One-sided reduction to bidiagonal form Original Research Article

We present an idea for reducing a rectangular matrix A to bidiagonal form which is based on the implicit reduction of the symmetric positive semidefinite matrix AtA to tridiagonal form. In other papers we have shown that a method based upon this idea may become a serious competitor (in terms of speed) for computing the singular values of large matrices and also that it is well suited for parallel processing. However, there are still some open questions related to the numerical stability of the method and these will be addressed in this paper. The algorithm, as it is at present, is not backward stable. Nevertheless, we give examples of ill-conditioned matrices for which we have been able to produce a bidiagonal form whose singular values are much more accurate than the ones computed with the standard bidiagonalization technique.