Matrix polynomials with partially prescribed eigenstructure: eigenvalue sensitivity and condition estimation.
In: Computational & Applied Mathematics, Jg. 24 (2005-09-01), Heft 3, S. 365-392
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Zugriff:
Let P m (z) be a matrix polynomial of degree m whose coefficients A t 2 ℂ qxq satisfy a recurrence relation of the form: h k A 0 + h k + 1 A 1 + … + h k+m-1 A m-1 = h k+m , k ≥ 0, where h k = RZ k L ∈ ℂ pxq , R ∈ ℂ pxn , Z = diag(z 1 , … , z n ) with z i ≠ z j for i ≠ j , 0 < |z j | ≤ 1, and L ∈ ℂ nxq . The coefficients are not uniquely determined from the recurrence relation but the polynomials are always guaranteed to have n fixed eigenpairs, {z j , l j }, where l j is the jth column of L * . In this paper, we show that the z j 's are also the n eigenvalues of an n x n matrix C A ; based on this result the sensitivity of the z j 's is investigated and bounds for their condition numbers are provided. The main result is that the z j 's become relatively insensitive to perturbations in C A provided that the polynomial degree is large enough, the number n is small, and the eigenvalues are close to the unit circle but not extremely close to each other. Numerical results corresponding to a matrix polynomial arising from an application in system theory show that low sensitivity is possible even if the spectrum presents clustered eigenvalues. [ABSTRACT FROM AUTHOR]
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Titel: |
Matrix polynomials with partially prescribed eigenstructure: eigenvalue sensitivity and condition estimation.
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Autor/in / Beteiligte Person: | Viloche Bazán, Fermin S. |
Zeitschrift: | Computational & Applied Mathematics, Jg. 24 (2005-09-01), Heft 3, S. 365-392 |
Veröffentlichung: | 2005 |
Medientyp: | academicJournal |
ISSN: | 0101-8205 (print) |
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