David Hartman, Milan Hladík, and David Říha. Computing the spectral decomposition of interval matrices and a study on interval matrix powers. Appl. Math. Comput., 403:126174:1–13, 2021.
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We present an algorithm for computing a spectral decomposition of an interval matrix as an enclosure of spectral decompositions of particular realizations of interval matrices. The algorithm relies on tight outer estimations of eigenvalues and eigenvectors of corresponding interval matrices, resulting in the total time complexity $O(n^4)$, where n is the order of the matrix. We present a method for general interval matrices as well as its modification for symmetric interval matrices. In the second part of the paper, we apply the spectral decomposition to computing powers of interval matrices, which is our second goal. Numerical results suggest that a simple binary exponentiation is more efficient for smaller exponents, but our approach becomes better when computing higher powers or powers of a special type of matrices. In particular, we consider symmetric interval and circulant interval matrices. In both cases we utilize some properties of the corresponding classes of matrices to make the power computation more efficient.
@article{HarHla2021a,
author = "David Hartman and Milan Hlad\'{\i}k and David \v{R}\'{\i}ha",
title = "Computing the spectral decomposition of interval matrices and a study on interval matrix powers",
journal = "Appl. Math. Comput.",
fjournal = "Applied Mathematics and Computation",
volume = "403",
pages = "126174:1-13",
year = "2021",
doi = "10.1016/j.amc.2021.126174",
issn = "0096-3003",
url = "https://www.sciencedirect.com/science/article/pii/S0096300321002642",
bib2html_dl_html = "http://dx.doi.org/10.1016/j.amc.2021.126174",
abstract = "We present an algorithm for computing a spectral decomposition of an interval matrix as an enclosure of spectral decompositions of particular realizations of interval matrices. The algorithm relies on tight outer estimations of eigenvalues and eigenvectors of corresponding interval matrices, resulting in the total time complexity $O(n^4)$, where n is the order of the matrix. We present a method for general interval matrices as well as its modification for symmetric interval matrices. In the second part of the paper, we apply the spectral decomposition to computing powers of interval matrices, which is our second goal. Numerical results suggest that a simple binary exponentiation is more efficient for smaller exponents, but our approach becomes better when computing higher powers or powers of a special type of matrices. In particular, we consider symmetric interval and circulant interval matrices. In both cases we utilize some properties of the corresponding classes of matrices to make the power computation more efficient.",
keywords = "Interval matrix; Spectral decomposition; Matrix power; Eigenvalues; Eigenvectors",
}
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