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Generalized companion matrix for approximate GCD

Paola Boito 1 Olivier Ruatta 1, * 
* Corresponding author
1 DMI
XLIM - XLIM
Abstract : We study a variant of the univariate approximate GCD problem, where the coefficients of one polynomial $f (x)$are known exactly, whereas the coefficients of the second polynomial $g (x)$may be perturbed. Our approach relies on the properties of the matrix which describes the operator of multiplication by $g$in the quotient ring $\mathbb{C}[x] / (f)$. In particular, the structure of the null space of the multiplication matrix contains all the essential information about GCD$(f, g)$. Moreover, the multiplication matrix exhibits a displacement structure that allows us to design a fast algorithm for approximate GCD computation with quadratic complexity w.r.t. polynomial degrees.
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https://hal.archives-ouvertes.fr/hal-00564448
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Submitted on : Tuesday, February 8, 2011 - 10:42:36 PM
Last modification on : Wednesday, October 20, 2021 - 1:31:17 AM
Long-term archiving on: : Monday, May 9, 2011 - 3:25:10 AM

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Paola Boito, Olivier Ruatta. Generalized companion matrix for approximate GCD. 2011. ⟨hal-00564448⟩

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