crucial for scientific computing and engineering applications where reliability is paramount.