Updating the singular value decomposition

The spectral decomposition of a symmetric matrix A with small off-diagonal and distinct diagonal elements can be approximated using a direct scheme of R. Davies and Modi (Linear Algebra Appl. 77 (1986) 61). In this paper a generalization of this method for computing the singular value decomposition of close-to-diagonal A∈Rm×n is presented. When A has repeated or “close” singular values it is possible to apply the direct method to split the problem in two with one part containing the well-separated singular values and one requiring the computation of the “close” singular values.