A new semilocal convergence theorem for the Weierstrass method for finding zeros of a polynomial simultaneously

Proinov, Petko D.; Petkova, Milena

doi:10.1016/j.jco.2013.11.002

Cited by 23 publications

(27 citation statements)

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“…It should be noted that all corollaries given in [24] can be improved using Theorem 8.8 instead of Theorem 1 of [24]. We end this section with a result which improves Corollary 2 of [24]. This result generalizes and improves the results of [2,7,17,19,21,22,38].…”

Section: )supporting

confidence: 52%

“…It should be noted that recently Proinov and Petkova [33] proved another semilocal convergence theorem for the Weierstrass method under initial conditions involving the Viète operator. To prove this, we choose a vector x 0 = (a + b, a − b), where b is an arbitrary element of K. It is easy to calculate that E(x 0 ) = R(2, ∞) = 1 4 , where E is defined by (8.1) with p = ∞.…”

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confidence: 99%

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General convergence theorems for iterative processes and applications to the Weierstrass root-finding method

Proinov

2016

Journal of Complexity

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In this paper, we prove some general convergence theorems for the Picard iteration in cone metric spaces over a solid vector space. As an application, we provide a detailed convergence analysis of the Weierstrass iterative method for computing all zeros of a polynomial simultaneously. These results improve and generalize existing ones in the literature.Keywords: iterative methods, cone metric space, convergence analysis, error estimates, Weierstrass method, polynomial zeros 2000 MSC: 65J15, 54H25, 65H04, 12Y05where T : D ⊂ X → X is an iteration function in a cone metric space (X, d) over a solid vector space (Y, ). Cone metric spaces have a long history (see Collatz [3], Zabrejko [43], Janković, Kadelburg and Radenović [10], Proinov [29] and references therein). For an overview of the theory of cone metric spaces over a solid vector space, we refer the reader to [29] and [31, Section 2].In the second part of the paper, we study the convergence of the famous Weierstrass method [39] for computing all zeros of a polynomial simultaneously. This method was introduced and studied for the first time by Weierstrass in 1891. In 1960-1966, the method was rediscovered by Durand [6] (in Email address: proinov@uni-plovdiv.bg (Petko D. Proinov)

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confidence: 52%

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confidence: 99%

General convergence theorems for iterative processes and applications to the Weierstrass root-finding method

Proinov

2016

Journal of Complexity

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“…On the basis of this theory lays the notion function of initial conditions of T since the convergence of any iterative method of the type (1) is studied with respect to some function of initial conditions (see [35,36]). Some applications of this theory can be found in [1,2,5,7,8,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][35][36][37][38]. Let R n be equipped with coordinate-wise ordering defined by:…”

Section: Preliminariesmentioning

confidence: 99%

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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“…Several authors have described conditions for the safe convergence of the classical method depending only on the initial approximations. The history of this problem can be found in Proinov() (see also Proinov and Petkova and Petković). This is a very interesting question that we intend to address in the near future, in the quaternionic case.…”

Section: Final Remarksmentioning

confidence: 99%

Weierstrass method for quaternionic polynomial root‐finding

Falcão

Miranda

Severino

et al. 2017

Math Methods in App Sciences

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MOS Classification: 65H04; 30G35; 12D10Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas that motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce.In this paper, we propose a Weierstrass-like method for finding simultaneously all the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.

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A new semilocal convergence theorem for the Weierstrass method for finding zeros of a polynomial simultaneously

Abstract: In this paper we present a new semilocal convergence theorem from data at one point for the Weierstrass iterative method for the simultaneous computation of polynomial zeros. The main result generalizes and improves all previous ones in this area.

Cited by 23 publications

References 14 publications

General convergence theorems for iterative processes and applications to the Weierstrass root-finding method

General convergence theorems for iterative processes and applications to the Weierstrass root-finding method

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

Weierstrass method for quaternionic polynomial root‐finding

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