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Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications pp 295–310Cite as

An Overview of Polynomially Computable Characteristics of Special Interval Matrices

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Part of the book series: Studies in Computational Intelligence ((SCI,volume 835))

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.

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Acknowledgements

The author was supported by the Czech Science Foundation Grant P403-18-04735S.

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Authors and Affiliations

  1. Department of Applied Mathematics, Charles University, Faculty of Mathematics and Physics, Malostranské nám. 25, 11800, Prague, Czech Republic

    Milan Hladík

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  1. Milan Hladík
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Correspondence to Milan Hladík .

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Editors and Affiliations

  1. Department of Teacher Education, University of Texas at El Paso, El Paso, TX, USA

    Olga Kosheleva

  2. Institute of Computational Technologies SB RAS, Novosibirsk, Russia

    Sergey P. Shary

  3. Johns Creek, GA, USA

    Gang Xiang

  4. Deptartment of Informatics, The State Russian Museum, Saint Petersburg, Russia

    Roman Zapatrin

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Hladík, M. (2020). An Overview of Polynomially Computable Characteristics of Special Interval Matrices. In: Kosheleva, O., Shary, S., Xiang, G., Zapatrin, R. (eds) Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications. Studies in Computational Intelligence, vol 835. Springer, Cham. https://doi.org/10.1007/978-3-030-31041-7_16

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