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## Golub-Kahan-Lanczos Method

We have seen that the Hermitian eigenvalue problem and the singular value decomposition are closely related. The singular values of a matrix are the square roots of the eigenvalues of Hermitian matrix . Consequently, we can calculate singular values by applying the Hermitian Lanczos method to . The matrix-vector product required by the algorithm can be computed in the form .

In this section we consider applying the Lanczos method of §4.4 to . The special structure of lets us choose a special starting vector that leads to a cheaper algorithm that produces two sequences of vectors, one intended to span the left singular vectors and one for the right singular vectors. In addition, it reduces to bidiagonal form ; i.e., is nonzero only on the main diagonal and first superdiagonal. We derive it from first principles and then show how it is related to Lanczos as described in §4.4.

Subsections

Next: Golub-Kahan-Lanczos Bidiagonalization Procedure. Up: Iterative Algorithms   J. Previous: Which Singular Values and   Contents   Index
Susan Blackford 2000-11-20