Options
Structured perturbations part II: componentwise distances
Publikationstyp
Journal Article
Date Issued
2004-03-09
Sprache
English
Author(s)
Institut
TORE-URI
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.
Subjects
Componentwise structured perturbations
Condition number
Distance to singularity
DDC Class
004: Informatik
510: Mathematik