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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.  ...  Section 4 extends the method to compute the Popov form of a non-singular matrix of Ore polynomials.  ... 
doi:10.1145/3087604.3087650 dblp:conf/issac/KhochtaliRS17 fatcat:feha4hlvdbeffjrcra4zyum2bm

Computing Valuation Popov Forms [chapter]

Mark Giesbrecht, George Labahn, Yang Zhang
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.  ... 
doi:10.1007/11428862_84 fatcat:otechopefbg3hb47p6eqpdgepi

Output-sensitive modular algorithms for polynomial matrix normal forms

Howard Cheng, George Labahn
2007 Journal of symbolic computation  
We give modular algorithms to 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) .  ... 
doi:10.1016/j.jsc.2007.03.001 fatcat:b7vgjskirbeglah7o7ghyyckiu

Computing Popov and Hermite forms of polynomial matrices

G. Villard
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  ... 
doi:10.1145/236869.237082 dblp:conf/issac/Villard96 fatcat:d4syt5g6hbgtpbvnfkhj54d5yq

Normalization of row reduced matrices

Soumojit Sarkar, Arne Storjohann
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 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.  ... 
doi:10.1145/1993886.1993931 dblp:conf/issac/SarkarS11 fatcat:m57bvy4bnjb4dd6lp3euvv2lre

Normal forms for general polynomial matrices

Bernhard Beckermann, George Labahn, Gilles Villard
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.  ... 
doi:10.1016/j.jsc.2006.02.001 fatcat:x6sig7gznzfttpe3ozobyglyju

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.  ...  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.  ... 
doi:10.1145/3208976.3208988 dblp:conf/issac/NeigerRS18 fatcat:cqnimb4e6fgihfddi7b2ur3ybu

A computational view on normal forms of matrices of Ore polynomials

Johannes Middeke
2012 ACM Communications in Computer Algebra  
Abstract This thesis treats normal 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.  ... 
doi:10.1145/2110170.2110182 fatcat:oi242aamhbhrzimvn5fojfcele

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

Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K [article]

Johannes Middeke
2017 arXiv   pre-print
We extend a method 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.  ... 
arXiv:1705.04188v1 fatcat:xpwhyzoptvddljd3ku3gumnmuu

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  
Tl ic d 1. t,t, cr Iut:t.lIotl gives a fractiorl-frw algorithm 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).  ... 
doi:10.1145/309831.309929 dblp:conf/issac/BeckermannLV99 fatcat:oqfrew4s45fmnlopmzhhvzqxju

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.  ...  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.  ... 
arXiv:2202.09329v1 fatcat:wryzq4dk2japvkt4yr6qh5gbei

A Practical Implementation of a Modular Algorithm for Ore Polynomial Matrices [chapter]

Howard Cheng, George Labahn
2014 Computer Mathematics  
The algorithm can be used 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  ... 
doi:10.1007/978-3-662-43799-5_5 dblp:conf/ascm/ChengL09 fatcat:rgnfx3yvgff4pc2aoujh6mioeq

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.  ...  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.  ... 
arXiv:1802.01928v2 fatcat:yc27t3aztzfl7n4s3h3xoktz7q

Row Reduction Applied to Decoding of Rank Metric and Subspace Codes [article]

Sven Puchinger and Johan Rosenkilde né Nielsen and Wenhui Li and Vladimir Sidorenko
2016 arXiv   pre-print
Inspired by row reduction 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.  ... 
arXiv:1510.04728v3 fatcat:67v6yro6czegbivydc72xh36f4
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