Approximating the leading singular triplets of a large matrix function

Given a large square matrix A and a sufficiently regular function f so that f(A) is well defined, we are interested in the approximation of the leading singular values and corresponding singular vectors of f(A), and in particular of ‖f(A)‖, where ‖ · ‖ is the matrix norm induced by the Euclidean vector norm. Since neither f(A) nor f(A)v can be computed exactly, we introduce and analyze an inexact Golub-Kahan-Lanczos bidiagonalization procedure, where the inexactness is related to the inaccuracy of the operations f(A)v, f(A)∗v. Particular outer and inner stopping criteria are devised so as to cope with the lack of a true residual. Numerical experiments with the new algorithm on typical application problems are reported.