Exponential splittings of products of matrices and accurately computing singular values of long products

• Published in: Linear algebra and its applications Vol. 309; no. 1; pp. 175 - 190
• Main Authors: Oliveira, Suely, Stewart, David E.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.04.2000
• Subjects: Lyapunov exponents; Products of matrices; Stability; SVD; SVD; 65F15; Stability; Products of matrices; Lyapunov exponents; 34D08
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/S0024-3795(99)00273-6

Accurately computing the singular values of long products of matrices is important for estimating Lyapunov exponents: λ i= lim n→∞(1/n) logσ i(A n⋯A 1) . Algorithms for computing singular values of products, in fact, compute the singular values of a perturbed product (A n+E n)⋯(A 1+E 1) . The question is how small are the relative errors of the singular values of the product with respect to these factorwise perturbations. In general, the relative errors in the singular values can be quite large. However, if the product has an exponential splitting, then the error in the singular values is O(n 2 max iκ 2(A i)∥E i∥ F) , uniformly in  n. The exponential splitting property is not directly comparable with the notion of hyperbolicity in dynamical systems, but is similar in philosophy.