Fast verified computation for real powers of large matrices with Kronecker structure

Shinya Miyajima

Applied Mathematics and Computation, 2023, vol. 453, issue C

Abstract: Let ⊗ be the Kronecker product, In be the n×n identity matrix, α∈R, A∈Cm×m, B∈Cn×n, C∈Cm×n, and vec(C) be a column vector by stacking the columns of C. We propose two fast numerical algorithms for computing interval vectors containing (In⊗A+B⊗Im)αvec(C), where C has rank one. Particular emphasis is put on the computational costs of these algorithms, which are only O(m3+n3) if |α|≪min(m,n). The first algorithm is based on numerical spectral decomposition of A and B. Radii given by the first algorithm are smaller than those by the second algorithm when numerically computed eigenvector matrices for A and B are well-conditioned. The second algorithm is based on numerical block diagonalization, and applicable even when the computed eigenvector matrices are singular or ill-conditioned. Numerical results show efficiency of the algorithms.

Keywords: Matrix real power; Kronecker product; Verified numerical computation (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:453:y:2023:i:c:s0096300323002242

DOI: 10.1016/j.amc.2023.128055

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