  We introduce a Generalized Randomized QR-decomposition that may be applied to
arbitrary products of matrices and their inverses, without needing to
explicitly compute the products or inverses. This factorization is a critical
part of a communication-optimal spectral divide-and-conquer algorithm for the
nonsymmetric eigenvalue problem. In this paper, we establish that this
randomized QR-factorization satisfies the strong rank-revealing properties. We
also formally prove its stability, making it suitable in applications. Finally,
we present numerical experiments which demonstrate that our theoretical bounds
capture the empirical behavior of the factorization.
