Local and Semilocal Convergence of a Family of Multi-point Weierstrass-Type Root-Finding Methods

Proinov, Petko D.; Petkova, Milena

doi:10.1007/s00009-020-01545-z

Cited by 10 publications

(31 citation statements)

References 21 publications

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“…Here, we compare the computational efficiency and convergence behaviour of our new fourteenth-order method MMN14M (11) with the Midrog Petkovic method [20] of order 10 and the Gargantini-Farmer-Loizou method [21][22][23][24] of order 15 (abbreviated as GFLM15M for multiple and GFLM15D for distinct roots). As presented in [20], the efficiency of an iterative method can be estimated using the efficiency index given by…”

Section: Computational Aspectmentioning

confidence: 99%

“…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%

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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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Section: Computational Aspectmentioning

confidence: 99%

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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“…B(E(x (0) )) > 0 , (33) where the functions E and B are defined by (15) and ( 22), respectively. Then Nourein's iteration (11) is well-defined and converges with fourth-order to the root vector ξ = (ξ 1 , .…”

Section: Lemmamentioning

confidence: 99%

“…For other local convergence theorems of the first kind for other simultaneous iterative methods, we refer to [5,9,[29][30][31][32][33]].…”

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

confidence: 99%

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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“…In the literature, there are multi-point iterative methods for finding a single zero of a nonlinear equation (see, e.g., [1][2][3][4][5][6][7]). This study is devoted to the multi-point iterative methods for approximating all the zeros of a polynomial simultaneously (see, e.g., [8][9][10][11]).…”

Section: Introductionmentioning

confidence: 99%

On the Convergence of a New Family of Multi-Point Ehrlich-Type Iterative Methods for Polynomial Zeros

Proinov

Petkova

2021

Mathematics

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In this paper, we construct and study a new family of multi-point Ehrlich-type iterative methods for approximating all the zeros of a uni-variate polynomial simultaneously. The first member of this family is the two-point Ehrlich-type iterative method introduced and studied by Trićković and Petković in 1999. The main purpose of the paper is to provide local and semilocal convergence analysis of the multi-point Ehrlich-type methods. Our local convergence theorem is obtained by an approach that was introduced by the authors in 2020. Two numerical examples are presented to show the applicability of our semilocal convergence theorem.

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Local and Semilocal Convergence of a Family of Multi-point Weierstrass-Type Root-Finding Methods

Cited by 10 publications

References 21 publications

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

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

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

On the Convergence of a New Family of Multi-Point Ehrlich-Type Iterative Methods for Polynomial Zeros

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