Contents

# ?lasd6

Computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by
?bdsdc
.

## Syntax

Include Files
• mkl.h
Description
The routine
?lasd6
computes the
SVD
of an updated upper bidiagonal matrix
B
obtained by merging two smaller ones by appending a row. This routine is used only for the problem which requires all singular values and optionally singular vector matrices in factored form.
B
is an
n
-by-
m
matrix with
n
=
nl
+
nr
+ 1
and
m
=
n
+
sqre
. A related subroutine,
?lasd1
, handles the case in which all singular values and singular vectors of the bidiagonal matrix are desired.
?lasd6
computes the
SVD
as follows: =
U
(out)*(
D
(out)*
VT
(
out
)
where
Z
' = (
Z1
'
a
Z2
'
b
) =
u
'*
VT
'
, 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 singular values of
B
can be computed using
D1
,
D2
, the first components of all the right singular vectors of the lower block, and the last components of all the right singular vectors of the upper block. These components are stored and updated in
vf
and
vl
, respectively, in
?lasd6
. Hence
U
and
VT
are not explicitly referenced.
The singular values are stored in
D
. The algorithm consists of two stages:
1. The first stage consists of deflating the size of the problem when there are multiple singular values or if there is a zero 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
?lasd7
.
2. The second stage consists of calculating the updated singular values. This is done by finding the roots of the secular equation via the routine
?lasd4
(as called by
?lasd8
). This routine also updates
vf
and
vl
and computes the distances between the updated singular values and the old singular values.
?lasd6
is called from
?lasda
.
Input Parameters
icompq
Specifies whether singular vectors are to be computed in factored form:
= 0: Compute singular values only
= 1: Compute singular vectors in factored form as well.
nl
The row dimension of the upper block.
nl
1
.
nr
The row dimension of the lower block.
nr
1
.
sqre
= 0: the lower block is an
nr
-by-
nr
square matrix.
= 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 ). 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.
vf
Array,
dimension
(
m
).
On entry,
vf
(1:
nl
+1) contains the first components of all right singular vectors of the upper block; and
vf
(
nl
+2:
m
)
contains the first components of all right singular vectors of the lower block.
vl
Array,
dimension
(
m
).
On entry,
vl
(1:
nl
+1) contains the last components of all right singular vectors of the upper block; and
vl
(
nl
+2:
m
) contains the last components of all right singular vectors 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.
ldgcol
The leading dimension
of
the output array
givcol
, must be at least
n
.
ldgnum
The leading dimension of the output arrays
givnum
and
poles
, must be at least
n
.
work
Workspace array,
dimension
( 4
m
).
iwork
Workspace array,
dimension
( 3
n
).
Output Parameters
d
On exit
d
(1:
n
) contains the singular values of the modified matrix.
vf
On exit,
vf
contains the first components of all right singular vectors of the bidiagonal matrix.
vl
On exit,
vl
contains the last components of all right singular vectors of the bidiagonal 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
.
idxq
Array,
dimension
(
n
). This 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.
perm
Array,
dimension
(
n
). The permutations (from deflation and sorting) to be applied to each block. Not referenced if
icompq
= 0
.
givptr
The number of Givens rotations which took place in this subproblem. Not referenced if
icompq
= 0
.
givcol
Array,
dimension
(
ldgcol
, 2 ). Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Not referenced if
icompq
= 0
.
givnum
Array,
dimension
(
ldgnum
, 2 ). Each number indicates the
C
or
S
value to be used in the corresponding Givens rotation. Not referenced if
icompq
= 0
.
poles
Array,
dimension
(
ldgnum
, 2 ). On exit,
poles
(1,*) is an array containing the new singular values obtained from solving the secular equation, and
poles
(2,*) is an array containing the poles in the secular equation. Not referenced if
icompq
= 0
.
difl
Array,
dimension
(
n
). On exit,
difl
(
i
) is the distance between
i
-th updated (undeflated) singular value and the
i
-th (undeflated) old singular value.
difr
Array,
dimension
(
ldgnum
, 2 ) if
icompq
= 1
and
dimension
(
n
) if
icompq
= 0
.
On exit,
difr
(
i
, 1) is the distance between
i
-th updated (undeflated) singular value and the
i
+1-th (undeflated) old singular value. If
icompq
= 1,
difr
(1:
k
, 2)
is an array containing the normalizing factors for the right singular vector matrix.
See
?lasd8
for details on
difl
and
difr
.
z
Array,
dimension
(
m
).
The first elements of this array contain the components of the deflation-adjusted updating row vector.
k
Contains the dimension of the non-deflated matrix. This is the order of the related secular equation.
1 ≤
k
n
.
c
c
contains garbage if
sqre
=0
and the
C
-value of a Givens rotation related to the right null space if
sqre
= 1
.
s
s
contains garbage if
sqre
=0
and the
S
-value of a Givens rotation related to the right null space if
sqre
= 1
.
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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