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# Structured perturbations part II: componentwise distances

Publikationstyp

Journal Article

Publikationsdatum

2004-03-09

Sprache

English

Author

Institut

TORE-URI

Enthalten in

Volume

25

Issue

1

Start Page

31

End Page

56

Citation

SIAM Journal on Matrix Analysis and Applications 1 (25): 31-56 (2004-03-09)

Publisher DOI

Scopus ID

Publisher

Soc.

In the second part of this paper we study condition numbers with respect to componentwise perturbations in the input data for linear systems and for matrix inversion, and the distance to the nearest singular matrix. The structures under investigation are linear structures, namely symmetric, persymmetric, skewsymmetric, symmetric Toeplitz, general Toeplitz, circulant, Hankel, and persymmetric Hankel structures. We give various formulas and estimations for the condition numbers. For all structures mentioned except circulant structures we give explicit examples of linear systems Aεx = b with parameterized matrix Aε such that the unstructured componentwise condition number is script O sigh(ε-1) and the structured componentwise condition number is script O sign(1). This is true for the important case of componentwise relative perturbations in the matrix and in the right-hand side. We also prove corresponding estimations for circulant structures. Moreover, bounds for the condition number of matrix inversion are given. Finally, we give for all structures mentioned above explicit examples of parameterized (structured) matrices Aε such that the (componentwise) condition number of matrix inversion is script O sign(ε-1), but the componentwise distance to the nearest singular matrix is script O sign(1). This is true for componentwise relative perturbations. It shows that, unlike the normwise case, there is no reciprocal proportionality between the componentwise condition number and the distance to the nearest singular matrix.