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

@article{Storjohann2005ComputingTR,
  title={Computing the rank and a small nullspace basis of a polynomial matrix},
  author={Arne Storjohann and Gilles Villard},
  journal={Proceedings of the 2005 international symposium on Symbolic and algebraic computation},
  year={2005}
}
  • A. Storjohann, G. Villard
  • Published 11 May 2005
  • Mathematics, Computer Science
  • Proceedings of the 2005 international symposium on Symbolic and algebraic computation
We reduce the problem of computing the rank and a null-space 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)= O~(nωd operations, with ω the exponent of matrix multiplication over K, then the algorithm uses O~MM(n,d… 

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References

SHOWING 1-10 OF 42 REFERENCES

On the complexity of polynomial matrix computations

TLDR
Under the straight-line program model, it is shown that multiplication is reducible to the problem of computing the coefficient of degree <i>d</i> of the determinant and algorithms for minimal approximant computation and column reduction that are based on polynomial matrix multiplication are proposed.

Normal forms for general polynomial matrices

Essentially optimal computation of the inverse of generic polynomial matrices

On fast multiplication of polynomials over arbitrary algebras

TLDR
This paper generalizes the well-known Sch6nhage-Strassen algorithm for multiplying large integers to an algorithm for dividing polynomials with coefficients from an arbitrary, not necessarily commutative, not always associative, algebra d, and obtains a method not requiring division that is valid for any algebra.

A pencil approach for embedding a polynomial matrix into a unimodular matrix

TLDR
A new method for constructing the unimodular embedding of a polynomial matrix $P( \lambda )$ is derived, which uses a fast variant of the staircase algorithm and only requires $O( p^3 )$ operations in contrast to the methods proposed up to now.

On computing determinants of matrices without divisions

An algorithm ~ given that computes the determinant of an n x n matrix with entries fr~rn an arbitrary commutative ring in 0(n3~ ring additions, subtractions, and multiplications; the ‘soft-O” O

Minimal Bases of Rational Vector Spaces, with Applications to Multivariable Linear Systems

TLDR
It is shown how minimal bases can be used to factor a transfer function matrix G in the form $G = ND^{ - 1} $, where N and D are polynomial matrices that display the controllability indices of G and its controller canonical realization.

High-order lifting and integrality certification

On Wiedemann's Method of Solving Sparse Linear Systems

Douglas Wiedemann’s (1986) landmark approach to solving sparse linear systems over finite fields provides the symbolic counterpart to non-combinatorial numerical methods for solving sparse linear

Computation of structural invariants of generalized state-space systems