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Complexity of Computing Interval Matrix Powers for Special Classes of Matrices

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 kth power has components that can be expressed as polynomials in a fixed number of interval variables, the computation of the kth 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.

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Correspondence to David Hartman or Milan Hladík.

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The research has been supported by the Czech Science Foundation Grant P403-18-04735S.

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Hartman, D., Hladík, M. Complexity of Computing Interval Matrix Powers for Special Classes of Matrices. Appl Math 65, 645–663 (2020). https://doi.org/10.21136/AM.2020.0379-19

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  • DOI: https://doi.org/10.21136/AM.2020.0379-19

Keywords

  • matrix power
  • interval matrix
  • interval computations
  • NP-hardness

MSC 2020

  • 65G40
  • 15Bxx