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#### Golub-Kahan-Lanczos Bidiagonalization Procedure.

As discussed in §6.2, the first phase of a transformation method for the SVD is to compute unitary matrices and such that is in bidiagonal form. In fact, the first column of can be chosen as an arbitrary unit vector, after which the other columns of and are generally determined uniquely. We write this as
 (111)

All s and s are real even if was complex.

The constants and are given by

From the bidiagonal form (6.4) we may derive a double recursion for the columns and of and . Multiplying by , we have

Equating the th columns on both sides, we get

or
 (112)

On the other hand, from the relation

we get

or
 (113)

Since the columns of and are normalized, we must have

and

We summarize the recursion in the following algorithm.

Collecting the computed quantities from the first steps of the algorithm, we have the following important relations:

 (114) (115)

and
 (116)

where is the by leading principal submatrix of defined in (6.4).

Next: Relationship to Symmetric Lanczos. Up: Golub-Kahan-Lanczos Method Previous: Golub-Kahan-Lanczos Method   Contents   Index
Susan Blackford 2000-11-20