Applications of Mathematics, Vol. 65, No. 5, pp. 645-663, 2020


Complexity of computing interval matrix powers for special classes of matrices

David Hartman, Milan Hladík

Received December 31, 2019.   Published online September 7, 2020.

Abstract:  Computing powers of interval matrices is a computationally hard problem. Indeed, it is NP-hard even when the exponent is 3 and the matrices only have interval components in one row and one column. Motivated by this result, we consider special types of interval matrices where the interval components occupy specific positions. We show that computing the third power of matrices with only one column occupied by interval components can be solved in cubic time; so the asymptotic time complexity is the same as for the real case (considering the textbook matrix product method). We further show that for a fixed exponent $k$ and for each interval matrix (of an arbitrary size) whose $k$th power has components that can be expressed as polynomials in a fixed number of interval variables, the computation of the $k$th power is polynomial up to a given accuracy. Polynomiality is shown by using the Tarski method of quantifier elimination. This result is used to show the polynomiality of computing the cube of interval band matrices, among others. Additionally, we study parametric matrices and prove NP-hardness already for their squares. We also describe one specific class of interval parametric matrices that can be squared by a polynomial algorithm.
Keywords:  matrix power; interval matrix; interval computations; NP-hardness
Classification MSC:  65G40, 15Bxx
DOI:  10.21136/AM.2020.0379-19

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Affiliations:   David Hartman, Computer Science Institute of Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic, and Institute of Computer Science of the Czech Academy of Sciences, Pod Vodárenskou věží 2, 182 07 Praha 8, Czech Republic, e-mail: hartman@iuuk.mff.cuni.cz; Milan Hladík, Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Malostranské nám. 25, 118 00 Praha 1, Czech Republic, e-mail: hladik@kam.mff.cuni.cz


 
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