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
, r = min(m , n),
with 
. If A is complex, then
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 2.12, 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 m > = n and lower bidiagonal when m < n, so
that B is nonzero only on the main diagonal and either on the first
superdiagonal (if m > = n) 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 xGEBRD, or by xGBBRD for a band matrix.
The routine xGEBRD represents
and in factored form as products of elementary reflectors,
as described in section 5.4.
If A is real,
the matrices and may be computed explicitly using routine xORGBR,
or multiplied by other matrices without forming and using routine xORMBR .
If A is complex, one instead uses xUNGBR
and xUNMBR , respectively.
If A is banded and xGBBRD is used to reduce it to bidiagonal form,
and are determined as products of Givens rotations
, rather than
as products of elementary reflectors. If or is required, it
must be formed explicitly by xGBBRD. xGBBRD uses a vectorizable
algorithm, similar to that used by xSBTRD (see Kaufman [57]).
xGBBRD may be much faster than xGEBRD when the bandwidth is narrow.
The SVD of the bidiagonal matrix is computed by the subroutine xBDSQR. xBDSQR is more accurate than its counterparts in LINPACK and EISPACK: barring underflow and overflow, it computes all the singular values of A to nearly full relative precision, independent of their magnitudes. It also computes the singular vectors much more accurately. See section 4.9 and [41][16][22] for details.
If m >> n, it may be more efficient to first perform a QR factorization
of A, using the routine xGEQRF
,
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 m << n, 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 xGESVD.
The SVD may be used to find a minimum norm solution to a (possibly)
rank-deficient linear least squares
problem (2.1). 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 using xORMBR, and
xBDSQR has an option to multiply a vector by 
.
-----------------------------------------------------------------------------
Type of matrix Single precision Double precision
and storage scheme Operation real complex real complex
-----------------------------------------------------------------------------
general bidiagonal reduction SGEBRD CGEBRD DGEBRD ZGEBRD
-----------------------------------------------------------------------------
general band bidiagonal reduction SGBBRD CGBBRD DGBBRD ZGBBRD
-----------------------------------------------------------------------------
orthogonal/unitary generate matrix after SORGBR CUNGBR DORGBR ZUNGBR
bidiagonal reduction
multiply matrix after SORMBR CUNMBR DORMBR ZUNMBR
bidiagonal reduction
-----------------------------------------------------------------------------
bidiagonal singular values/ SBDSQR CBDSQR DBDSQR ZBDSQR
singular vectors
-----------------------------------------------------------------------------
Table 2.12: Computational routines for the singular value decomposition