Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials
This paper revisits the Bézout, Sylvester, and power-basis matrix representations of the greatest common divisor (GCD) of sets of several polynomials. Furthermore, the present work introduces the application of the QR decomposition with column pivoting to a Bézout matrix achieving the computation of...
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2017-10-01
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Online Access: | https://doi.org/10.1515/spma-2017-0015 |
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doaj-6fa06bdda3b942ffa196d203ce82e56f2021-10-02T19:25:55ZengDe GruyterSpecial Matrices2300-74512017-10-015120222410.1515/spma-2017-0015spma-2017-0015Structured Matrix Methods Computing the Greatest Common Divisor of PolynomialsChristou Dimitrios0Mitrouli Marilena1Triantafyllou Dimitrios2Department of Mathematics and Engineering Sciences, Hellenic Military Academy, GR-16673, Vari, GreeceDepartment of Mathematics, National and Kapodistrian University of Athens, Panepistemiopolis GR-15773, Athens, GreeceDepartment of Mathematics and Engineering Sciences, Hellenic Military Academy, GR-16673, Vari, GreeceThis paper revisits the Bézout, Sylvester, and power-basis matrix representations of the greatest common divisor (GCD) of sets of several polynomials. Furthermore, the present work introduces the application of the QR decomposition with column pivoting to a Bézout matrix achieving the computation of the degree and the coeffcients of the GCD through the range of the Bézout matrix. A comparison in terms of computational complexity and numerical effciency of the Bézout-QR, Sylvester-QR, and subspace-SVD methods for the computation of theGCDof sets of several polynomials with real coeffcients is provided.Useful remarks about the performance of the methods based on computational simulations of sets of several polynomials are also presented.https://doi.org/10.1515/spma-2017-0015sylvester matrixbézout matrixqr decompositionsingular value decomposition |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Christou Dimitrios Mitrouli Marilena Triantafyllou Dimitrios |
spellingShingle |
Christou Dimitrios Mitrouli Marilena Triantafyllou Dimitrios Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials Special Matrices sylvester matrix bézout matrix qr decomposition singular value decomposition |
author_facet |
Christou Dimitrios Mitrouli Marilena Triantafyllou Dimitrios |
author_sort |
Christou Dimitrios |
title |
Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials |
title_short |
Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials |
title_full |
Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials |
title_fullStr |
Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials |
title_full_unstemmed |
Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials |
title_sort |
structured matrix methods computing the greatest common divisor of polynomials |
publisher |
De Gruyter |
series |
Special Matrices |
issn |
2300-7451 |
publishDate |
2017-10-01 |
description |
This paper revisits the Bézout, Sylvester, and power-basis matrix representations of the greatest common divisor (GCD) of sets of several polynomials. Furthermore, the present work introduces the application of the QR decomposition with column pivoting to a Bézout matrix achieving the computation of the degree and the coeffcients of the GCD through the range of the Bézout matrix. A comparison in terms of computational complexity and numerical effciency of the Bézout-QR, Sylvester-QR, and subspace-SVD methods for the computation of theGCDof sets of several polynomials with real coeffcients is provided.Useful remarks about the performance of the methods based on computational simulations of sets of several polynomials are also presented. |
topic |
sylvester matrix bézout matrix qr decomposition singular value decomposition |
url |
https://doi.org/10.1515/spma-2017-0015 |
work_keys_str_mv |
AT christoudimitrios structuredmatrixmethodscomputingthegreatestcommondivisorofpolynomials AT mitroulimarilena structuredmatrixmethodscomputingthegreatestcommondivisorofpolynomials AT triantafylloudimitrios structuredmatrixmethodscomputingthegreatestcommondivisorofpolynomials |
_version_ |
1716846827710447616 |