Exponential splittings of products of matrices and accurately computing singular values of long products Original Research Article

Accurately computing the singular values of long products of matrices is important for estimating Lyapunov exponents: λi=limn→∞(1/n)logσi(Ancdots, three dots, centeredA1). Algorithms for computing singular values of products, in fact, compute the singular values of a perturbed product (An+En)cdots, three dots, centered(A1+E1). 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(n2maxiκ2(Ai)short parallelEishort parallelF), uniformly in n. The exponential splitting property is not directly comparable with the notion of hyperbolicity in dynamical systems, but is similar in philosophy.