Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

Contents

?ggsvd

Computes the generalized singular value decomposition of a pair of general rectangular matrices (deprecated).

Syntax

lapack_int LAPACKE_sggsvd
(
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_dggsvd
(
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_cggsvd
(
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_zggsvd
(
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
This routine is deprecated; use
ggsvd3
.
The routine computes the generalized singular value decomposition (GSVD) of an
m
-by-
n
 real/complex matrix
A
 and
p
-by-
n
 real/complex matrix
B
:
U'
*
A
*
Q
 =
D
1
*(0
R
)
,
V'
*
B
*
Q
 =
D
2
*(0
R
)
,
where
U
,
V
 and
Q
 are orthogonal/unitary matrices and
U'
,
V'
 mean transpose/conjugate transpose of
U
 and
V
 respectively.
Let
k
+
l
 = the effective numerical rank of the matrix (
A'
,
B'
)', 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,
Equation
Equation
Equation
where
C
 = diag(
alpha
[
k
],...,
alpha
[
k
 +
l
 - 1]
)
S
 = diag(
beta
[
k
],...,
beta
[
k
 +
l
 - 1]
)
C
2
 +
S
2
 = I
Nonzero element
r
i j
 (1
≤
i
≤
j
≤
k
 +
l
) of
R
 is stored in
a
[(
i
 - 1) + (
n
 -
k
 -
l
 +
j
 - 1)*
lda
]
 for column major layout and in
a
[(
i
 - 1)*
lda
 + (
n
 -
k
 -
l
 +
j
 - 1)]
 for row major layout.
If
m
-
k
-l
 < 0
,
Equation
Equation
Equation
where
C
 = diag(
alpha
[
k
],...,
alpha
(
m
)
)
,
S
 = diag(
beta
[
k
],...,
beta
[
m
 - 1]
)
,
C
2
 +
S
2
 = I
On exit, the location of nonzero element
r
i j
 (1
≤
i
≤
j
≤
k
 +
l
) of
R
 depends on the value of
i
. For
i
≤
m
 this element is stored in
a
[(
i
 - 1) + (
n
 -
k
 -
l
 +
j
 - 1)*
lda
]
 for column major layout and in
a
[(
i
 - 1)*
lda
 + (
n
 -
k
 -
l
 +
j
 - 1)]
 for row major layout. For
m
 <
i
≤
k
 +
l
 it is stored in
b
[(
i
 -
k
 - 1) + (
n
 -
k
 -
l
 +
j
 - 1)*
ldb
]
 for column major layout and in
b
[(
i
 -
k
 - 1)*
ldb
 + (
n
 -
k
 -
l
 +
j
 - 1)]
 for row major layout.
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
*
B
-1
:
A
*
B
-1
 =
U
*(
D
1
*
D
2
-1
)*
V'
.
If (
A
',
B
')' 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'
*
*A
*
x
 =
λ
*
B'
*
B
*
x
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
jobu
Must be
'U'
 or
'N'
.
If
jobu
 =
'U'
, orthogonal/unitary matrix
U
 is computed.
If
jobu
 =
'N'
,
U
 is not computed.
jobv
Must be
'V'
 or
'N'
.
If
jobv
 =
'V'
, orthogonal/unitary matrix
V
 is computed.
If
jobv
 =
'N'
,
V
 is not computed.
jobq
Must be
'Q'
 or
'N'
.
If
jobq
 =
'Q'
, orthogonal/unitary matrix
Q
 is computed.
If
jobq
 =
'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
,
b
Arrays:
a
(size at least max(1,
lda
*
n
) for column major layout and max(1,
lda
*
m
) for row major layout)
 contains the
m
-by-
n
 matrix
A
.
b
(size at least max(1,
ldb
*
n
) for column major layout and max(1,
ldb
*
p
) for row major layout)
 contains the
p
-by-
n
 matrix
B
.
lda
The leading dimension of
a
; at least max(1,
m
)
for column major layout and max(1,
n
) for row major layout
.
ldb
The leading dimension of
b
; at least max(1,
p
)
for column major layout and max(1,
n
) for row major layout
.
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.
Output Parameters
k
,
l
On exit,
k
 and
l
 specify the dimension of the subblocks. The sum
k
+l
 is equal to the effective numerical rank of (
A'
,
B'
)'.
a
On exit,
a
 contains the triangular matrix
R
 or part of
R
.
b
On exit,
b
 contains part of the triangular matrix R if
m
-
k
-
l
 < 0
.
alpha
,
beta
Arrays, size at least max(1,
n
) each.
Contain the generalized singular value pairs of
A
 and
B
:
alpha
(1:
k
) = 1
,
beta
(1:
k
) = 0
,
and if
m
-
k
-
l
≥
 0
,
alpha
(
k
+1:
k
+
l
) =
C
,
beta
(
k
+1:
k
+
l
) =
S
,
or if
m
-
k
-
l
 < 0
,
alpha
(
k
+1:
m
)=
C
,
alpha
(
m
+1: