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Error Estimate of Solving Polynomial...

Zhu, Haiyang.

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Error Estimate of Solving Polynomial Equations and the Modified Durand-Kerner Iteration.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Error Estimate of Solving Polynomial Equations and the Modified Durand-Kerner Iteration./
Author:	Zhu, Haiyang.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2019,
Description:	65 p.
Notes:	Source: Masters Abstracts International, Volume: 80-12.
Contained By:	Masters Abstracts International80-12.
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=13815271
ISBN:	9781392163795

Error Estimate of Solving Polynomial Equations and the Modified Durand-Kerner Iteration.
Zhu, Haiyang.

Error Estimate of Solving Polynomial Equations and the Modified Durand-Kerner Iteration. - Ann Arbor : ProQuest Dissertations & Theses, 2019 - 65 p.

Source: Masters Abstracts International, Volume: 80-12.

Thesis (M.S.)--Northeastern Illinois University, 2019.

This item must not be sold to any third party vendors.

In this thesis, we study the accuracy of a computed polynomial root and we construct a modification of the Durand-Kerner method for computing multiple roots. For the first topic, we focus on the error estimate of solving polynomial equations. An important notion of the numerical solution is its backward error along with the condition number and forward error. The basic tenet of the backward error analysis may be summarized in one sentence: A stable algorithm calculates the exact solution of a nearby problem or the same problem at nearby data. We formulate the backward error as a constrained minimization problem and apply the classical method of Lagrange multipliers. By solving this optimization problem, we obtain a precise formula of the backward error. Using this formula, we can estimate the accuracy of a computed root of a polynomial and decide if it is an acceptable solution. For the second objective, we concentrate on developing a new algorithm for computing multiple roots. The Durand-Kerner iteration is one of the widely used root-finding methods due to its simplicity and the theoretical global convergence. From our experiment, however, the Durand-Kerner iteration is inaccurate and inefficient when the polynomial possesses multiple roots. We construct a new algorithm to compute multiple roots accurately by using a similar approach for developing the Durand-Kerner iteration. We assume the multiplicities of the roots are known in the Vieta's equation and use only the distinct roots as variables. The resulting Vieta's equation is an overdetermined nonlinear system. The Gauss-Newton algorithm is then applied to solve for the least squares solution. In this way, we obtain a modified Durand-Kerner iteration method for finding the polynomial roots. From our computing experiment on polynomials possesses multiple roots, it appears that our new iteration is substantially more accurate than the original Durand-Kerner iteration.

ISBN: 9781392163795Subjects--Topical Terms:

515831
Mathematics.

Error Estimate of Solving Polynomial Equations and the Modified Durand-Kerner Iteration.
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245 1 0 $a Error Estimate of Solving Polynomial Equations and the Modified Durand-Kerner Iteration.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2019
300 $a 65 p.
500 $a Source: Masters Abstracts International, Volume: 80-12.
500 $a Publisher info.: Dissertation/Thesis.
500 $a Advisor: Zeng, Zhonggang.
502 $a Thesis (M.S.)--Northeastern Illinois University, 2019.
506 $a This item must not be sold to any third party vendors.
520 $a In this thesis, we study the accuracy of a computed polynomial root and we construct a modification of the Durand-Kerner method for computing multiple roots. For the first topic, we focus on the error estimate of solving polynomial equations. An important notion of the numerical solution is its backward error along with the condition number and forward error. The basic tenet of the backward error analysis may be summarized in one sentence: A stable algorithm calculates the exact solution of a nearby problem or the same problem at nearby data. We formulate the backward error as a constrained minimization problem and apply the classical method of Lagrange multipliers. By solving this optimization problem, we obtain a precise formula of the backward error. Using this formula, we can estimate the accuracy of a computed root of a polynomial and decide if it is an acceptable solution. For the second objective, we concentrate on developing a new algorithm for computing multiple roots. The Durand-Kerner iteration is one of the widely used root-finding methods due to its simplicity and the theoretical global convergence. From our experiment, however, the Durand-Kerner iteration is inaccurate and inefficient when the polynomial possesses multiple roots. We construct a new algorithm to compute multiple roots accurately by using a similar approach for developing the Durand-Kerner iteration. We assume the multiplicities of the roots are known in the Vieta's equation and use only the distinct roots as variables. The resulting Vieta's equation is an overdetermined nonlinear system. The Gauss-Newton algorithm is then applied to solve for the least squares solution. In this way, we obtain a modified Durand-Kerner iteration method for finding the polynomial roots. From our computing experiment on polynomials possesses multiple roots, it appears that our new iteration is substantially more accurate than the original Durand-Kerner iteration.
590 $a School code: 1696.
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710 2 $a Northeastern Illinois University. $b Mathematics, Applied Mathematics Concentration. $3 3545187
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