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Popov Form Computation for Matrices of Ore Polynomials

2017
*
Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '17
*

Let F[∂; σ, δ] be a ring

doi:10.1145/3087604.3087650
dblp:conf/issac/KhochtaliRS17
fatcat:feha4hlvdbeffjrcra4zyum2bm
*of**Ore**polynomials*over a field. We give a new deterministic algorithm*for**computing*the*Popov**form*P*of*a non-singular matrix A ∈ F[∂; σ, δ] n×n . ... The resulting bit-complexity*for*the differential and shift*polynomial*case over Q(z) improves upon the previous best. ... Section 4 extends the method to*compute*the*Popov**form**of*a non-singular matrix*of**Ore**polynomials*. ...##
###
Computing Valuation Popov Forms
[chapter]

2005
*
Lecture Notes in Computer Science
*

*Popov*

*forms*and weak

*Popov*

*forms*

*of*

*matrices*over noncommutative valuation domains are defined and discussed. ... Two new algorithms to construct these

*Popov*

*forms*are given, along with a description

*of*some

*of*their applications. ... Acknowledgements The authors are grateful to the anonymous referees

*for*their helpful comments. ...

##
###
Output-sensitive modular algorithms for polynomial matrix normal forms

2007
*
Journal of symbolic computation
*

We give modular algorithms to

doi:10.1016/j.jsc.2007.03.001
fatcat:b7vgjskirbeglah7o7ghyyckiu
*compute*row-reduced*forms*, weak*Popov**forms*, and*Popov**forms**of**polynomial**matrices*, as well as the corresponding unimodular transformation*matrices*. ... Our algorithms can be used to*compute*normalized one-sided greatest common divisors and least common multiples*of**polynomial**matrices*, along with irreducible matrix-fraction descriptions*of*matrix rational ... Normal*forms**for**polynomial**matrices*are used in such*computations*as one-sided greatest common divisor*or*least common multiple*of*two*polynomial**matrices*(Kailath, 1980) . ...##
###
Computing Popov and Hermite forms of polynomial matrices

1996
*
Proceedings of the 1996 international symposium on Symbolic and algebraic computation - ISSAC '96
*

*For*a

*polynomial*matrix P(z)

*of*degree d in M~,~(K[z]) where K is a commutative field, a reduction to the Hermite normal

*form*can be

*computed*in O (ndM(n) + M(nd)) arithmetic operations if M(n) is the ... These results are obtamed by applying in the matrix case, the techniques used in the scalar case

*of*the gcd

*of*

*polynomials*. ... Introduction The problem

*of*

*computing*the greatest common divisor (gcd)

*of*scalar

*polynomials*in K[z] (K is a commutative field)

*or*

*of*

*polynomial*

*matrices*in M~,~(lY[z]) has attracted a lot

*of*attention ...

##
###
Normalization of row reduced matrices

2011
*
Proceedings of the 36th international symposium on Symbolic and algebraic computation - ISSAC '11
*

Given as input a row reduced matrix R over K[x], K a field, our algorithm

doi:10.1145/1993886.1993931
dblp:conf/issac/SarkarS11
fatcat:m57bvy4bnjb4dd6lp3euvv2lre
*computes*the*Popov**form*in about the same time as required to multiply together over K[x] two*matrices**of*the same dimension and ... We also show that the problem*of*transforming a row reduced matrix to*Popov**form*is at least as hard as*polynomial*matrix multiplication. ... INTRODUCTION This paper considers the problem*of*lattice reduction,*or*row reduction,*for**matrices*over the ring K[x]*of*univariate*polynomials*with coefficients from a field K. ...##
###
Normal forms for general polynomial matrices

2006
*
Journal of symbolic computation
*

*For*specific input shifts, we obtain methods

*for*

*computing*the matrix greatest common divisor

*of*two matrix

*polynomials*(in normal

*form*)

*or*such

*polynomial*normal

*form*

*computation*as the classical

*Popov*... We present an algorithm

*for*the

*computation*

*of*a shifted

*Popov*Normal

*Form*

*of*a rectangular

*polynomial*matrix. ... 38

*for*the

*computation*

*of*

*matrices*

*of*linear difference and differential operators. ...

##
###
Computing Popov and Hermite Forms of Rectangular Polynomial Matrices

2018
*
Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '18
*

We consider the

doi:10.1145/3208976.3208988
dblp:conf/issac/NeigerRS18
fatcat:cqnimb4e6fgihfddi7b2ur3ybu
*computation**of*two normal*forms**for**matrices*over the univariate*polynomials*: the*Popov**form*and the Hermite*form*. ... The obtained cost bound*for*the*Popov**form*matches the previous best known randomized algorithm, while the cost bound*for*the Hermite*form*improves on the previous best known ones by a factor which is ... ACKNOWLEDGMENTS The authors are grateful to Clément Pernet*for*pointing at the notion*of*saturation. ...##
###
A computational view on normal forms of matrices of Ore polynomials

