Point estimation of simultaneous methods for solving polynomial equations: A survey (II)

Petković, Marko D.; Petković, Ljiljana D.; Rančić, L.

doi:10.1016/j.cam.2006.04.039

Cited by 18 publications

(12 citation statements)

References 32 publications

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“…Initial conditions of the type (66) were considered for the first time by Proinov [14,15]. Our semilocal convergence results improved the result of Petković, Petković and Rančić [13] (Theorem 2). The advantages of this result are:…”

Section: Discussionsupporting

confidence: 67%

“…In 2007, Petković, Petković, and Rančić [13] gave a semilocal convergence result for Wang-Zheng's method (1), improving the previous result due to Petković and Herceg [12].…”

Section: Introductionmentioning

confidence: 63%

“…This yields that φ is a quasi-homogeneous function of degree m = 3 on [0, µ). In the following lemma, we prove that the function φ satisfies the assumption (13) of Theorem 3 for the iteration function T :…”

Section: Local Convergence Analysis Of the First Typementioning

confidence: 96%

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Local and Semilocal Convergence of Wang-Zheng’s Method for Simultaneous Finding Polynomial Zeros

Cholakov

2019

Symmetry

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In 1984, Wang and Zheng (J. Comput. Math. 1984, 1, 70–76) introduced a new fourth order iterative method for the simultaneous computation of all zeros of a polynomial. In this paper, we present new local and semilocal convergence theorems with error estimates for Wang–Zheng’s method. Our results improve the earlier ones due to Wang and Wu (Computing 1987, 38, 75–87) and Petković, Petković, and Rančić (J. Comput. Appl. Math. 2007, 205, 32–52).

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Section: Discussionsupporting

confidence: 67%

“…In 2007, Petković, Petković, and Rančić [13] gave a semilocal convergence result for Wang-Zheng's method (1), improving the previous result due to Petković and Herceg [12].…”

Section: Introductionmentioning

confidence: 63%

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Local and Semilocal Convergence of Wang-Zheng’s Method for Simultaneous Finding Polynomial Zeros

Cholakov

2019

Symmetry

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“…, z (0) n and the polynomial degree n. Certainly, this is of great importance in practice since these conditions are computationally verifiable. More details on the point estimation theory concerning iterative methods for the simultaneous determination of polynomial zeros can be found in [8][9][10][11][12][13], and the references cited therein.…”

Section: Convergence Analysis-guaranteed Convergencementioning

confidence: 99%

On the new fourth-order methods for the simultaneous approximation of polynomial zeros

Petković

Rančić

Milošević

2011

Journal of Computational and Applied Mathematics

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a b s t r a c tA new iterative method of the fourth-order for the simultaneous determination of polynomial zeros is proposed. This method is based on a suitable zero-relation derived from the fourth-order method for a single zero belonging to the Schröder basic sequence. One of the most important problems in solving polynomial equations, the construction of initial conditions that enable both guaranteed and fast convergence, is studied in detail for the proposed method. These conditions are computationally verifiable since they depend only on initial approximations, the polynomial coefficients and the polynomial degree, which is of practical importance. The construction of improved methods in ordinary complex arithmetic and complex circular arithmetic is discussed. Finally, numerical examples and the comparison with existing fourth-order methods are given.

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“…They are computationally verifiable, which is of great importance in practice. More details on the point estimation theory concerning iterative methods for the simultaneous determination of polynomial zeros can be found in [6,[8][9][10], and the references cited there.…”

Section: Convergence Analysis-guaranteed Convergencementioning

confidence: 99%

Chebyshev‐like root‐finding methods with accelerated convergence

Petković¹,

Rančić²,

Petković³

et al. 2009

Numerical Linear Algebra App

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SUMMARYIterative methods for the simultaneous determination of simple or multiple complex zeros of a polynomial, based on a cubically convergent Chebyshev method, are considered. Using Newton's and Halley's corrections the convergence of the basic method of the fourth order is increased to five and six, respectively. The improved convergence is achieved with negligible number of additional calculations, which significantly increases the computational efficiency of the accelerated methods. One of the most important problems in solving polynomial equations, the construction of initial conditions that enable both guaranteed and fast convergence, is also studied for the proposed methods. These conditions are computationally verifiable since they depend only on initial approximations, the polynomial coefficients and the polynomial degree, which is of practical importance. Finally, modified methods of Chebyshev's type for finding multiple zeros and single-step methods based on the Gauss-Seidel approach are constructed.

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Point estimation of simultaneous methods for solving polynomial equations: A survey (II)

Cited by 18 publications

References 32 publications

Local and Semilocal Convergence of Wang-Zheng’s Method for Simultaneous Finding Polynomial Zeros

Local and Semilocal Convergence of Wang-Zheng’s Method for Simultaneous Finding Polynomial Zeros

On the new fourth-order methods for the simultaneous approximation of polynomial zeros

Chebyshev‐like root‐finding methods with accelerated convergence

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