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GolubKahanLanczos 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:
and

(116) 
where is the by leading principal submatrix of
defined in (6.4).
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Susan Blackford
20001120