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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