Polynomial Root-Finding and Polynomiography

Kalantari, Bahman

doi:10.1142/6265

Cited by 66 publications

(123 citation statements)

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“…Analogous to the case of Newton's function, the notion of an orbit, basin of attraction, Julia and Fatou sets can be defined with respect to iterations of R(z), see, e.g., [3,10,15]. The degree of R(z) is the maximum of the degrees of P (z) and Q(z).…”

Section: Impossibility Of Rational Iterations With Voronoi Cell As Bamentioning

confidence: 99%

“…It is well known that the boundary of A(θ ) coincides with the Julia set of R, denoted as J (R). Furthermore, J (R) coincides with the basin of attraction of any other root θ of p(z), see, e.g., [3,10,15]. Thus to prove the theorem, it suffices to show that the boundary of V (θ) does not coincide with the boundary of other roots of p(z).…”

Section: Theorem 22 Let R(z) Be a Rational Iteration Function For A mentioning

confidence: 99%

“…These two by themselves posses a rich and interesting history, see [18], also [10]. The Basic Family was studied by Schröder [17], see also [10,18] for several different but equivalent formulations, as well as many other properties.…”

Section: Basic Family Of Iteration Functionsmentioning

confidence: 99%

“…The boundary of the Fatou components forms the Julia set, known to exhibit fractal behavior for general polynomials. For precise definition of Fatou and Julia sets with respect to general rational functions and their properties, see, e.g., [3,10,15].…”

Section: Introductionmentioning

confidence: 99%

“…While the first result is a negative result, the second result, our main result, establishes a strong connection between polynomial root-finding and Voronoi cells, while proving a powerful and novel property of the Basic Family. For a study of many properties of this family and their applications, see [10].…”

Section: Introductionmentioning

confidence: 99%

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Polynomial Root-Finding Methods Whose Basins of Attraction Approximate Voronoi Diagram

Kalantari

2011

Discrete Comput Geom

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Given a complex polynomial p(z) with at least three distinct roots, we first prove that no rational iteration function exists where the basin of attraction of a root coincides with its Voronoi cell. In spite of this negative result, we prove that the Voronoi diagram of the roots can be well approximated through a high order sequence of iteration functions, the Basic Family, B m (z), m ≥ 2. Let θ be a simple root of p(z), V (θ) its Voronoi cell, and A m (θ ) its basin of attraction with respect to B m (z). We prove that given any closed subset C of V (θ), including any homothetic copy of V (θ), there exists m 0 such that for all m ≥ m 0 , C is also a subset of A m (θ ). This implies that when all roots of p(z) are simple, the basins of attraction of B m (z) uniformly approximate the Voronoi diagram of the roots to within any prescribed tolerance. Equivalently, the Julia set of B m (z), and hence the chaotic behavior of its iterations, will uniformly lie to within prescribed strip neighborhood of the boundary of the Voronoi diagram. In a sense, this is the strongest property a rational iteration function can exhibit for polynomials. Next, we use the results to define and prove an infinite layering within each Voronoi cell of a given set of points, whether known implicitly as roots of a polynomial equation, or explicitly via their coordinates. We discuss potential application of our layering in computational geometry.

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Section: Impossibility Of Rational Iterations With Voronoi Cell As Bamentioning

confidence: 99%

Section: Theorem 22 Let R(z) Be a Rational Iteration Function For A mentioning

confidence: 99%

Section: Basic Family Of Iteration Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Polynomial Root-Finding Methods Whose Basins of Attraction Approximate Voronoi Diagram

Kalantari

2011

Discrete Comput Geom

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A novel Noor iterative method of operators with property (E) as concerns convex programming applicable in signal recovery and polynomiography

Paimsang,

Yambangwai,

Thianwan

2024

Math Methods in App Sciences

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This work proposes a novel Noor iterative scheme, called CT‐iteration, to approximate the fixed points in the new context of generalized nonexpansive mappings with property . We establish the strong and weak convergence results in a uniformly convex Banach space. Additionally, numerical examples of the iterative technique are demonstrated using a signal recovery application in a compressed sensing situation. Furthermore, we show the use of the proposed method to generate polynomiographs. The proposed algorithm has been implemented and tested via numerical simulation in MATLAB. The simulation results show the effectiveness of the proposed algorithm.

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From Halley to Secant: Redefining root finding with memory‐based methods including convergence and stability

Qureshi,

Soomro,

Naseem

et al. 2024

Math Methods in App Sciences

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Root‐finding methods solve equations and identify unknowns in physics, engineering, and computer science. Memory‐based root‐seeking algorithms may look back to expedite convergence and enhance computational efficiency. Real‐time systems, complicated simulations, and high‐performance computing demand frequent, large‐scale calculations. This article proposes two unique root‐finding methods that increase the convergence order of the classical Newton–Raphson (NR) approach without increasing evaluation time. Taylor's expansion uses the classical Halley method to create two memory‐based methods with an order of 2.4142 and an efficiency index of 1.5538. We designed a two‐step memory‐based method with the help of Secant and NR algorithms using a backward difference quotient. We demonstrate memory‐based approaches' robustness and stability using visual analysis via polynomiography. Local and semilocal convergence are thoroughly examined. Finally, proposed memory‐based approaches outperform several existing memory‐based methods when applied to models including a thermistor, path traversed by an electron, sheet‐pile wall, adiabatic flame temperature, and blood rheology nonlinear equation.

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Polynomial Root-Finding and Polynomiography

Cited by 66 publications

References 0 publications

Polynomial Root-Finding Methods Whose Basins of Attraction Approximate Voronoi Diagram

Polynomial Root-Finding Methods Whose Basins of Attraction Approximate Voronoi Diagram

A novel Noor iterative method of operators with property (E) as concerns convex programming applicable in signal recovery and polynomiography

From Halley to Secant: Redefining root finding with memory‐based methods including convergence and stability

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