On the Julia set of König’s root–finding algorithms

Honorato, Gerardo

doi:10.1090/s0002-9939-2013-11636-9

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Introduction2

Dynamics Of C λ1

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2022

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Cited by 10 publications

(3 citation statements)

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“…2. If λ = 5 then Equation (6) gives that a = 0 and p(z) = z 3 + b becomes an unicritical polynomial.…”

Section: Dynamics Of C λmentioning

confidence: 99%

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The Julia sets of Chebyshev's method with small degrees

Nayak¹,

Pal²

2022

Preprint

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Given a polynomial p, the degree of its Chebyshev's method C p is determined. If p is cubic then the degree of C p is found to be 4, 6 or 7 and we investigate the dynamics of C p in these cases. If a cubic polynomial p is unicritical or non-generic then, it is proved that the Julia set of C p is connected. The family of all rational maps arising as the Chebyshev's method applied to a cubic polynomial which is non-unicritical and generic is parametrized by the multiplier of one of its extraneous fixed points. Denoting a member of this family with an extraneous fixed point with multiplier λ by C λ , we have shown that the Julia set of C λ is connected whenever λ ∈ [−1, 1].

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“…2. If λ = 5 then Equation (6) gives that a = 0 and p(z) = z 3 + b becomes an unicritical polynomial.…”

Section: Dynamics Of C λmentioning

confidence: 99%

“…Though the Julia set of Newton method (applied to a polynomial) is always connected [9], there are other members of Konig's methods with a disconnected Julia set [6]. The Chebyshev's method applied to non-generic cubic polynomials is dealt in [4], where the connectivity question of their Julia sets remain to be answered.…”

Section: Introductionmentioning

confidence: 99%

The Julia sets of Chebyshev's method with small degrees

Nayak¹,

Pal²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Buff and Henriken [6] established some local and global properties of König's methods. Though the Julia set of Newton's method for any given polynomial is connected [26], there exists a polynomial f n such that the Julia set of K fn,n is not connected for any given integer n ≥ 3 [10].…”

Section: Introductionmentioning

confidence: 99%