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Computing the singular values of a bidiagonal matrix is the final phase of the standard algorithm for the singular value decomposition of a general matrix. A new algorithm that computes all the singular values of a bidiagonal matrix to high relative accuracy independent of their magnitudes is presented. In contrast, the standard algorithm for bidiagonal matrices may compute small singular values with no relative accuracy at all. Numerical experiments show that the new algorithm is comparable in speed to the standard algorithm, and frequently faster.
Demmel et al. (Sat,) studied this question.