Contents

# ?ggsvd3

Computes generalized SVD.

## Syntax

Include Files
• mkl.h
Description
?ggsvd3
computes the generalized singular value decomposition (GSVD) of an
m
-by-
n
real or complex matrix
A
and
p
-by-
n
real or complex matrix
B
:
U
T
*
A
*
Q
=
D
1
*( 0
R
),
V
T
*
B
*
Q
=
D
2
*( 0
R
) for real flavors
or
U
H
*
A
*
Q
=
D
1
*( 0
R
),
V
H
*
B
*
Q
=
D
2
*( 0
R
) for complex flavors
where
U
,
V
and
Q
are orthogonal/unitary matrices.
Let
k
+
l
= the effective numerical rank of the matrix (
A
T
B
T
)
T
for real flavors or the matrix (
A
H
,
B
H
)
H
for complex flavors, then
R
is a (
k
+
l
)-by-(
k
+
l
) nonsingular upper triangular matrix,
D
1
and
D
2
are
m
-by-(
k
+
l
) and
p
-by-(
k
+
l
) "diagonal" matrices and of the following structures, respectively:
If
m
-
k
-
l
0,
where
C
= diag(
alpha
(
k
+1), ... ,
alpha
(
k
+
l
) ),
S
= diag(
beta
(
k
+1), ... ,
beta
(
k
+
l
) ),
C
2
+
S
2
=
I
.
If
m
-
k
-
l
< 0,
where
C
= diag(
alpha
(
k
+ 1), ... ,
alpha
(
m
)),
S
= diag(
beta
(
k
+ 1), ... ,
beta
(
m
)),
C
2
+
S
2
=
I
.
The routine computes
C
,
S
,
R
, and optionally the orthogonal/unitary transformation matrices
U
,
V
and
Q
.
In particular, if
B
is an
n
-by-
n
nonsingular matrix, then the GSVD of
A
and
B
implicitly gives the SVD of
A
*inv(
B
):
A
*inv(
B
) =
U
*(
D
1
*inv(
D
2
))*
V
T
for real flavors
or
A
*inv(
B
) =
U
*(
D
1
*inv(
D
2
))*
V
H
for complex flavors.
If (
A
T
,
B
T
)
T
for real flavors or (
A
H
,
B
H
)
H
for complex flavors has orthonormal columns, then the GSVD of
A
and
B
is also equal to the CS decomposition of
A
and
B
. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem:
A
T
*
A
X
=
λ
*
B
T
*
B
X
for real flavors
or
A
H
*
A
X
=
λ
*
B
H
*
B
X
for complex flavors
In some literature, the GSVD of
A
and
B
is presented in the form
U
T
*
A
*
X
= ( 0
D
1
),
V
T
*
B
*
X
= ( 0
D
2
) for real (
A
,
B
)
or
U
H
*
A
*
X
= ( 0
D
1
),
V
H
*
B
*
X
= ( 0
D
2
) for complex (
A
,
B
)
where
U
and
V
are orthogonal and
X
is nonsingular,
D
1
and
D
2
are "diagonal''. The former GSVD form can be converted to the latter form by taking the nonsingular matrix
X
as
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
jobu
= 'U': Orthogonal/unitary matrix
U
is computed;
= 'N':
U
is not computed.
jobv
= 'V': Orthogonal/unitary matrix
V
is computed;
= 'N':
V
is not computed.
jobq
= 'Q': Orthogonal/unitary matrix
Q
is computed;
= 'N':
Q
is not computed.
m
The number of rows of the matrix
A
.
m
0.
n
The number of columns of the matrices
A
and
B
.
n
0.
p
The number of rows of the matrix
B
.
p
0.
a
Array, size
(
lda
*
n
)
.
On entry, the
m
-by-
n
matrix
A
.
lda
The leading dimension of the array
a
.
lda
max(1,
m
).
b
Array, size
(
ldb
*
n
)
.
On entry, the
p
-by-
n
matrix
B
.
ldb
The leading dimension of the array
b
.
ldb
max(1,
p
).
ldu
The leading dimension of the array
u
.
ldu
max(1,
m
) if
jobu
= 'U';
ldu
1 otherwise.
ldv
The leading dimension of the array
v
.
ldv
max(1,
p
) if
jobv
= 'V';
ldv
1 otherwise.
ldq
The leading dimension of the array
q
.
ldq
max(1,
n
) if
jobq
= 'Q';
ldq
1 otherwise.
iwork
Array, size (
n
).
Output Parameters
k
,
l
On exit,
k
and
l
specify the dimension of the subblocks described in the Description section.
k
+
l
= effective numerical rank of (
A
T
,
B
T
)
T
for real flavors or (
A
H
,
B
H
)
H
for complex flavors.
a
On exit,
a
contains the triangular matrix
R
, or part of
R
.
If
m
-
k
-
l
0,
R
is stored in
the elements of array
a
corresponding to
A
1:
k
+
l
,
n
-
k
-
l
+ 1:
n
.
If
m
-
k
-
l
< 0,
is stored in
the elements of array
a
corresponding to
A
(1:
m
,
n
-
k
-
l
+ 1:
n
, and
R33
is stored in
b