Bourne, M., Winkler, J. and Su, Y. (2020) The computation of multiple roots of a Bernstein basis polynomial. SIAM Journal on Scientific Computing, 42 (1). A452-A476. ISSN 1064-8275
Abstract
This paper describes the algorithms of Musser and Gauss for the computation of multiple roots of a theoretically exact Bernstein basis polynomial ˆ 5 f(y) when the coefficients of its given form f(y) are corrupted by noise. The exact roots of f(y) can therefore be assumed to be simple, and thus the problem reduces to the calculation of multiple roots of a polynomial f˜(y) that is near f(y), such that the backward error is small. The algorithms require many greatest common divisor (GCD) computations and polynomial deconvolutions, both of which are implemented by a structure-preserving matrix method. The motivation of these algorithms arises from the unstructured and structured condition numbers of a multiple root of a polynomial. These condition numbers have an elegant interpretation in terms of the pejorative manifold of ˆ 12 f(y), which allows the geometric significance of the GCD computations and polynomial deconvolutions to be considered. A variant of the Sylvester resultant matrix is used for the GCD computations because it yields better results than the standard form of this matrix, and the polynomial deconvolutions can be computed in several different ways, sequentially or simultaneously, and with the inclusion or omission of the preservation of the structure of the coefficient matrix. It is shown that Gauss’ algorithm yields better results than Musser’s algorithm, and the reason for these superior results is explained.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2020 Society for Industrial and Applied Mathematics. Reproduced in accordance with the publisher's self-archiving policy. |
Keywords: | Bernstein basis polynomials; Sylvester resultant matrix; Sylvester subresultant matrices; greatest common divisor; multiple roots |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 04 Dec 2019 12:02 |
Last Modified: | 21 Feb 2020 15:49 |
Status: | Published |
Publisher: | Society for Industrial and Applied Mathematics |
Refereed: | Yes |
Identification Number: | 10.1137/18M1219904 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:154150 |