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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*. ... For*matrices*which are square*and*nonsingular, deterministic algorithms with satisfactory cost bounds are known. Here, we present deterministic, fast algorithms for*rectangular*input*matrices*. ... ACKNOWLEDGMENTS The authors are grateful to Clément Pernet for pointing at the notion*of*saturation. ...##
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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*. ... For*matrices*which are square*and*nonsingular, deterministic algorithms with satisfactory cost bounds are known. Here, we present deterministic, fast algorithms for*rectangular*input*matrices*. ... ACKNOWLEDGMENTS The authors are grateful to Clément Pernet for pointing at the notion*of*saturation. ...##
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Normal forms for general polynomial matrices

2006
*
Journal of symbolic computation
*

We present an algorithm for the

doi:10.1016/j.jsc.2006.02.001
fatcat:x6sig7gznzfttpe3ozobyglyju
*computation**of*a shifted*Popov*Normal*Form**of*a*rectangular**polynomial*matrix. ...*form**and*the*Hermite*Normal*Form*. ... 38 for the*computation**of**matrices**of*linear difference*and*differential operators. ...##
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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. ...*POPOV**FORM**OF**MATRICES*OVER F[∂; σ, δ] VIA LINEARIZATION In this section we apply the linearization technique to*compute*the*Popov**form**of*F[∂; σ, δ]*polynomials*. ...##
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Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations

2016
*
Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '16
*

We give a Las Vegas algorithm which

doi:10.1145/2930889.2930936
dblp:conf/issac/Neiger16
fatcat:zv6bihlnh5f4pgtyfb27pflk5y
*computes*the shifted*Popov**form**of*an m × m nonsingular*polynomial*matrix*of*degree d in expected O(m^ω d) field operations, where ω is the exponent*of*matrix multiplication ... Our algorithm first builds a system*of*modular equations whose solution set is the row space*of*the input matrix,*and*then finds the basis in shifted*Popov**form**of*this set. ... The author sincerely thanks the anonymous reviewers for their careful reading*and*detailed comments, which were very helpful for preparing the final version*of*this paper. He also thanks C.-P. ...##
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On lattice reduction for polynomial matrices

2003
*
Journal of symbolic computation
*

The algorithm is adapted

doi:10.1016/s0747-7171(02)00139-6
fatcat:d3uu2fktmnbzfie67lcdn3z6cu
*and*applied to various tasks, including rank profile*and*determinant*computation*, transformation to*Hermite**and**Popov*canonical*form*,*polynomial*linear system solving*and*short ... A simple algorithm for lattice reduction*of**polynomial**matrices*is described*and*analysed. ... Acknowledgement The work for this paper was mostly done during both authors' stay at the Institute*of*Scientific*Computing*, Department*of**Computer*Science, ETH Zurich, Switzerland. ...##
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Shifted normal forms of polynomial matrices

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

The notion

doi:10.1145/309831.309929
dblp:conf/issac/BeckermannLV99
fatcat:oqfrew4s45fmnlopmzhhvzqxju
*of*a shifted*form*is basidl~ OIW*of*altering t,lic tlcgrcc st~riic:t.uros*of*I,lic rows*of*a niatris*and*then*computing**forms**of*t,lie resulting nlatris. ... Tl ic d 1. t,t, cr Iut:t.lIotl gives a fractiorl-frw algorithm for*computing*niatris riormal*forms*. Key words:*Popov**Form*. ... The full row rank*rectangular*cast is important. for*computing*mat.rix*polynomial*GCDs in normal*form*. ...##
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A computational view on normal forms of matrices of Ore polynomials

