The spectral properties of Vandermonde matrices with clustered nodes
We study rectangular Vandermonde matrices V with N+1 rows and s irregularly spaced nodes on the unit circle, in cases where some of the nodes are “clustered” together – the elements inside each cluster being separated by at most h≲1N, and the clusters being separated from each other by at least θ≳1N...
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Published in | Linear algebra and its applications Vol. 609; pp. 37 - 72 |
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Main Authors | , , , |
Format | Journal Article |
Language | English |
Published |
Amsterdam
Elsevier Inc
15.01.2021
American Elsevier Company, Inc |
Subjects | |
Online Access | Get full text |
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Summary: | We study rectangular Vandermonde matrices V with N+1 rows and s irregularly spaced nodes on the unit circle, in cases where some of the nodes are “clustered” together – the elements inside each cluster being separated by at most h≲1N, and the clusters being separated from each other by at least θ≳1N. We show that any pair of column subspaces corresponding to two different clusters are nearly orthogonal: the minimal principal angle between them is at mostπ2−c1Nθ−c2Nh, for some constants c1,c2 depending only on the multiplicities of the clusters. As a result, spectral analysis of VN is significantly simplified by reducing the problem to the analysis of each cluster individually. Consequently we derive accurate estimates for 1) all the singular values of V, and 2) componentwise condition numbers for the linear least squares problem. Importantly, these estimates are exponential only in the local cluster multiplicities, while changing at most linearly with s. |
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ISSN: | 0024-3795 1873-1856 |
DOI: | 10.1016/j.laa.2020.08.034 |