Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

Contents

?ggsvd3

Computes generalized SVD.

Syntax

lapack_int
LAPACKE_sggsvd3
(
int
matrix_layout
,
char
jobu
,
char
jobv
,
char
jobq
,
lapack_int
m
,
lapack_int
n
,
lapack_int
p
,
lapack_int
*
k
,
lapack_int
*
l
,
float
*
a
,
lapack_int
lda
,
float
*
b
,
lapack_int
ldb
,
float
*
alpha
,
float
*
beta
,
float
*
u
,
lapack_int
ldu
,
float
*
v
,
lapack_int
ldv
,
float
*
q
,
lapack_int
ldq
,
lapack_int
*
iwork
);
lapack_int
LAPACKE_dggsvd3
(
int
matrix_layout
,
char
jobu
,
char
jobv
,
char
jobq
,
lapack_int
m
,
lapack_int
n
,
lapack_int
p
,
lapack_int
*
k
,
lapack_int
*
l
,
double
*
a
,
lapack_int
lda
,
double
*
b
,
lapack_int
ldb
,
double
*
alpha
,
double
*
beta
,
double
*
u
,
lapack_int
ldu
,
double
*
v
,
lapack_int
ldv
,
double
*
q
,
lapack_int
ldq
,
lapack_int
*
iwork
);
lapack_int
LAPACKE_cggsvd3
(
int
matrix_layout
,
char
jobu
,
char
jobv
,
char
jobq
,
lapack_int
m
,
lapack_int
n
,
lapack_int
p
,
lapack_int
*
k
,
lapack_int
*
l
,
lapack_complex_float
*
a
,
lapack_int
lda
,
lapack_complex_float
*
b
,
lapack_int
ldb
,
float
*
alpha
,
float
*
beta
,
lapack_complex_float
*
u
,
lapack_int
ldu
,
lapack_complex_float
*
v
,
lapack_int
ldv
,
lapack_complex_float
*
q
,
lapack_int
ldq
,
lapack_int
*
iwork
);
lapack_int
LAPACKE_zggsvd3
(
int
matrix_layout
,
char
jobu
,
char
jobv
,
char
jobq
,
lapack_int
m
,
lapack_int
n
,
lapack_int
p
,
lapack_int
*
k
,
lapack_int
*
l
,
lapack_complex_double
*
a
,
lapack_int
lda
,
lapack_complex_double
*
b
,
lapack_int
ldb
,
double
*
alpha
,
double
*
beta
,
lapack_complex_double
*
u
,
lapack_int
ldu
,
lapack_complex_double
*
v
,
lapack_int
ldv
,
lapack_complex_double
*
q
,
lapack_int
ldq
,
lapack_int
*
iwork
);
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