2012
*
ACM Communications in Computer Algebra
*

Abstract This thesis treats normal

doi:10.1145/2110170.2110182
fatcat:oi242aamhbhrzimvn5fojfcele
*forms**of**matrices*over rings*of**Ore**polynomials*. ... Second, in the main part we present one-and two-sided normal*forms**of**matrices*. More precisely, we deal with the*Popov*normal*form*, Hermite normal*form*and the Jacobson normal*form*. ... Page 53*of*102 Normal*forms**of**Ore**polynomial**matrices*A*Popov*normal*form*may be*computed*analogously to row-reduction in Algorithm 5.13. ...##
###
Deterministic computation of the characteristic polynomial in the time of matrix multiplication
[article]

2021
*
arXiv
*
pre-print

Our algorithm

arXiv:2010.04662v2
fatcat:cevitv5t5bbljopvhvgasqbh2i
*computes*more generally the determinant*of*a univariate*polynomial*matrix in reduced*form*, and relies on new subroutines*for*transforming shifted reduced*matrices*into shifted weak*Popov*...*matrices*, and shifted weak*Popov**matrices*into shifted*Popov**matrices*. ... Leading*matrices*and reduced*forms**of**polynomial**matrices*We will often*compute*with*polynomial**matrices*that have a special*form*, called the (shifted) reduced*form*. ...##
###
Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K
[article]

2017
*
arXiv
*
pre-print

We extend a method

arXiv:1705.04188v1
fatcat:xpwhyzoptvddljd3ku3gumnmuu
*for*finding a bound on the maximal power*of*t in the denominator*of*arbitrary rational solutions y(t) as well as a method*for*bounding the degree*of**polynomial*solutions from the scalar ... The approach is direct and does not rely on uncoupling*or*reduction to a first order system. Unlike in the scalar case this usually requires an initial transformation*of*the system. ... at matrix normal*forms**for**polynomial**matrices*. ...##
###
Shifted normal forms of polynomial matrices

1999
*
Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99
*

Tl ic d 1. t,t, cr Iut:t.lIotl gives a fractiorl-frw algorithm

doi:10.1145/309831.309929
dblp:conf/issac/BeckermannLV99
fatcat:oqfrew4s45fmnlopmzhhvzqxju
*for**computing*niatris riormal*forms*. Key words:*Popov**Form*. ... -determining it shifted*Popov*nornia.1*form**for*a shift tleterniined from the degree structure*of*t.lw input, nlat.ris polynonlid (cf. Esimlple 2.5). ... A( a) into a11 ii-*Popov**form*(*or*some similar*form*) T(z). ...##
###
Rank-Sensitive Computation of the Rank Profile of a Polynomial Matrix
[article]

2022
*
arXiv
*
pre-print

Consider a matrix 𝐅∈𝕂^m × n

arXiv:2202.09329v1
fatcat:wryzq4dk2japvkt4yr6qh5gbei
*of*univariate*polynomials*over a field 𝕂. We study the problem*of**computing*the column rank profile*of*𝐅. ... We then provide a second algorithm which*computes*the column rank profile*of*𝐅 with a rank-sensitive complexity*of*O(r^ω-2 n (m+D)) operations in 𝕂. ...*for**computing*normal*forms*such as Hermite*or**Popov*. ...##
###
A Practical Implementation of a Modular Algorithm for Ore Polynomial Matrices
[chapter]

2014
*
Computer Mathematics
*

The algorithm can be used

doi:10.1007/978-3-662-43799-5_5
dblp:conf/ascm/ChengL09
fatcat:rgnfx3yvgff4pc2aoujh6mioeq
*for*finding the rank, left nullspace, and the*Popov**form**of*such*matrices*. ... We briefly review a modular algorithm to perform row reduction*of*a matrix*of**Ore**polynomials*with coefficients in Z[t], and describe a practical implementation in Maple that improves over previous modular ... This algorithm*computes*the rank and left nullspace*of*these*matrices*, and can be used to*compute*the row-reduced and weak*Popov**forms**of*shift*polynomial**matrices*[2] , as well as the*Popov**form**of*general ...##
###
Computing Popov and Hermite forms of rectangular polynomial matrices
[article]

2018
*
arXiv
*
pre-print

We consider the

arXiv:1802.01928v2
fatcat:yc27t3aztzfl7n4s3h3xoktz7q
*computation**of*two normal*forms**for**matrices*over the univariate*polynomials*: the*Popov**form*and the Hermite*form*. ... The obtained cost bound*for*the*Popov**form*matches the previous best known randomized algorithm, while the cost bound*for*the Hermite*form*improves on the previous best known ones by a factor which is ... ACKNOWLEDGMENTS The authors are grateful to Clément Pernet*for*pointing at the notion*of*saturation. ...##
###
Row Reduction Applied to Decoding of Rank Metric and Subspace Codes
[article]

2016
*
arXiv
*
pre-print

Inspired by row reduction

arXiv:1510.04728v3
fatcat:67v6yro6czegbivydc72xh36f4
*of*[x]*matrices*, we develop a general and flexible approach*of*transforming*matrices*over skew*polynomial*rings into a certain reduced*form*. ... We show that decoding*of*ℓ-Interleaved Gabidulin codes, as well as list-ℓ decoding*of*Mahdavifar--Vardy codes can be performed by row reducing skew*polynomial**matrices*. ... Acknowledgements The authors would like to thank the anonymous reviewers*for*suggestions that have substantially improved the clarity*of*the paper. ...
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