Shifted normal forms of polynomial matrices

@inproceedings{Beckermann1999ShiftedNF,
  title={Shifted normal forms of polynomial matrices},
  author={Bernhard Beckermann and George Labahn and Gilles Villard},
  booktitle={International Symposium on Symbolic and Algebraic Computation},
  year={1999}
}
In t,his paper we st,ucly the problen~ of transforrniug, via invertible colu1tln opcrat.ious~ it matrix polyioruial into a varicty Of .shiftcd forms. Esarnplcs of forms c:overed in out frmwa-ork include a colunm rctluccd form: il triangular fornlz R I%!rInite IlOrInd fOrll1 or it Popov IlorIllal fOrIll alollg wit,11 their shifted courltcrpart,s. I3y obt.aiuiug tlcgrvc bounds for uuiniodiilar niiill,iplicrs of shifted Popor fornis we are able t,o c11lbct1 tlic probleni of conqmtiug il normal… 

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References

SHOWING 1-10 OF 72 REFERENCES

Computing Popov and Hermite forms of polynomial matrices

These results are obtamed by applying in the matrix case, the techniques used in the scalar case of the gcd of polynomials to the Hermite normal form.

Parallel algorithms for matrix normal forms

Asymptotically fast computation of Hermite normal forms of integer matrices

This paper presents a new algorithm for computing the Hermite normal form H of an A Z n m of rank m to gether with a unimodular pre multiplier matrix U such that UA H Our algorithm requires O m nM m

Preconditioning of rectangular polynomial matrices for efficient Hermite normal form computation

A Las Vegas probabilistic algorithm for reducing the computation of Hermite normal forms of rectangular polynomial matrices allows for the efficient computation of one-sided GCD’S of two matrix polynomials along with the solution of the matrix diophantine equation associated to such a GCD.

Rational matrix structure

  • G. VergheseT. Kailath
  • Mathematics
    1979 18th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes
  • 1979
Recent work1-9, has brought out the importance of a closer examination of the pole/zero and vector-space structure of rational matrices G(s). Results developed by several people are brought together

Solving Systems of Linear Equations over Polynomials

  • R. Kannan
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 1985

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

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.

A Uniform Approach for the Fast Computation of Matrix-Type Pade Approximants

A recurrence relation is presented for the computation of a basis for the corresponding linear solution space of these approximants, and these methods result in fast (and superfast) reliable algorithms for the inversion of stripedHankel, layered Hankel, and (rectangular) block-Hankel matrices.

Extended GCD and Hermite Normal Form Algorithms via Lattice Basis Reduction

An algorithm which uses lattice basis reduction to produce small integer multipliers for the equation s = gcd (s(1), ..., s(m) = x(1)s (1) + ... + x( m)s(m), where s1, ... , s (m) are given integers.
...