Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

Contents

?tgsja

Computes the generalized SVD of two upper triangular or trapezoidal matrices.

Syntax

lapack_int LAPACKE_stgsja
(
int
matrix_layout
,
char
jobu
,
char
jobv
,
char
jobq
,
lapack_int
m
,
lapack_int
p
,
lapack_int
n
,
lapack_int
k
,
lapack_int
l
,
float*
a
,
lapack_int
lda
,
float*
b
,
lapack_int
ldb
,
float
tola
,
float
tolb
,
float*
alpha
,
float*
beta
,
float*
u
,
lapack_int
ldu
,
float*
v
,
lapack_int
ldv
,
float*
q
,
lapack_int
ldq
,
lapack_int*
ncycle
);
lapack_int LAPACKE_dtgsja
(
int
matrix_layout
,
char
jobu
,
char
jobv
,
char
jobq
,
lapack_int
m
,
lapack_int
p
,
lapack_int
n
,
lapack_int
k
,
lapack_int
l
,
double*
a
,
lapack_int
lda
,
double*
b
,
lapack_int
ldb
,
double
tola
,
double
tolb
,
double*
alpha
,
double*
beta
,
double*
u
,
lapack_int
ldu
,
double*
v
,
lapack_int
ldv
,
double*
q
,
lapack_int
ldq
,
lapack_int*
ncycle
);
lapack_int LAPACKE_ctgsja
(
int
matrix_layout
,
char
jobu
,
char
jobv
,
char
jobq
,
lapack_int
m
,
lapack_int
p
,
lapack_int
n
,
lapack_int
k
,
lapack_int
l
,
lapack_complex_float*
a
,
lapack_int
lda
,
lapack_complex_float*
b
,
lapack_int
ldb
,
float
tola
,
float
tolb
,
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*
ncycle
);
lapack_int LAPACKE_ztgsja
(
int
matrix_layout
,
char
jobu
,
char
jobv
,
char
jobq
,
lapack_int
m
,
lapack_int
p
,
lapack_int
n
,
lapack_int
k
,
lapack_int
l
,
lapack_complex_double*
a
,
lapack_int
lda
,
lapack_complex_double*
b
,
lapack_int
ldb
,
double
tola
,
double
tolb
,
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*
ncycle
);
Include Files
  • mkl.h
Description
The routine computes the generalized singular value decomposition (GSVD) of two real/complex upper triangular (or trapezoidal) matrices
A
 and
B
. On entry, it is assumed that matrices
A
 and
B
 have the following forms, which may be obtained by the preprocessing subroutine ggsvp from a general
m
-by-
n
 matrix
A
 and
p
-by-
n
 matrix
B
:
Equation
Equation
Equation
where the
k
-by-
k
 matrix
A
12
 and
l
-by-
l
 matrix
B
13
 are nonsingular upper triangular;
A
23
 is
l
-by-
l
 upper triangular if
m
-
k
-l
≥
0
, otherwise
A
23
 is (
m
-
k
)-by-
l
 upper trapezoidal.
On exit,
U
H
*
A
*
Q
 =
D
1
*(0
R
)
,
V
H
*
B
*
Q
 =
D
2
*(0
R
)
,
where
U
,
V
 and
Q
 are orthogonal/unitary matrices,
R
 is a nonsingular upper triangular matrix, and
D
1
 and
D
2
 are "diagonal" matrices, which are of the following structures:
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
R
 is stored in
a
(1:
k
+
l
,
n
-
k
-
l
+1:
n
 ) on exit.
If
m
-
k
-l
 < 0
,
Equation
Equation
Equation
where
C
 = diag(
alpha
[
k
],...,
alpha
[
m
-1]
)
,
S
 = diag(
beta
[
k
],...,
beta
[
m
-1]
)
,
C
2
 +
S
2
 = I
On exit, Equation is stored in
a
(1:
m
,
n
-
k
-
l
+1:
n
 ) and
R
33
 is stored
in
b
(
m
-
k
+1:
l
,
n
+
m
-
k
-
l
+1:
n
 ).
The computation of the orthogonal/unitary transformation matrices
U
,
V
 or
Q
 is optional. These matrices may either be formed explicitly, or they
may
 be postmultiplied into input matrices
U
1
,
V
1
, or
Q
1
.
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'
,
'I'
, or
'N'
.
If
jobu
 =
'U'
,
u
 must contain an orthogonal/unitary matrix
U
1
 on entry.
If
jobu
 =
'I'
,
u
 is initialized to the unit matrix.
If
jobu
 =
'N'
,
u
 is not computed.
jobv
Must be
'V'
,
'I'
, or
'N'
.
If
jobv
 =
'V'
,
v
 must contain an orthogonal/unitary matrix
V
1
 on entry.
If
jobv
 =
'I'
,
v
 is initialized to the unit matrix.
If
jobv
 =
'N'
,
v
 is not computed.
jobq
Must be
'Q'
,
'I'
, or
'N'
.
If
jobq
 =
'Q'
,
q
 must contain an orthogonal/unitary matrix
Q
1
 on entry.
If
jobq
 =
'I'
,
q
 is initialized to the unit matrix.
If
jobq
 =
'N'
,
q
 is not computed.
m
The number of rows of the matrix
A
 (
m
≥
 0).
p
The number of rows of the matrix
B
 (
p
≥
 0).
n
The number of columns of the matrices
A
 and
B
 (
n
≥
 0).
k
,
l
Specify the subblocks in the input matrices
A
 and
B
, whose GSVD is computed.
a
,
b
,
u
,
v
,
q
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
.
If
jobu
 =
'U'
,
u
(size max(1,
ldu
*
m
))
 must contain a matrix
U
1
 (usually the orthogonal/unitary matrix returned by
?ggsvp
).
If
jobv
 =
'V'
,
v
(size at least max(1,
ldv
*
p
))
 must contain a matrix
V
1
 (usually the orthogonal/unitary matrix returned by
?ggsvp
).
If
jobq
 =
'Q'
,
q
(size at least max(1,
ldq
*
n
))
 must contain a matrix
Q
1
 (usually the orthogonal/unitary matrix returned by
?ggsvp
).
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.
tola
,
tolb
tola
 and
tolb
 are the convergence criteria for the Jacobi-Kogbetliantz iteration procedure. Generally, they are the same as used in
?ggsvp
:
tola
 = max(
m
,
n
)*|
A
|*MACHEPS
,
tolb
 = max(
p
,
n
)*|
B
|*MACHEPS
.
Output Parameters
a
On exit,
a
(
n
-
k
+1:
n
, 1:min(
k
+
l
,
m
)) contains the triangular matrix
R
 or part of
R
.
b
On exit, if necessary,
b
(
m
-
k
+1:
l
,
n
+
m
-
k
-
l
+1:
n
)) contains a part of
R
.