On the convergence of high-order Gargantini–Farmer–Loizou type iterative methods for simultaneous approximation of polynomial zeros

Proinov, Petko D.; Vasileva, Maria T.

doi:10.1016/j.amc.2019.05.026

Cited by 10 publications

(10 citation statements)

References 16 publications

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“…Using ( 20), we get that (30) holds true if and only if σ i = 1, where σ i is given by (19). By the triangle inequality, Lemma 3, the second part of Lemma 1, the inequality ( 29), Hölder's inequality and condition (25), we obtain for σ i the following estimate:…”

Section: Lemmamentioning

confidence: 93%

“…, ξ s ). The function of initial conditions (15) has been used in [23][24][25] for studying the local convergence of the first kind of some iterative methods for simultaneous approximation of multiple polynomial zeros. We define the quantities m = m(m 1 , .…”

Section: Local Convergence Theorem Of the First Kind For Multiple Zerosmentioning

confidence: 99%

“…Recall that D denotes the set of all vectors in K s with pairwise distinct coordinates. We note that the function of initial conditions (35) has been used in [23][24][25] to study the local convergence of the second kind of some iterative methods for finding simultaneously multiple polynomial zeros.…”

Section: Local Convergence Theorem Of the Second Kind For Multiple Zerosmentioning

confidence: 99%

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Local and Semilocal Convergence of Nourein’s Iterative Method for Finding All Zeros of a Polynomial Simultaneously

Proinov

Vasileva

2020

Symmetry

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In 1977, Nourein (Intern. J. Comput. Math. 6:3, 1977) constructed a fourth-order iterative method for finding all zeros of a polynomial simultaneously. This method is also known as Ehrlich’s method with Newton’s correction because it is obtained by combining Ehrlich’s method (Commun. ACM 10:2, 1967) and the classical Newton’s method. The paper provides a detailed local convergence analysis of a well-known but not well-studied generalization of Nourein’s method for simultaneous finding of multiple polynomial zeros. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with verifiable initial condition and a posteriori error bound) for the classical Nourein’s method. Each of the new semilocal convergence results improves the result of Petković, Petković and Rančić (J. Comput. Appl. Math. 205:1, 2007) in several directions. The paper ends with several examples that show the applicability of our semilocal convergence theorems.

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Section: Lemmamentioning

confidence: 93%

Section: Local Convergence Theorem Of the First Kind For Multiple Zerosmentioning

confidence: 99%

Section: Local Convergence Theorem Of the Second Kind For Multiple Zerosmentioning

confidence: 99%

See 1 more Smart Citation

Local and Semilocal Convergence of Nourein’s Iterative Method for Finding All Zeros of a Polynomial Simultaneously

Proinov

Vasileva

2020

Symmetry

Self Cite

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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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“…A very high computational efficiency for the newly constructed scheme for finding distinct as well as multiple roots is achieved by using a suitable corrections [19] which enable us to achieve fourteenth-order convergence with minimal number of functional evaluations in each step. So far among the higher order simultaneous methods, only the Midrog Petkovic method [20] of order ten and the Gargantini-Farmer-Loizou method of 2N + 1 convergence order (where N is positive integer) [21][22][23][24] exist in the literature. Consider nonlinear polynomial equation of degree m:…”

Section: Introductionmentioning

confidence: 99%

A Highly Efficient Computer Method for Solving Polynomial Equations Appearing in Engineering Problems

Rafiq

Shams

Mir

et al. 2021

Mathematical Problems in Engineering

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A highly efficient two-step simultaneous iterative computer method is established here for solving polynomial equations. A suitable special type of correction helps us to achieve a very high computational efficiency as compared to the existing methods so far in the literature. Analysis of simultaneous scheme proves that its convergence order is 14. Residual graphs are also provided to demonstrate the efficiency and performance of the newly constructed simultaneous computer method in comparison with the methods given in the literature. In the end, some engineering problems and some higher degree complex polynomials are solved numerically to validate its numerical performance.

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On the convergence of high-order Gargantini–Farmer–Loizou type iterative methods for simultaneous approximation of polynomial zeros

Cited by 10 publications

References 16 publications

Local and Semilocal Convergence of Nourein’s Iterative Method for Finding All Zeros of a Polynomial Simultaneously

Local and Semilocal Convergence of Nourein’s Iterative Method for Finding All Zeros of a Polynomial Simultaneously

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

A Highly Efficient Computer Method for Solving Polynomial Equations Appearing in Engineering Problems

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