This project develops perturbation techniques for deriving bounds on the relative error of singular values and eigenvalues, as well as bounds on the error angles of singular vectors and eigenvectors in terms of a relative gap. Deflation and convergence criteria based on these bounds have the potential to lead to highly accurate and efficient QR-type and Jacobi algorithms for computing eigenvalue, singular value and URV decompositions. In the context of the singular value decomposition of a matrix B, a class of perturbations is investigated which represents all possible that do not change the rank of B. In particular, it includes componentwise relative perturbations of bidiagonal and biacyclic matrices, and perturbations that annihilate an offdiagonal block in a block triangular matrix. So far, many existing relative perturbations bounds have been derived for singular values and vectors of bidiagonal matrices from general results for this class of perturbations. Also, new perturbation results have been proved for singular values and vectors of biacyclic, triangular and shifted triangular matrices. These results are extended to the generalized eigenvalue problem, and to the eigenvalue problems for structured and non-Hermitian matrices.
