An improved algorithm for the computation of Kronecker's canonical form of a singular pencil

Beelen, T.G.J.; Dooren, Paul Van

doi:10.1016/0024-3795(88)90003-1

Cited by 135 publications

(87 citation statements)

References 13 publications

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“…The computational burden lies with Step 1 in which the pencil is brought to the generalized Schur form. The computation of this form requires O(m 2 n) operations for the m×n pencil zM P −N P , provided the efficient algorithm in [3] is used. Moreover, the algorithm in [3] uses solely unitary transformations leading to numerical backward stability.…”

Section: Basic Notationmentioning

confidence: 99%

“…The computation of this form requires O(m 2 n) operations for the m×n pencil zM P −N P , provided the efficient algorithm in [3] is used. Moreover, the algorithm in [3] uses solely unitary transformations leading to numerical backward stability. Reordering of eigenvalues or generalized eigenvalues is potentially needed in each of the first three steps of the algorithm (provided at Step 2 the generalized Schur type algorithm of [34] is used).…”

Section: Basic Notationmentioning

confidence: 99%

“…(see for example (2.5) in [3]), where zM s −N s (with dimensions n s ×n s ) and zM a −N a (with dimensions n a × n a ) are regular pencils containing the stable and antistable…”

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confidence: 99%

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General Matrix Pencil Techniques for Solving Discrete-Time Nonsymmetric Algebraic Riccati Equations

Jungers¹,

Oară²,

Abou-Kandil³

et al. 2010

SIAM J. Matrix Anal. & Appl.

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Abstract. A discrete-time nonsymmetric algebraic Riccati system which incorporates as special cases various discrete-time nonsymmetric algebraic Riccati equations is introduced and studied without any restrictive assumptions on the matrix coefficients. Necessary and sufficient existence conditions together with computable formulas for the stabilizing solution are given in terms of proper deflating subspaces of an associated matrix pencil. The theory is applied in the framework of game theory with an open-loop information structure to design Nash strategy without the classical assumptions on the invertibility of some matrix coefficients.

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Section: Basic Notationmentioning

confidence: 99%

Section: Basic Notationmentioning

confidence: 99%

See 1 more Smart Citation

General Matrix Pencil Techniques for Solving Discrete-Time Nonsymmetric Algebraic Riccati Equations

Jungers¹,

Oară²,

Abou-Kandil³

et al. 2010

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

show abstract

“…So we assume that f is a problem data, of the same nature as n and d. A similar analysis on complexity can be done for the pencil algorithm to obtain the staircase form (3). From [4,47] the complexity of the pencil algorithm is in O(n 3 d 3 ) when considering an average value for the null-space degree f . Our analysis is finer in the sense that we incorporate a term depending on the nullspace degree.…”

Section: Algorithmic Complexitymentioning

confidence: 99%

An improved Toeplitz algorithm for polynomial matrix null-space computation

Anaya

Henrion

2009

Applied Mathematics and Computation

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In this paper we present an improved algorithm to compute the minimal null-space basis of polynomial matrices, a problem which has many applications in control and systems theory. This algorithm takes advantage of the block Toeplitz structure of the Sylvester matrix associated with the polynomial matrix. The analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well-known pencil (state-space) algorithms found in the literature.

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“…This problem was solved by Kronecker [19] (see also [16,21]). Nevertheless, the analysis of matrix pencils is still an active research area (see, e.g., the recent paper [18]), mainly because of numerical issues, or to find ways to obtain the Kronecker canonical form efficiently (see, e.g., [36,37,8,12,13,38]). Our main goal in this paper is to highlight the importance of the Wong sequences [40] for the analysis of matrix pencils.…”

Section: Introduction We Study (Singular) Linear Matrix Pencilsmentioning

confidence: 99%

Addition to “The Quasi-Kronecker Form for Matrix Pencils''

Berger¹,

Trenn²

2013

SIAM J. Matrix Anal. & Appl.

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Abstract. We study singular matrix pencils and show that the so-called Wong sequences yield a quasi-Kronecker form. This form decouples the matrix pencil into an underdetermined part, a regular part, and an overdetermined part. This decoupling is sufficient to fully characterize the solution behavior of the differential-algebraic equations associated with the matrix pencil. Furthermore, we show that the minimal indices of the pencil can be determined with only the Wong sequences and that the Kronecker canonical form is a simple corollary of our result; hence, in passing, we also provide a new proof for the Kronecker canonical form. The results are illustrated with an example given by a simple electrical circuit.

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An improved algorithm for the computation of Kronecker's canonical form of a singular pencil

Cited by 135 publications

References 13 publications

General Matrix Pencil Techniques for Solving Discrete-Time Nonsymmetric Algebraic Riccati Equations

General Matrix Pencil Techniques for Solving Discrete-Time Nonsymmetric Algebraic Riccati Equations

An improved Toeplitz algorithm for polynomial matrix null-space computation

Addition to “The Quasi-Kronecker Form for Matrix Pencils''

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