Abstract
We investigate the complex dynamics of a triparametric family of optimal fourth-order multiple-root solvers by analyzing their basins of attraction along with extensive study of Möbius conjugacy maps and extraneous fixed points applied to a prototype quadratic polynomial raised to the power of the known integer multiplicitym. A600×600uniform grid centered at the origin covering6×6square region is chosen to display the initial points on each basin of attraction according to a coloring scheme based on their orbit behavior. With illustrative basins of attractions applied to various test polynomials and the corresponding statistical data for convergence as well as a number of comparisons made among the listed methods, we confirm our investigation and analysis developed in this paper.
Highlights
We investigate the complex dynamics of a triparametric family of optimal fourth-order multiple-root solvers by analyzing their basins of attraction along with extensive study of Mobius conjugacy maps and extraneous fixed points applied to a prototype quadratic polynomial raised to the power of the known integer multiplicity m
Many researchers [1,2,3,4,5,6,7] have shown their interest in the dynamics of iterative methods locating the multiple roots [8, 9] of a nonlinear equation
To ensure the convergence of an iterative method in a root-finding problem [10], it is very important to take a good initial value [11,12,13,14,15,16,17] close to the desired zero of the given nonlinear equation under consideration. In connection with such a choice of a good initial value, we pay a special attention to the complex dynamics for a number of optimal fourth-order multiple-root finders by investigating their basins of attraction
Summary
Many researchers [1,2,3,4,5,6,7] have shown their interest in the dynamics of iterative methods locating the multiple roots [8, 9] of a nonlinear equation. To ensure the convergence of an iterative method in a root-finding problem [10], it is very important to take a good initial value [11,12,13,14,15,16,17] close to the desired zero of the given nonlinear equation under consideration In connection with such a choice of a good initial value, we pay a special attention to the complex dynamics for a number of optimal fourth-order multiple-root finders by investigating their basins of attraction. They are listed below in their respective order. Typical cases of methods Yk’s are presented in Table 1 for 1 ≤ k ≤ 6 with selected parameters λ, ρ, and d
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