The eigenstructure of an arbitrary polynomial matrix: computational aspects

Dooren, Paul Van; Dewilde, P.

doi:10.1016/0024-3795(83)90069-1

Cited by 121 publications

(107 citation statements)

References 11 publications

Supporting

Mentioning

102

Contrasting

Unclassified

Order By: Relevance

“…This algorithm can naturally be extended to obtain the whole eigenstructure of a polynomial matrix [25], and thus it can be seen as a reliable alternative to the pencil methods presented in [15]. Our algorithm processes numerically block Toeplitz matrices readily constructed from polynomial matrix coefficients, and no elementary operations over polynomials are needed.…”

Section: Discussionmentioning

confidence: 99%

“…the multiplicities of the finite and infinite zeros, and the degrees of the vectors in a minimal null-space basis, are contained in its Kronecker canonical form, and reliable algorithms to compute this canonical form are developed. In [15] it is proved that the structural indices of an arbitrary polynomial matrix can be recovered from the Kronecker canonical form of a related pencil, a companion matrix associated to A(s). So, reliable algorithms to obtain the eigenstructure of A(s) were presented.…”

Section: Brief Review Of Existing Algorithmsmentioning

confidence: 99%

“…So, reliable algorithms to obtain the eigenstructure of A(s) were presented. Seminal works [4,15] are currently the basis of many algorithms for polynomial matrix analysis. However, a drawback of this approach called in the following the pencil approach, is that it only returns the structural indices.…”

Section: Brief Review Of Existing Algorithmsmentioning

confidence: 99%

See 2 more Smart Citations

An improved Toeplitz algorithm for polynomial matrix null-space computation

Anaya

Henrion

2009

Applied Mathematics and Computation

View full text Add to dashboard Cite

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.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Brief Review Of Existing Algorithmsmentioning

confidence: 99%

See 1 more Smart Citation

An improved Toeplitz algorithm for polynomial matrix null-space computation

Anaya

Henrion

2009

Applied Mathematics and Computation

View full text Add to dashboard Cite

show abstract

“…Computing the zeros of polynomials can be posed as an eigenvalue problem [33,34]. Consider the matrix pair (A,B) where…”

Section: Zeros Findermentioning

confidence: 99%

Identifying transitions in finite systems by means of partition function zeros and microcanonical inflection-point analysis: A comparison for elastic flexible polymers

et al. 2014

View full text Add to dashboard Cite

For the estimation of transition points of finite elastic, flexible polymers with chain lengths from 13 to 309 monomers, we compare systematically transition temperatures obtained by the Fisher partition function zeros approach with recent results from microcanonical inflection-point analysis. These methods rely on accurate numerical estimates of the density of states, which have been obtained by advanced multicanonical Monte Carlo sampling techniques. Both the Fisher zeros method and microcanonical inflection-point analysis yield very similar results and enable the unique identification of transition points in finite systems, which is typically impossible in the conventional canonical analysis of thermodynamic quantities.

show abstract

“…a classical approach to perform step (1) consists in linearizing the polynomial matrix, namely, constructing a pencil matrix F (s) = F 1 s + F 0 containing the same structural properties as A(s) [Van Dooren and Dewilde, 1983]. For example, the most accepted method to solve the polynomial eigenvalue problem of A(s) consists in solving the equivalent generalized eigenvalue problem on pencil F (s) via the QZ algorithm [Moler and Stewart, 1973].…”

Section: Introductionmentioning

confidence: 99%