Contents

# ?tgsyl

Solves the generalized Sylvester equation.

## Syntax

Include Files
• mkl.h
Description
The routine solves the generalized Sylvester equation:
A
*
R
-
L
*
B
=
scale
*
C
D
*
R
-
L
*
E
=
scale
*
F
where
R
and
L
are unknown
m
-by-
n
matrices, (
A
,
D
), (
B
,
E
) and (
C
,
F
) are given matrix pairs of size
m
-by-
m
,
n
-by-
n
and
m
-by-
n
, respectively, with real/complex entries. (
A
,
D
) and (
B
,
E
) must be in generalized real-Schur/Schur canonical form, that is,
A
,
B
are upper quasi-triangular/triangular and
D
,
E
are upper triangular.
The solution (
R
,
L
) overwrites (
C
,
F
). The factor
scale
,
0
scale
1
, is an output scaling factor chosen to avoid overflow.
In matrix notation the above equation is equivalent to the following: solve
Z
*
x
=
scale
*
b
, where
Z
is defined as
Here
I
k
is the identity matrix of size
k
and
X
T
is the transpose/conjugate-transpose of
X
.
kron
(
X
,
Y
)
is the Kronecker product between the matrices
X
and
Y
.
If
trans
=
'T'
(for real flavors), or
trans
=
'C'
(for complex flavors), the routine
?tgsyl
solves the transposed/conjugate-transposed system
Z
T
*
y
=
scale
*
b
, which is equivalent to solve for
R
and
L
in
A
T
*
R
+
D
T
*
L
=
scale
*
C
R
*
B
T
+
L
*
E
T
=
scale
*(-
F
)
This case (
trans
=
'T'
for
stgsyl
/
dtgsyl
or
trans
=
'C'
for
ctgsyl
/
ztgsyl
) is used to compute an one-norm-based estimate of
Dif[(
A
,
D
), (
B
,
E
)]
, the separation between the matrix pairs (
A
,
D
) and (
B
,
E
).
If
ijob
≥ 1
,
?tgsyl
computes a Frobenius norm-based estimate of
Dif[(
A
,
D
), (
B
,
E
)]
. That is, the reciprocal of a lower bound on the reciprocal of the smallest singular value of
Z
. This is a level 3 BLAS algorithm.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
trans
Must be
'N'
,
'T'
, or
'C'
.
If
trans
=
'N'
, solve the generalized Sylvester equation.
If
trans
=
'T'
, solve the 'transposed' system (for real flavors only).
If
trans
=
'C'
, solve the ' conjugate transposed' system (for complex flavors only).
ijob
Specifies what kind of functionality to be performed:
If
ijob
=0
, solve the generalized Sylvester equation only;
If
ijob
=1
, perform the functionality of
ijob
=0
and
ijob
=3
;
If
ijob
=2
, perform the functionality of
ijob
=0
and
ijob
=4
;
If
ijob
=3
, only an estimate of Dif[(
A
,
D
), (
B
,
E
)] is computed (look ahead strategy is used);
If
ijob
=4
, only an estimate of Dif[(
A
,
D
), (
B
,
E
)] is computed (?gecon on sub-systems is used). If
trans
=
'T'
or
'C'
,
ijob
is not referenced.
m
The order of the matrices
A
and
D
, and the row dimension of the matrices
C
,
F
,
R
and
L
.
n
The order of the matrices
B
and
E
, and the column dimension of the matrices
C
,
F
,
R
and
L
.
a
,
b
,
c
,
d
,
e
,
f
Arrays:
a
(size max(1,
lda
*
m
))
contains the upper quasi-triangular (for real flavors) or upper triangular (for complex flavors) matrix
A
.
b
(size max(1,
ldb
*
n
))
contains the upper quasi-triangular (for real flavors) or upper triangular (for complex flavors) matrix
B
.
c
(size max(1,
ldc
*
n
) for column major layout and max(1,
ldc
*
m
) for row major layout)
contains the right-hand-side of the first matrix equation in the generalized Sylvester equation (as defined by
trans
)
d
(size max(1,
ldd
*
m
))
contains the upper triangular matrix
D
.
e
(size max(1,
lde
*
n
))
contains the upper triangular matrix
E
.
f
(size max(1,
ldf
*
n
) for column major layout and max(1,
ldf
*
m
) for row major layout)
contains the right-hand-side of the second matrix equation in the generalized Sylvester equation (as defined by
trans
)
lda
a
; at least max(1,
m
).
ldb
b
; at least max(1,
n
).
ldc
c
; at least max(1,
m
)
for column major layout and at least max(1,
n
) for row major layout
.
ldd
d
; at least max(1,
m
).
lde
e
; at least max(1,
n
).
ldf
f
; at least max(1,
m
)
for column major layout and at least max(1,
n
) for row major layout
.
Output Parameters
c
If
ijob
=0
, 1, or 2, overwritten by the solution
R
.
If
ijob
=3
or 4 and
trans
=
'N'
,
c
holds
R
, the solution achieved during the computation of the
Dif
-estimate.
f
If
ijob
=0
, 1, or 2, overwritten by the solution
L
.
If
ijob
=3
or 4 and
trans
=
'N'
,
f
holds
L
, the solution achieved during the computation of the
Dif
-estimate.
dif
On exit,
dif
is the reciprocal of a lower bound of the reciprocal of the Dif-function, that is,
dif
is an upper bound of
Dif[(
A
,
D
), (
B
,
E
)] = sigma_min(
Z
)
, where
Z
as defined in the description.
If
ijob
= 0
, or
trans
=
'T'
(for real flavors), or
trans
=
'C'
(for complex flavors),
dif
is not touched.
scale
On exit,
scale
is the scaling factor in the generalized Sylvester equation.
If
0 <
scale
< 1
,
c
and
f
hold the solutions
R
and
L
, respectively, to a slightly perturbed system but the input matrices
A
,
B
,
D
and
E
have not been changed.
If
scale
= 0
,
c
and
f
hold the solutions
R
and
L
, respectively, to the homogeneous system with
C
=
F
= 0
. Normally,
scale
= 1
.
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
> 0
, (
A
,
D
) and (
B
,
E
) have common or close eigenvalues.

#### 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