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Computing the rank and a small nullspace basis of a polynomial matrix

Λεπτομέρειες βιβλιογραφικής εγγραφής
Τίτλος: Computing the rank and a small nullspace basis of a polynomial matrix
Συγγραφείς: Laboratoire de l'informatique du parallélisme, Storjohann, Arne, Villard, Gilles
Πηγή: http://www.ens-lyon.fr/LIP/Pub/Rapports/RR/RR2005/RR2005-03.pdf.
Έτος έκδοσης: 2005
Συλλογή: LARA - Libre accès aux rapports scientifiques et techniques (INIST - Institut de l'Information Scientifique et Technique / CNRS)
Θεματικοί όροι: Linear algebra, polynomial matrix, matrix multiplication, nullspace basis, minimal polynomial basis, randomized algorithm
Περιγραφή: (eng) We reduce the problem of computing the rank and a nullspace basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input n x n matrix of degree d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension n and degree d. If the latter multiplication is done in MM(n,d)=softO((n^omega)d) operations, with omega the exponent of matrix multiplication over K, then the algorithm uses softO(MM(n,d)) operations in K. For m x n matrices of rank r and degree d, the cost expression is softO(nmr^(omega -2)d). The softO notation indicates some missing logarithmic factors. The method is randomized with Las Vegas certification. We achieve our results in part through a combination of matrix Hensel high-order lifting and matrix minimal fraction reconstruction, and through the computation of minimal or small degree vectors in the nullspace seen as a K[x]-module.
Τύπος εγγράφου: report
Περιγραφή αρχείου: 2+24p; 420284 bytes; 23 bytes; application/pdf; application/octet-stream
Γλώσσα: English
Relation: LIP-RR - 2005-03; http://hdl.handle.net/2332/512
Διαθεσιμότητα: http://hdl.handle.net/2332/512
Αριθμός Καταχώρησης: edsbas.3BA387B8
Βάση Δεδομένων: BASE
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  – Url: http://hdl.handle.net/2332/512#
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  Data: Computing the rank and a small nullspace basis of a polynomial matrix
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  Data: <searchLink fieldCode="AR" term="%22Laboratoire+de+l'informatique+du+parallélisme%22">Laboratoire de l'informatique du parallélisme</searchLink><br /><searchLink fieldCode="AR" term="%22Storjohann%2C+Arne%22">Storjohann, Arne</searchLink><br /><searchLink fieldCode="AR" term="%22Villard%2C+Gilles%22">Villard, Gilles</searchLink>
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  Data: <i>http://www.ens-lyon.fr/LIP/Pub/Rapports/RR/RR2005/RR2005-03.pdf</i>.
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  Data: 2005
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  Data: LARA - Libre accès aux rapports scientifiques et techniques (INIST - Institut de l'Information Scientifique et Technique / CNRS)
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  Data: <searchLink fieldCode="DE" term="%22Linear+algebra%22">Linear algebra</searchLink><br /><searchLink fieldCode="DE" term="%22polynomial+matrix%22">polynomial matrix</searchLink><br /><searchLink fieldCode="DE" term="%22matrix+multiplication%22">matrix multiplication</searchLink><br /><searchLink fieldCode="DE" term="%22nullspace+basis%22">nullspace basis</searchLink><br /><searchLink fieldCode="DE" term="%22minimal+polynomial+basis%22">minimal polynomial basis</searchLink><br /><searchLink fieldCode="DE" term="%22randomized+algorithm%22">randomized algorithm</searchLink>
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  Label: Description
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  Data: (eng) We reduce the problem of computing the rank and a nullspace basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input n x n matrix of degree d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension n and degree d. If the latter multiplication is done in MM(n,d)=softO((n^omega)d) operations, with omega the exponent of matrix multiplication over K, then the algorithm uses softO(MM(n,d)) operations in K. For m x n matrices of rank r and degree d, the cost expression is softO(nmr^(omega -2)d). The softO notation indicates some missing logarithmic factors. The method is randomized with Las Vegas certification. We achieve our results in part through a combination of matrix Hensel high-order lifting and matrix minimal fraction reconstruction, and through the computation of minimal or small degree vectors in the nullspace seen as a K[x]-module.
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  BibEntity:
    Languages:
      – Text: English
    Subjects:
      – SubjectFull: Linear algebra
        Type: general
      – SubjectFull: polynomial matrix
        Type: general
      – SubjectFull: matrix multiplication
        Type: general
      – SubjectFull: nullspace basis
        Type: general
      – SubjectFull: minimal polynomial basis
        Type: general
      – SubjectFull: randomized algorithm
        Type: general
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      – TitleFull: Computing the rank and a small nullspace basis of a polynomial matrix
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