Source-linked AI summary
Drawing dynamical and parameter planes of iterative families and methods
Francisco I. Chicharro, Alicia Cordero, Juan R. Torregrosa
TL;DR
The paper studies how complex dynamics characterizes the behavior of Kim’s parametric fourth-order iterative family on quadratic polynomials. It analyzes critical points, parameter and dynamical planes, and MATLAB generation procedures, identifying particularly stable parameter choices including λ = 0 and λ = 1.
Problem
The paper examines how parameter choice affects the dynamical stability and numerical quality of Kim’s fourth-order iterative family.
Method
The study analyzes Kim’s family on quadratic polynomials using fixed-point and critical-point dynamics, parameter and dynamical planes, and MATLAB code for their generation.
Results
λ = 0 gives Ostrowski’s most stable family member, while λ = 1 raises convergence order to five and remains very stable for quadratic polynomials.
Takeaways & Limitations
Parameter-plane regions can identify family members with similar dynamics and guide selection toward numerically stable schemes.
Abstract
from arXiv · showhide
In this paper the complex dynamical analysis of the parametric fourth-order Kim's iterative family is made on quadratic polynomials, showing the Matlab codes generated to draw the fractal images necessary to complete the study. The parameter spaces associated to the free critical points have been analyzed, showing the stable (and unstable) regions where the selection of the parameter will provide us excellent schemes (or dreadful ones).
1 Introduction
The paper situates iterative methods and complex dynamics within the study of nonlinear equations, then introduces its analysis of Kim’s fourth-order family and supporting computational tools.
- Iterative methods are widely used to solve nonlinear equations arising in scientific and engineering problems.
- Complex dynamics helps analyze rational functions generated when iterative schemes solve f(z) = 0 over the complex plane.
- Rational-map dynamics provides information about numerical features such as method stability and reliability.
- For rational functions, the Fatou set contains points whose orbits tend toward attracting fixed points, while the Julia set forms boundaries between attraction basins.
- The paper analyzes Kim’s fourth-order family through fixed-point conjugacy, strange-fixed-point stability, free critical points, parameter and dynamical planes, and MATLAB code.
2 Complex dynamics features of Kim’s family
The paper characterizes the complex dynamics of Kim’s parametric fourth-order family through fixed points, critical points, conjugacy, and parameter-dependent stability. It then applies these results to quadratic polynomials, revealing stable regions, bifurcation structures, and parameter-dependent dynamical behavior.
- Kim’s family yields a fourth-order one-parameter scheme with no convergence-order conditions on λ, after setting η = µ = 0.
- Quadratic-polynomial dynamics: Affine conjugacy reduces the analysis of polynomial families to suitably scaled representative cases, and the quadratic-polynomial study produces parameter planes from free-critical-point orbits.The parameter planes group λ-values into connected components with similar dynamical behavior.
- Fixed points and stability: The operator has superattracting fixed points at 0 and ∞, while z = 1 and six additional strange fixed points have parameter-dependent dynamics.The strange fixed point z = 1 is defined for λ ≠ 1 and λ ≠ 16.
- Critical points: The free critical-point structure depends on λ, with special cases at λ = 0, 1, 16, and −4, and at most three independent free critical points in general.The relations cr1(λ) = 1/cr2(λ) and cr3(λ) = 1/cr4(λ) reduce the number of independent critical points.
- Conjugacy and special parameters: λ = 0 is the unique family member conjugated to z^4 and corresponds to Ostrowski’s method, with no free critical points and the most stable behavior described.Its iterations can only converge to images of the polynomial roots.
- Conjugacy and special parameters: λ = 1 becomes a fifth-order method whose three free critical points lie in the basins of 0 and ∞, while the strange fixed points are repulsive.The paper identifies this parameter as a very stable family member with increased convergence for quadratic polynomials.
- Quadratic-polynomial dynamics: The parameter planes contain a repulsive disk for z = 1, white regions of convergence to other fixed points, Mandelbrot-like figures, and bulbs associated with periodic orbits.Reported bulbs include periods two, three, and four, while Sharkovsky’s theorem is used to affirm that arbitrary periods can occur.
3 MATLAB c ⃝planes code
The section presents MATLAB code for generating dynamical and parameter planes, explaining how fixed-point operators, attracting points, complex meshes, basin colors, and image displays are constructed.
- Dynamical planes: The code generates dynamical planes by computing basins of attraction for the Kim iterative method applied to a quadratic polynomial.Each starting point is iterated until convergence or the maximum iteration count, and the resulting image records both colors and iteration counts.
- Code structure: The implementation is organized into values, fixed-point operators, attractive fixed points, image creation, and image display.Image creation establishes the complex mesh and determines basins after the operator and attracting points are set.
- Dynamical planes: Each starting point receives a color according to its attracting basin, while iteration speed is represented by the color intensity.Points converging to different attracting points are colored orange, blue, or green; nonconvergent points remain black, and slower convergence moves colors toward white.
- Image display: The image display uses imshow with Cartesian orientation, and the iteration-count output can be visualized with surf and shading.The program outputs the image I and the iteration-count matrix it.
- Code adaptation: To adapt the code to other fixed-point operators, only the corresponding operator section must change, with additional branches for more than three attracting points.Each added branch assigns another color to the corresponding basin.
- Parameter planes: Parameter-plane code defines a rectangular λ region, samples it with a points-by-points mesh, and limits iterations for each parameter value.The parameter-plane implementation repeats the same five-part organization and associates the code with the free critical point cr_2.