Milan Hladík. An overview of polynomially computable characteristics of special interval matrices. In Olga Kosheleva et al., editor, Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications, Studies in Computational Intelligence, pp. 295–310, Springer, Cham, 2020.
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It is well known that many problems in interval computation are intractable, which restricts our attempts to solve large problems in reasonable time. This does not mean, however, that all problems are computationally hard. Identifying polynomially solvable classes thus belongs to important current trends. The purpose of this paper is to review some of such classes. In particular, we focus on several special interval matrices and investigate their convenient properties. We consider tridiagonal matrices, \M, H, P, Bmatrices, inverse M-matrices, inverse nonnegative matrices, nonnegative matrices, totally positive matrices and some others. We focus in particular on computing the range of the determinant, eigenvalues, singular values, and selected norms. Whenever possible, we state also formulae for determining the inverse matrix and the hull of the solution set of an interval system of linear equations. We survey not only the known facts, but we present some new views as well.
The reasoning on page 302 regarding the inner eigenvalue intervals is not correct; there might be zero entries in some eigenvectors. Thanks to Juergen Garloff for correcting me.
@inCollection{Hla2020a,
author = "Milan Hlad\'{\i}k",
title = "An overview of polynomially computable characteristics of special interval matrices",
editor = "Kosheleva et al., Olga",
feditor = "Kosheleva, Olga and Shary, Sergey P. and Xiang, Gang and Zapatrin, Roman",
booktitle = "Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications",
publisher = "Springer",
address = "Cham",
series = "Studies in Computational Intelligence",
volume = "835",
pages = "295-310",
year = "2020",
doi = "10.1007/978-3-030-31041-7_16",
isbn = "978-3-030-31041-7",
url = "https://doi.org/10.1007/978-3-030-31041-7_16",
bib2html_dl_html = "https://link.springer.com/chapter/10.1007/978-3-030-31041-7_16",
bib2html_errata = "The reasoning on page 302 regarding the inner eigenvalue intervals is not correct; there might be zero entries in some eigenvectors. Thanks to Juergen Garloff for correcting me.",
abstract = "It is well known that many problems in interval computation are intractable, which restricts our attempts to solve large problems in reasonable time. This does not mean, however, that all problems are computationally hard. Identifying polynomially solvable classes thus belongs to important current trends. The purpose of this paper is to review some of such classes. In particular, we focus on several special interval matrices and investigate their convenient properties. We consider tridiagonal matrices, {\{}M, H, P, B{\}}-matrices, inverse M-matrices, inverse nonnegative matrices, nonnegative matrices, totally positive matrices and some others. We focus in particular on computing the range of the determinant, eigenvalues, singular values, and selected norms. Whenever possible, we state also formulae for determining the inverse matrix and the hull of the solution set of an interval system of linear equations. We survey not only the known facts, but we present some new views as well.",
keywords = "Interval computation; Computational complexity; Tridiagonal matrix; M-matrix; H-matrix; P-matrix; Inverse nonnegative matrix",
}
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