A high order iteration formula for the simultaneous inclusion of polynomial zeros

Zhang, Xin; Peng, Hong; Hu, Guiwu

doi:10.1016/j.amc.2005.11.117

Cited by 15 publications

(7 citation statements)

References 4 publications

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“…Weierstrass' two-step and Ehrlich-Alberth's two-step simultaneous methods are special cases of our family of methods. From Tables 1-3, we observe that our methods are very effective and efficient as compared to fifth-order simultaneous method of Zhang et al 8 . Our results can be considered as an improvement and generalization of the previously known results in the existing literature.…”

Section: Discussionmentioning

confidence: 75%

Some Families of Two-Step Simultaneous Methods for Determining Zeros of Nonlinear Equations

Mir

Muneer

Jabeen

2011

ISRN Applied Mathematics

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Section: Discussionmentioning

confidence: 75%

Some Families of Two-Step Simultaneous Methods for Determining Zeros of Nonlinear Equations

Mir

Muneer

Jabeen

2011

ISRN Applied Mathematics

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“…This relation was again derived in [9] by a more complicated procedure using Lagrange interpolation with special nodes. Taking certain approximations c (k) j of ζ j in the sum on the right-hand side of the fixed point relation (6), the following methods were developed in [6]:…”

Section: Accelerated Simultaneous Methodsmentioning

confidence: 98%

“…We observe that the family of iterative methods (7) contains the method proposed in [9] with c (2) i = z i − W i , so that the method (2) can be regarded as a rediscovered method. Furthermore, let us note that the family of methods (7) does not contain derivatives of the polynomial P.…”

Section: Accelerated Simultaneous Methodsmentioning

confidence: 99%

“…The main goal of this work is to state computationally verifiable initial conditions which provide guaranteed convergence of simultaneous methods presented in [6] and improve some results given recently in [9]. Let P(z) = z n + a n−1 z n−1 + · · · + a 1 z + a 0 (a i ∈ C) be a monic polynomial of degree n having only simple (real or complex) zeros ζ 1 , .…”

Section: Introductionmentioning

confidence: 99%

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A note on the improved derivative free root-solvers

Rančić

Petković

2009

Journal of Computational and Applied Mathematics

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Using a fixed point relation of the square-root type and the basic fourth-order method, improved methods of fifth and sixth order for the simultaneous determination of simple zeros of a polynomial are obtained. An increase in convergence is achieved without additional numerical operations, which points to high computational efficiency of the accelerated methods. The main aim of this work is the convergence analysis of improved simultaneous methods given under computationally verifiable initial conditions in the spirit of Smale's point estimation theory.

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“…are given to 6 decimal places. which are taken from Zhang et al [23]. We have R = 0.125 and E(x (0) ) = 0.506619.…”

Section: Numerical Examplesmentioning

confidence: 99%

On the convergence of high-order Ehrlich-type iterative methods for approximating all zeros of a polynomial simultaneously

Proinov

Vasileva

2015

J Inequal Appl

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We study a family of high-order Ehrlich-type methods for approximating all zeros of a polynomial simultaneously. Let us denote by T(1) the famous Ehrlich method (1967). Starting from T(1) , Kjurkchiev and Andreev (1987) have introduced recursively a sequence (T (N) ) ∞ N=1 of iterative methods for simultaneous finding polynomial zeros. For given N ≥ 1, the Ehrlich-type method T (N) has the order of convergence 2N + 1. In this paper, we establish two new local convergence theorems as well as a semilocal convergence theorem (under computationally verifiable initial conditions and with an a posteriori error estimate) for the Ehrlich-type methods T (N) . Our first local convergence theorem generalizes a result of Proinov (2015) and improves the result of Kjurkchiev and Andreev (1987). The second local convergence theorem generalizes another recent result of , but only in the case of the maximum norm. Our semilocal convergence theorem is the first result in this direction. MSC: Primary 65H04; secondary 12Y05

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A high order iteration formula for the simultaneous inclusion of polynomial zeros

Cited by 15 publications

References 4 publications

Some Families of Two-Step Simultaneous Methods for Determining Zeros of Nonlinear Equations

Some Families of Two-Step Simultaneous Methods for Determining Zeros of Nonlinear Equations

A note on the improved derivative free root-solvers

On the convergence of high-order Ehrlich-type iterative methods for approximating all zeros of a polynomial simultaneously

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