Computing the spectral decomposition of interval matrices and a study on interval matrix power

TL;DR

This paper introduces an algorithm for spectral decomposition of interval matrices, providing enclosures of eigenvalues and eigenvectors, and demonstrates its application in efficiently computing matrix powers, especially for symmetric and circulant matrices.

Contribution

The paper presents a novel algorithm for spectral decomposition of interval matrices with tight enclosures, including modifications for symmetric matrices and applications to matrix power computations.

Findings

Binary exponentiation is efficient for small powers.

The proposed method outperforms simple methods for high powers.

Special properties of symmetric and circulant matrices improve efficiency.

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. We present a method for general interval matrices as well as its modification for symmetric interval matrices. As an illustration, we apply the spectral decomposition to computing powers of interval matrices. 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.