Contents

# ?lasda

Computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal
d
and off-diagonal
e
. Used by
?bdsdc
.

## Syntax

Include Files
• mkl.h
Description
Using a divide and conquer approach,
?lasda
computes the singular value decomposition (
SVD
) of a real upper bidiagonal
n
-by-
m
matrix
B
with diagonal
d
and off-diagonal
e
, where
m
=
n
+
sqre
.
The algorithm computes the singular values in the
SVD
B
=
U
*
S
*
VT
. The orthogonal matrices
U
and
VT
are optionally computed in compact form. A related subroutine
?lasd0
computes the singular values and the singular vectors in explicit form.
Input Parameters
icompq
Specifies whether singular vectors are to be computed in compact form, as follows:
= 0: Compute singular values only.
= 1: Compute singular vectors of upper bidiagonal matrix in compact form.
smlsiz
The maximum size of the subproblems at the bottom of the computation tree.
n
The row dimension of the upper bidiagonal matrix. This is also the dimension of the main diagonal array
d
.
sqre
Specifies the column dimension of the bidiagonal matrix.
If
sqre
= 0
: the bidiagonal matrix has column dimension
m
=
n
If
sqre
= 1
: the bidiagonal matrix has column dimension
m
=
n
+ 1
.
d
Array,
DIMENSION
(
n
). On entry,
d
contains the main diagonal of the bidiagonal matrix.
e
Array,
DIMENSION
(
m
- 1 ). Contains the subdiagonal entries of the bidiagonal matrix. On exit,
e
is destroyed.
ldu
u
,
vt
,
difl
,
difr
,
poles
,
givnum
, and
z
.
ldu
n
.
ldgcol
givcol
and
perm
.
ldgcol
n
.
work
Workspace array,
DIMENSION
(6
n
+(
smlsiz
+1)
2
)
.
iwork
Workspace array,
Dimension
must be at least (7
n
).
Output Parameters
d
On exit
d
, if
info
= 0
, contains the singular values of the bidiagonal matrix.
u
Array,
DIMENSION
(
ldu
,
smlsiz
) if
icompq
=1
.
Not referenced if
icompq
= 0
.
If
icompq
= 1
, on exit,
u
contains the left singular vector matrices of all subproblems at the bottom level.
vt
Array,
DIMENSION
(
ldu
,
smlsiz
+1 ) if
icompq
= 1
, and not referenced if
icompq
= 0
. If
icompq
= 1
, on exit,
vt
' contains the right singular vector matrices of all subproblems at the bottom level.
k
Array,
DIMENSION
(
n
) if
icompq
= 1
and
DIMENSION
(1) if
icompq
= 0
.
If
icompq
= 1
, on exit,
k
(
i
) is the dimension of the
i
-th secular equation on the computation tree.
difl
REAL
for
slasda
DOUBLE PRECISION
for
dlasda
.
Array,
DIMENSION
(
ldu
,
nlvl
),
where
nlvl
= floor(log
2
(
n
/
smlsiz
))
.
difr
Array,
DIMENSION
(
ldu
, 2
nlvl
) if
icompq
= 1
and
DIMENSION
(
n
) if
icompq
= 0
.
If
icompq
= 1
, on exit,
difl
(1:
n
,
i
) and
difr
(1:
n
,2
i
-1) record distances between singular values on the
i
-th level and singular values on the (
i
-1)-th level, and
difr
(1:
n
, 2i ) contains the normalizing factors for the right singular vector matrix. See
?lasd8
for details.
z
Array,
DIMENSION
(
ldu
,
nlvl
) if
icompq
= 1
and
DIMENSION
(
n
) if
icompq
= 0
. The first
k
elements of
z
(1,
i
) contain the components of the deflation-adjusted updating row vector for subproblems on the
i
-th level.
poles
Array,
DIMENSION
(
ldu
, 2*
nlvl
)
if
icompq
= 1
, and not referenced if
icompq
= 0
. If
icompq
= 1
, on exit,
poles
(1, 2
i
- 1) and
poles
(1, 2
i
) contain the new and old singular values involved in the secular equations on the
i
-th level.
givptr
Array,
DIMENSION
(
n
) if
icompq
= 1
, and not referenced if
icompq
= 0
. If
icompq
= 1
, on exit,
givptr
(
i
) records the number of Givens rotations performed on the
i
-th problem on the computation tree.
givcol
Array,
DIMENSION
(
ldgcol
, 2*
nlvl
)
if
icompq
= 1
, and not referenced if
icompq
= 0
. If
icompq
= 1
, on exit, for each
i
,
givcol
(1, 2
i
- 1) and
givcol
(1, 2
i
) record the locations of Givens rotations performed on the
i
-th level on the computation tree.
perm
Array,
DIMENSION
(
ldgcol
,
nlvl
) if
icompq
= 1
, and not referenced if
icompq
= 0
. If
icompq
= 1
, on exit,
perm
(1,
i
) records permutations done on the
i
-th level of the computation tree.
givnum
Array
DIMENSION
(
ldu
, 2*
nlvl
) if
icompq
= 1
, and not referenced if
icompq
= 0
. If
icompq
= 1
, on exit, for each
i
,
givnum
(1, 2
i
- 1) and
givnum
(1, 2
i
) record the
C
- and
S
-values of Givens rotations performed on the
i
-th level on the computation tree.
c
Array,
DIMENSION
(
n
) if
icompq
= 1
, and
DIMENSION
(1) if
icompq
= 0
.
If
icompq
= 1
and the
i
-th subproblem is not square, on exit,
c
(
i
) contains the
C
-value of a Givens rotation related to the right null space of the
i
-th subproblem.
s
Array,
DIMENSION
(
n
)
icompq
= 1
, and
DIMENSION
(1) if
icompq
= 0
.
If
icompq
= 1
and the
i
-th subproblem is not square, on exit,
s
(
i
) contains the
S
-value of a Givens rotation related to the right null space of the
i
-th subproblem.
info
= 0: successful exit.
< 0: if
info
= -
i
, the
i
-th argument had an illegal value
> 0: If
info
= 1
, an singular value did not converge

#### Product and Performance Information

1

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Notice revision #20110804