Computing minimal nullspace bases

@inproceedings{Zhou2012ComputingMN,
  title={Computing minimal nullspace bases},
  author={Wei Zhou and George Labahn and Arne Storjohann},
  booktitle={ISSAC},
  year={2012}
}
In this paper we present a deterministic algorithm for the computation of a minimal nullspace basis of an <i>m</i> x <i>n</i> input matrix of univariate polynomials over a field K with <i>m</i> ≤ <i>n</i>. This algorithm computes a minimal nullspace basis of a degree <i>d</i> input matrix with a cost of <i>O</i>~ (<i>n</i><sub>ω</sub> ⌈<i>md</i>/<i>n</i>⌉) field operations in K. Here the soft-<i>O</i> notation is Big-<i>O</i> with log factors removed while ω is the exponent of matrix… 
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References

SHOWING 1-10 OF 20 REFERENCES
Efficient computation of order bases
TLDR
The algorithm extends earlier work of Storjohann, whose method can be used to find a subset of an order basis that is within a specified degree bound δ using <i>O</i>~(MM(<i>n,δ</i))) field operations for δ≥⌈ <i-m</i><sup>ω</sup>⌉/<i-n</i>, and order σ.
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.
Computing the rank and a small nullspace basis of a polynomial matrix
TLDR
A rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension, n and degree, d, and the soft-O notation O~ indicates some missing logarithmic factors.
Efficient algorithms for order basis computation
Normal forms for general polynomial matrices
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.
Essentially optimal computation of the inverse of generic polynomial matrices
Computation of structural invariants of generalized state-space systems
Recursiveness in matrix rational interpolation problems
Shifted normal forms of polynomial matrices
TLDR
lIotl gives a fractiorl-frw algorithm for computing niatris riormal forms and is able to c11lbct1 tlic probleni of conqmtiug il normal forni into 0Iic of deterniiJIing a sliift.
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