Contents

# ?lasd1

Computes the SVD of an upper bidiagonal matrix B of the specified size. Used by
?bdsdc
.

## Syntax

Include Files
• mkl.h
Description
The routine computes the SVD of an upper bidiagonal
n
-by-
m
matrix
B
, where
n
=
nl
+
nr
+ 1
and
m
=
n
+
sqre
.
The routine
?lasd1
is called from
?lasd0
.
A related subroutine
?lasd7
handles the case in which the singular values (and the singular vectors in factored form) are desired.
?lasd1
computes the SVD as follows: =
U
(
out
)*(
D
(
out
) 0)*
VT
(
out
)
where
Z
T
= (
Z1
T
a
Z2
T
b
) =
u
T
*
VT
T
, and
u
is a vector of dimension
m
with
alpha
and
beta
in the
nl
+1 and
nl
+2-th entries and zeros elsewhere; and the entry
b
is empty if
sqre
= 0
.
The left singular vectors of the original matrix are stored in
u
, and the transpose of the right singular vectors are stored in
vt
, and the singular values are in
d
. The algorithm consists of three stages:
1. The first stage consists of deflating the size of the problem when there are multiple singular values or when there are zeros in the
Z
vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine
?lasd2
.
2. The second stage consists of calculating the updated singular values. This is done by finding the square roots of the roots of the secular equation via the routine
?lasd4
(as called by
?lasd3
). This routine also calculates the singular vectors of the current problem.
3. The final stage consists of computing the updated singular vectors directly using the updated singular values. The singular vectors for the current problem are multiplied with the singular vectors from the overall problem.
Input Parameters
nl
The row dimension of the upper block.
nl
1
.
nr
The row dimension of the lower block.
nr
1
.
sqre
If
sqre
= 0
: the lower block is an
nr
-by-
nr
square matrix.
If
sqre
= 1
: the lower block is an
nr
-by-(
nr
+1) rectangular matrix. The bidiagonal matrix has row dimension
n
=
nl
+
nr
+ 1
, and column dimension
m
=
n
+
sqre
.
d
Array,
DIMENSION
(
nl
+
nr
+1
).
n
=
nl
+
nr
+1
. On entry
d
(1:
nl
,1:
nl
)
contains the singular values of the upper block; and
d
(
nl
+2:
n
)
contains the singular values of the lower block.
alpha
Contains the diagonal element associated with the added row.
beta
Contains the off-diagonal element associated with the added row.
u
Array,
DIMENSION
(
ldu
,
n
). On entry
u
(1:
nl
, 1:
nl
)
contains the left singular vectors of the upper block;
u
(
nl
+2:
n
,
nl
+2:
n
)
contains the left singular vectors of the lower block.
ldu
The leading dimension of the array
U
.
ldu
max(1,
n
)
.
vt
Array,
DIMENSION
(
ldvt
,
m
), where
m
=
n
+
sqre
.
On entry
vt
(1:
nl
+1, 1:
nl
+1)
T
contains the right singular vectors of the upper block;
vt
(
nl
+2:
m
,
nl
+2:
m
)
T
contains the right singular vectors of the lower block.
ldvt
The leading dimension of the array
vt
.
ldvt
max(1,
M
)
.
iwork
Workspace array,
DIMENSION
(4
n
).
work
Workspace array,
DIMENSION
(
3
m
2
+ 2
m
).
Output Parameters
d
On exit
d
(1:
n
) contains the singular values of the modified matrix.
alpha
On exit, the diagonal element associated with the added row deflated by
max( abs(
alpha
), abs(
beta
), abs( D(I) ) )
,
I = 1,n
.
beta
On exit, the off-diagonal element associated with the added row deflated by
max( abs(
alpha
), abs(
beta
), abs( D(I) ) )
,
I = 1,n
.
u
On exit
u
contains the left singular vectors of the bidiagonal matrix.
vt
On exit
vt
T
contains the right singular vectors of the bidiagonal matrix.
idxq
Array,
DIMENSION
(
n
). Contains the permutation which will reintegrate the subproblem just solved back into sorted order, that is,
d
(
idxq
(
i
= 1,
n
))
will be in ascending order.
info
If
info
= 0
: successful exit.
If
info
= -
i
< 0
, the
i
-th argument had an illegal value.
If
info
= 1
, a singular value did not converge.

#### Product and Performance Information

1

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Notice revision #20110804