Bourne, M., Winkler, J.R. and Su, Y. (2021) An approximate factorisation of three bivariate Bernstein basis polynomials defined in a triangular domain. Journal of Computational and Applied Mathematics, 390. 113381. ISSN 0377-0427
Abstract
This paper considers an approximate factorisation of three bivariate Bernstein basis polynomials that are defined in a triangular domain. This problem is important for the computation of the intersection points of curves in computer-aided design systems, and it reduces to the determination of an approximate greatest common divisor (AGCD) d (y) of the polynomials. The Sylvester matrix and its subresultant matrices of these three polynomials are formed and it is shown that there are four forms of these matrices. The most difficult part of the computation is the determination of the degree of d (y) because it reduces to the determination of the rank loss of these matrices. This computation is made harder by the presence of trinomial terms in the Bernstein basis functions because they cause the entries of the matrices to span many orders of magnitude. The adverse numerical effects of this wide range of magnitudes of the entries of the four forms of the Sylvester matrix and its subresultant matrices are mitigated by processing the polynomials before these matrices are formed. It is shown that significantly improved results are obtained if the polynomials are processed before computations are performed on their Sylvester matrices and subresultant matrices.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2021 Elsevier. This is an author produced version of a paper subsequently published in Journal of Computational and Applied Mathematics. Uploaded in accordance with the publisher's self-archiving policy. Article available under the terms of the CC-BY-NC-ND licence (https://creativecommons.org/licenses/by-nc-nd/4.0/). |
Keywords: | Bivariate Bernstein basis polynomial; Sylvester resultant matrix and subresultant matrices; Approximate greatest common divisor |
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: | 05 Feb 2021 09:25 |
Last Modified: | 12 Jan 2022 01:38 |
Status: | Published |
Publisher: | Elsevier BV |
Refereed: | Yes |
Identification Number: | 10.1016/j.cam.2020.113381 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:170804 |