A QR–method for computing the singular values via semiseparable matrices. Abstract The standard procedure to compute the singular value decomposition of a dense matrix, first reduces it into a bidiagonal one by means of orthogonal transformations. Once the bidiagonal matrix has been computed, the QR–method is applied to reduce the latter matrix into a diagonal one. In this paper we propose a ne...

This paper presents a generalization of incremental condition estimation, a technique for tracking the extremal singular values of a triangular matrix. While the original approach allowed for the estimation of the largest or smallest singular value, the generalized scheme allows for the estimation of any number of extremal singular values. For example, we can derive estimates for the three smal...

Accurately computing the singular values of long products of matrices is important for estimating Lyapunov exponents: λi = limn→∞(1/n) log σi(An · · ·A1). Algorithms for computing singular values of products in fact compute the singular values of a perturbed product (An +En) · · · (A1 +E1). The question is how small are the relative errors of the singular values of the product with respect to t...