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

• Published in: Applied mathematics and computation Vol. 453; p. 128055
• Main Author: Miyajima, Shinya
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.09.2023
• Subjects: Kronecker product; Matrix real power; Verified numerical computation; Kronecker product; 15A16; 65F60; 65G20; Matrix real power; Verified numerical computation
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2023.128055

•This manuscript gives two fast numerical algorithms for computing interval vectors containing (B⊕A)αvec(c(1)(c(2))T).•The computational costs of these algorithms are only O(m3+n3) if |α|≪min(m,n).•This manuscript moreover provides “explicit” representations for (tImn+TB⊗TA)−1, and (TB⊗TA)α, where t∈[0;α), and TA∈Cm×n and TB∈Cn×n are block diagonal. 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.