Convergence of Newton, Halley and Chebyshev iterative methods as methods for simultaneous determination of multiple polynomial zeros

Kyncheva, Veselina K.; Yotov, Viktor V.; Ivanov, Stoil I.

doi:10.1016/j.apnum.2016.10.013

Cited by 23 publications

(19 citation statements)

References 22 publications

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“…From the numerical and analytical point of view, there are several methods to compute transcendental functions like this; for example, Chapeau-Blondeau [10] and Corless et al [5] proposed to evaluate the Lambert W branches using approximate analytical expressions and increasing accuracy with the Halley numerical method [35,[63][64][65]. Other proposals to obtain an analytic approximation of the Lambert W function are [11,[66][67][68].…”

Section: = +mentioning

confidence: 99%

PSEM Approximations for Both Branches of Lambert W Function with Applications

Vázquez-Leal

Sandoval-Hernandez

Garcia-Gervacio

et al. 2019

Discrete Dynamics in Nature and Society

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Transcendental functions are a fundamental building block of science and engineering. Among them, a relatively new function denominated as Lambert W is highlighted. The importance of such function relies on the fact that it can perform novel isolation of variables. In this work, we propose two accurate piece-wise approximate solutions, one for the lower branch and another one for the upper branch, respectively. The proposed analytic approximations are obtained by using the power series extender method (PSEM) in combination with asymptotic solutions. In addition, we will compare some published approximations with our proposal, highlighting our advantages in terms of significant digits and speed of evaluation. Furthermore, the approximations are validated by the successful simulation of a problem of economy and other acoustic waves of nonlinear ions.

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Section: = +mentioning

confidence: 99%

PSEM Approximations for Both Branches of Lambert W Function with Applications

Vázquez-Leal

Sandoval-Hernandez

Garcia-Gervacio

et al. 2019

Discrete Dynamics in Nature and Society

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“…where R n depends on n and the functions W f and d are defined by (7) and (11), respectively. Initial conditions of the type (66) were considered for the first time by Proinov [14,15].…”

Section: Discussionmentioning

confidence: 99%

“…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%

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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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“…ere is another class of derivative-free iterative methods which approximates all roots of (1) simultaneously. e simultaneous iterative methods for approximating all roots of (1) are very popular due to their global convergence and parallel implementation on computer (see, e.g., Weierstrass [3], Kanno [4], Proinov [5], Petkovi´c [6], Mir [7], Nourein [8], Aberth [9], and reference cited there in [10][11][12][13][14][15][16][17][18][19][20][21][22]).…”

Section: (2)mentioning

confidence: 99%

Inverse Numerical Iterative Technique for Finding all Roots of Nonlinear Equations with Engineering Applications

Shams

Rafiq

Ahmad

et al. 2021

Journal of Mathematics

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We introduce here a new two-step derivate-free inverse simultaneous iterative method for estimating all roots of nonlinear equation. It is proved that convergence order of the newly constructed method is four. Lower bound of the convergence order is determined using Mathematica and verified with theoretical local convergence order of the method introduced. Some nonlinear models which are taken from physical and engineering sciences as numerical test examples to demonstrate the performance and efficiency of the newly constructed modified inverse simultaneous methods as compared to classical methods existing in literature are presented. Dynamical planes and residual graphs are drawn using MATLAB to elaborate efficiency, robustness, and authentication in its domain.

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Convergence of Newton, Halley and Chebyshev iterative methods as methods for simultaneous determination of multiple polynomial zeros

Cited by 23 publications

References 22 publications

PSEM Approximations for Both Branches of Lambert W Function with Applications

PSEM Approximations for Both Branches of Lambert W Function with Applications

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

Inverse Numerical Iterative Technique for Finding all Roots of Nonlinear Equations with Engineering Applications

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