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
.
alpha
,
beta
Arrays, size at least max(1,
n
). 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
) = diag(
C
)
,
beta
(
k
+1:
k
+
l
) = diag(
S
)
,
or if
m
-
k
-
l
< 0
,
alpha
(
k
+1:
m
)= diag(
C
)
,
alpha
(
m
+1:
k
+
l
)=0
beta
(
k
+1:
m
) = diag(
S
)
,
beta
(
m
+1:
k
+
l
) = 1
.
Furthermore, if
k
+
l
<
n
,
alpha
(
k
+
l
+1:
n
)= 0
and
beta
(
k
+
l
+1:
n
) = 0
.
u
If
jobu
=
'I'
,
u
contains the orthogonal/unitary matrix
U
.
If
jobu
=
'U'
,
u
contains the product
U
1
*U
.
If
jobu
=
'N'
,
u
is not referenced.
v
If
jobv
=
'I'
,
v
contains the orthogonal/unitary matrix
U
.
If
jobv
=
'V'
,
v
contains the product
V
1
*V
.
If
jobv
=
'N'
,
v
is not referenced.
q
If
jobq
=
'I'
,
q
contains the orthogonal/unitary matrix
U
.
If
jobq
=
'Q'
,
q
contains the product
Q
1
*Q
.
If
jobq
=
'N'
,
q
is not referenced.
ncycle
The number of cycles required for convergence.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
=
-i
, the
i
-th parameter had an illegal value.
If
info
= 1
, the procedure does not converge after MAXIT cycles.

Product and Performance Information

1

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804