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Computing Popov and Hermite forms of rectangular polynomial matrices [article]

Vincent Neiger and Johan Rosenkilde and Grigory Solomatov
2018 arXiv   pre-print
We consider the 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.  ... 
arXiv:1802.01928v2 fatcat:yc27t3aztzfl7n4s3h3xoktz7q

Computing Popov and Hermite Forms of Rectangular Polynomial Matrices

Vincent Neiger, Johan Rosenkilde, Grigory Solomatov
2018 Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '18  
We consider the 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.  ... 
doi:10.1145/3208976.3208988 dblp:conf/issac/NeigerRS18 fatcat:cqnimb4e6fgihfddi7b2ur3ybu

Normal forms for general polynomial matrices

Bernhard Beckermann, George Labahn, Gilles Villard
2006 Journal of symbolic computation  
We present an algorithm for the 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.  ... 
doi:10.1016/j.jsc.2006.02.001 fatcat:x6sig7gznzfttpe3ozobyglyju

Popov Form Computation for Matrices of Ore Polynomials

Mohamed Khochtali, Johan Rosenkilde né Nielsen, Arne Storjohann
2017 Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '17  
Let F[∂; σ, δ] be a ring 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.  ... 
doi:10.1145/3087604.3087650 dblp:conf/issac/KhochtaliRS17 fatcat:feha4hlvdbeffjrcra4zyum2bm

Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations

Vincent Neiger
2016 Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '16  
We give a Las Vegas algorithm which 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.  ... 
doi:10.1145/2930889.2930936 dblp:conf/issac/Neiger16 fatcat:zv6bihlnh5f4pgtyfb27pflk5y

On lattice reduction for polynomial matrices

T. Mulders, A. Storjohann
2003 Journal of symbolic computation  
The algorithm is adapted 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.  ... 
doi:10.1016/s0747-7171(02)00139-6 fatcat:d3uu2fktmnbzfie67lcdn3z6cu

Shifted normal forms of polynomial matrices

Bernhard Beckermann, George Labahn, Gilles Villard
1999 Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99  
The notion 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.  ... 
doi:10.1145/309831.309929 dblp:conf/issac/BeckermannLV99 fatcat:oqfrew4s45fmnlopmzhhvzqxju

A computational view on normal forms of matrices of Ore polynomials

Johannes Middeke
2012 ACM Communications in Computer Algebra  
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.  ...  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  ... 
doi:10.1145/2110170.2110182 fatcat:oi242aamhbhrzimvn5fojfcele

Rank-Sensitive Computation of the Rank Profile of a Polynomial Matrix [article]

George Labahn, Vincent Neiger, Thi Xuan Vu, Wei Zhou
2022 arXiv   pre-print
Consider a matrix 𝐅∈𝕂^m × n 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).  ... 
arXiv:2202.09329v1 fatcat:wryzq4dk2japvkt4yr6qh5gbei

Verification Protocols with Sub-Linear Communication for Polynomial Matrix Operations [article]

David Lucas, Vincent Neiger, Clément Pernet, Daniel S. Roche, Johan Rosenkilde
2019 arXiv   pre-print
We design 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.  ... 
arXiv:1807.01272v2 fatcat:fcuqa3x2xva6ldp6qo34nopmp4

Fraction-Free Computation of Matrix Rational Interpolants and Matrix GCDs

Bernhard Beckermann, George Labahn
2000 SIAM Journal on Matrix Analysis and Applications  
We present a new set 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 ?  ... 
doi:10.1137/s0895479897326912 fatcat:jwa6tvzvrjdiva75mor64ryyai

Implementations of Efficient Univariate Polynomial Matrix Algorithms and Application to Bivariate Resultants

Seung Gyu Hyun, Vincent Neiger, Éric Schost
2019 Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation - ISSAC '19  
codes 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.  ... 
doi:10.1145/3326229.3326272 dblp:conf/issac/HyunNS19 fatcat:iblkn4rlizfnfef6ndyt7lpnta

Implementations of efficient univariate polynomial matrix algorithms and application to bivariate resultants [article]

Seung Gyu Hyun, Vincent Neiger, Éric Schost
2019 arXiv   pre-print
codes 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.  ... 
arXiv:1905.04356v1 fatcat:wvdw3dqohfdjfcbx2jmf2zir7q

Deterministic computation of the characteristic polynomial in the time of matrix multiplication [article]

Vincent Neiger, Clément Pernet
2021 arXiv   pre-print
Our algorithm 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.  ... 
arXiv:2010.04662v2 fatcat:cevitv5t5bbljopvhvgasqbh2i

Faster Modular Composition [article]

Vincent Neiger, Bruno Salvy, Éric Schost, Gilles Villard
2021 arXiv   pre-print
Our approach relies on the 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  ... 
arXiv:2110.08354v1 fatcat:r4abnf5vundljczjbfdfh75axu
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