2012
*
ACM Communications in Computer Algebra
*

Second, in the main part we present one-

doi:10.1145/2110170.2110182
fatcat:oi242aamhbhrzimvn5fojfcele
*and*two-sided normal*forms**of**matrices*. More precisely, we deal with the*Popov*normal*form*,*Hermite*normal*form**and*the Jacobson normal*form*. ... Abstract This thesis treats normal*forms**of**matrices*over rings*of*Ore*polynomials*. ... Normal*forms**of*Ore*polynomial**matrices*6*Hermite*normal*form*,*Popov*normal*form**and*their connection to Gröbner bases The*Popov*normal*form**and*the shifted*Popov*normal*form*Just as in the previous ...##
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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. ... Here, D is the sum*of*row degrees*of*𝐅, ω is the exponent*of*matrix multiplication,*and*O(·) hides logarithmic factors. ... Rosenkilde,*and*G. Solomatov. 2018.*Computing**Popov**and**Hermite**Forms**of**Rectangular**Polynomial**Matrices*. In Proceedings ISSAC’18 (New York, NY, USA). ...##
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Verification Protocols with Sub-Linear Communication for Polynomial Matrix Operations
[article]

2019
*
arXiv
*
pre-print

We design

arXiv:1807.01272v2
fatcat:fcuqa3x2xva6ldp6qo34nopmp4
*and*analyze new protocols to verify the correctness*of*various*computations*on*matrices*over the ring F[x]*of*univariate*polynomials*over a field F. ... For the sake*of*efficiency,*and*because many*of*the properties we verify are specific to*matrices*over a principal ideal domain, we cannot simply rely on previously-developed linear algebra protocols for ... Franais du Danemark,*and*by the CNRS-INS2I Institute through its program for young researchers. ...##
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Fraction-Free Computation of Matrix Rational Interpolants and Matrix GCDs

2000
*
SIAM Journal on Matrix Analysis and Applications
*

We present a new set

doi:10.1137/s0895479897326912
fatcat:jwa6tvzvrjdiva75mor64ryyai
*of*algorithms for*computation**of*matrix rational interpolants*and*one-sided matrix greatest common divisors. ... Examples*of*these interpolants include Pad e approximants, Newton-Pad e, Pad e-*Hermite*, simultaneous Pad e approximants*and*more generally M-Pad e approximants along with their matrix generalizations. ... De nition 7.1 (ñ{*Popov**form*,*Popov*{basis) A m m matrix*polynomial*M(z) 2 Q m m z] is inñ{*Popov**form*(with row degree~ ) if there exists a multi-index~ such that M(z) satis es the degree constraints z ? ...##
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Implementations of Efficient Univariate Polynomial Matrix Algorithms and Application to Bivariate Resultants

2019
*
Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation - ISSAC '19
*

codes

doi:10.1145/3326229.3326272
dblp:conf/issac/HyunNS19
fatcat:iblkn4rlizfnfef6ndyt7lpnta
*and*solving*polynomial*systems or structured linear systems. ... Complexity bounds for many problems on*matrices*with univariate*polynomial*entries have been improved in the last few years. ... This*form**of*the output*of*M-Basis-1 suffices to ensure that M-Basis*and*PM-Basis return bases in s-ordered weak*Popov**form*. ...##
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Implementations of efficient univariate polynomial matrix algorithms and application to bivariate resultants
[article]

2019
*
arXiv
*
pre-print

codes

arXiv:1905.04356v1
fatcat:wvdw3dqohfdjfcbx2jmf2zir7q
*and*solving*polynomial*systems or structured linear systems. ... Complexity bounds for many problems on*matrices*with univariate*polynomial*entries have been improved in the last few years. ... This*form**of*the output*of*M-Basis-1 suffices to ensure that M-Basis*and*PM-Basis return bases in s-ordered weak*Popov**form*. ...##
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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*. ...##
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Faster Modular Composition
[article]

2021
*
arXiv
*
pre-print

Our approach relies on the

arXiv:2110.08354v1
fatcat:r4abnf5vundljczjbfdfh75axu
*computation**of*a matrix*of*algebraic relations that is typically*of*small size. Randomization is used to reduce arbitrary input to this favorable situation. ... A new Las Vegas algorithm is presented for the composition*of*two*polynomials*modulo a third one, over an arbitrary field. ... -shifted*Popov*) normal*form*, then: • 11*and*22 are in*Hermite*normal*form*(resp. in shifted*Popov*normal*form*with respect to the corresponding subtuple*of*); • each column*of*12 (resp. 21 ) is its own ...
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