Let A be a general real m-by-n matrix. The singular value
decomposition (SVD) of A is the factorization 
, where
U and V are orthogonal and
, 
,
with 
. If A is complex,
its SVD is 
, where U and V are unitary
and 
is as before with real
diagonal elements.
The 
are called the singular values ,
the first r columns of V
the right singular vectors , and
the first r columns of U
the left singular vectors .
The routines described in this section, and listed in table 3.10, are used to compute this decomposition. The computation proceeds in the following stages:
if
A is real (
if A is complex), where 
and 
are orthogonal (unitary if A is complex), and B is real and
upper-bidiagonal when 
and lower bidiagonal when m < n, so
that B is nonzero only on the main diagonal and either on the first
superdiagonal (if ) or the first subdiagonal (if m<n).
,
where 
and 
are orthogonal and is diagonal as described
above. The singular vectors of A are then 
and
.
The reduction to bidiagonal form is performed by the subroutine PxGEBRD and the SVD of B is computed using the LAPACK routine xBDSQR.
The routine PxGEBRD represents
and in factored form as products of elementary reflectors,
as described in section 3.4.
If A is real,
the matrices and may be multiplied by other matrices without
forming and using routine PxORMBR .
If A is complex, one instead uses PxUNMBR .
If 
, it may be more efficient to first perform a QR factorization
of A, using the routine
PxGEQRF ,
and then to compute the SVD of the n-by-n matrix R, since
if A = QR and 
, then the SVD of A is given by
.
Similarly, if 
, it may be more efficient to first perform an
LQ factorization of A, using xGELQF. These preliminary QR and LQ
factorizations are performed by the driver PxGESVD.
The SVD may be used to find a minimum norm solution to a (possibly)
rank-deficient linear least squares
problem (3.1). The effective rank, k, of A can be determined as the
number of singular values which exceed a suitable threshold.
Let 
be the leading k-by-k submatrix of , and
be the matrix consisting of the first k columns of V.
Then the solution is given by
where 
consists of the first k elements of 
. 
can be computed by using PxORMBR.
Table 3.10: Computational routines for the singular value decomposition