A Note on Multiplicative Backward Errors of Accurate SVD Algorithms

Multiplicative backward stability results are presented for two algorithms which compute the singular value decomposition of dense matrices. These algorithms are the classical onesided Jacobi algorithm, with a stringent stopping criterion, and an algorithm which uses one-sided Jacobi to compute high accurate singular value decompositions of matrices given as rank-revealing factorizations. When multiplicative backward errors are small, the multiplicative perturbation theory for the singular value decomposition developed in the last decade can be applied to get high accuracy bounds on the errors of the computed singular values and vectors.