A new method for computing a column reduced polynomial matrix

Beelen, T.G.J.; Hurk, G. J. van den; Praagman, C.

doi:10.1016/0167-6911(88)90010-2

Cited by 19 publications

(13 citation statements)

References 3 publications

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“…This solution has been found without iterations, i.e., for b ϭ 1, and equals the solution found in Beelen et al [1988].…”

Section: Examplesmentioning

confidence: 94%

“…The first observation is that if (UЈ, RЈ)Ј is such a basis, with U ʦ R nϫn [s], then U is unimodular; see Beelen et al [1988] and Neven and Praagman [1993]. Of course, R ϭ PU, but although (UЈ, RЈ)Ј is minimal and hence column reduced, this does not necessarily hold for R. Take for example the following:…”

Section: Preliminariesmentioning

confidence: 99%

“…The first example is taken from the book of Kailath [1980] and has been discussed before in Beelen et al [1988] and Neven and Praagman [1993].…”

Section: Examplesmentioning

confidence: 99%

“…The algorithm underlying the subroutine has been developed in several stages: the part of the algorithm in which a minimal basis of the right null space of a polynomial matrix is calculated is an adaptation of the algorithm described in Beelen [1987]. The original idea of the algorithm has been described in Beelen et al [1988], and successive improvements have been reported in Neven [1988] andPraagman [1985;. The paper by Neven and Praagman [1993] gives the algorithm in its most general form, exploiting the special structure of the problem in the calculation of the kernel of a polynomial matrix.…”

Section: Introductionmentioning

confidence: 99%

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Algorithm 767: a Fortran 77 package for column reduction of polynomial matrices

Geurts

Praagman

1997

ACM Trans. Math. Softw.

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A polynomial matrix is called column reduced if its column degrees are as low as possible in some sense. Two polynomial matrices P and R are called unimodularly equivalent if there exists a unimodular polynomial matrix U such that PU ϭ R. Every polynomial matrix is unimodularly equivalent to a column-reduced polynomial matrix. In this article a subroutine is described that takes a polynomial matrix P as input and yields on output a unimodular matrix U and a column-reduced matrix R such that PU ϭ R; actually PU Ϫ R is near zero. The subroutine is based on an algorithm, described in a paper by Neven and Praagman. The subroutine has been tested with a number of examples on different computers, with comparable results. The performance of the subroutine on every example tried is satisfactory in the sense that the magnitude of the elements of the residual matrix PU Ϫ R is about ʈPʈ ʈUʈEPS, where EPS is the machine precision. To obtain these results a tolerance, used to determine the rank of some (sub)matrices, has to be set properly. The influence of this tolerance on the performance of the algorithm is discussed, from which a guideline for the usage of the subroutine is derived.

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“…This solution has been found without iterations, i.e., for b ϭ 1, and equals the solution found in Beelen et al [1988].…”

Section: Examplesmentioning

confidence: 94%

Section: Preliminariesmentioning

confidence: 99%

“…The first example is taken from the book of Kailath [1980] and has been discussed before in Beelen et al [1988] and Neven and Praagman [1993].…”

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Algorithm 767: a Fortran 77 package for column reduction of polynomial matrices

Geurts

Praagman

1997

ACM Trans. Math. Softw.

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“…See for instance Van DOOREN [4], BEELEN [1], BEELEN-VELTKAMP [2], BEELEN-van den HURK-PRAAGMAN [3]. In this paper I shall apply these recently achieved results to obtain various representations for families of time series, specified a priori by AR-equations.…”

Section: Introductionmentioning

confidence: 99%

Inputs, outputs and states in the representation of time series

Praagman

Analysis and Optimization of Systems

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Column and diagonal‐reduction of polynomial matrices by orthogonal transformation

Stefanovski

2008

Asian Journal of Control

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Many of the applications of polynomial matrices in real world systems require column‐ or diagonally‐reduced polynomial matrices. If a given polynomial matrix is not column‐ or diagonally‐reduced, Callier or Wolowich algorithms, which use unimodular transformations, can be applied for column‐ or diagonal‐reduction, respectively, as a pre‐processing step in the applications. However, Callier and Wolowich algorithms may be unstable, from a numerical viewpoint, because they use elementary column and row operations. The purpose of this paper is to present sufficient conditions for existence of a constant orthogonal transformation of the given polynomial matrix so that it becomes column‐ or diagonally‐reduced. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

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A new method for computing a column reduced polynomial matrix

Cited by 19 publications

References 3 publications

Algorithm 767: a Fortran 77 package for column reduction of polynomial matrices

Algorithm 767: a Fortran 77 package for column reduction of polynomial matrices

Inputs, outputs and states in the representation of time series

Column and diagonal‐reduction of polynomial matrices by orthogonal transformation

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