Contents

# ?trsyl

Solves Sylvester equation for real quasi-triangular or complex triangular matrices.

## Syntax

Include Files
• mkl.h
Description
The routine solves the Sylvester matrix equation
op(
A
)*
X
±
X
*op(
B
) =
α
*
C
, where
op(
A
) =
A
or
A
H
, and the matrices
A
and
B
are upper triangular (or, for real flavors, upper quasi-triangular in canonical Schur form);
α
1
is a scale factor determined by the routine to avoid overflow in
X
;
A
is
m
-by-
m
,
B
is
n
-by-
n
, and
C
and
X
are both
m
-by-
n
. The matrix
X
is obtained by a straightforward process of back substitution.
The equation has a unique solution if and only if
α
i
±
β
i
0
, where {
α
i
} and {
β
i
} are the eigenvalues of
A
and
B
, respectively, and the sign (+ or -) is the same as that used in the equation to be solved.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
trana
Must be
'N'
or
'T'
or
'C'
.
If
trana
=
'N'
, then
op(
A
) =
A
.
If
trana
=
'T'
, then
op(
A
) =
A
T
(real flavors only).
If
trana
=
'C'
then
op(
A
) =
A
H
.
tranb
Must be
'N'
or
'T'
or
'C'
.
If
tranb
=
'N'
, then
op(
B
) =
B
.
If
tranb
=
'T'
, then
op(
B
) =
B
T
(real flavors only).
If
tranb
=
'C'
, then
op(
B
) =
B
H
.
isgn
Indicates the form of the Sylvester equation.
If
isgn
= +1
,
op(
A
)*
X
+
X
*op(
B
) =
alpha
*
C
.
If
isgn
= -1
,
op(
A
)*
X
-
X
*op(
B
) =
alpha
*
C
.
m
The order of
A
, and the number of rows in
X
and
C
(
m
0).
n
The order of
B
, and the number of columns in
X
and
C
(
n
0).
a
,
b
,
c
Arrays:
a
(size max(1,
lda
*
m
))
contains the matrix
A
.
b
(size max(1,
ldb
*
n
))
contains the matrix
B
.
c
(size max(1,
ldc
*
n
) for column major layout and max(1,
ldc
*
m
for row major layout)
contains the matrix
C
.
lda
a
; at least max(1,
m
)
for column major layout and max(1,
n
) for row major layout
.
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
.
Output Parameters
c
Overwritten by the solution matrix
X
.
scale
The value of the scale factor
α
.
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
,
A
and
B
have common or close eigenvalues; perturbed values were used to solve the equation.
Application Notes
Let
X
be the exact,
Y
the corresponding computed solution, and
R
the residual matrix:
R
=
C
- (
AY
±
YB
). Then the residual is always small:
||
R
||
F
=
O
(
ε
)*(||
A
||
F
+||
B
||
F
)*||
Y
||
F
.
However,
Y
is not necessarily the exact solution of a slightly perturbed equation; in other words, the solution is not backwards stable.
For the forward error, the following bound holds:
||
Y
-
X
||
F
||
R
||
F
/sep(
A
,
B
)
but this may be a considerable overestimate. See [Golub96] for a definition of sep(
A
,
B
).
The approximate number of floating-point operations for real flavors is
m
*
n
*(
m
+
n
)
. For complex flavors it is 4 times greater.

#### Product and Performance Information

1

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Notice revision #20110804