Contents

# ?tgevc

Computes some or all of the right and/or left generalized eigenvectors of a pair of upper triangular matrices.

## Syntax

Include Files
• mkl.h
Description
The routine computes some or all of the right and/or left eigenvectors of a pair of real/complex matrices (
S
,
P
), where
S
is quasi-triangular (for real flavors) or upper triangular (for complex flavors) and
P
is upper triangular.
Matrix pairs of this type are produced by the generalized Schur factorization of a real/complex matrix pair (
A
,
B
):
A
=
Q
*
S
*
Z
H
,
B
=
Q
*
P
*
Z
H
as computed by
?gghrd
plus
?hgeqz
.
The right eigenvector
x
and the left eigenvector
y
of (
S
,
P
) corresponding to an eigenvalue
w
are defined by:
S
*
x
=
w
*
P
*
x
,
y
H
*
S
=
w
*
y
H
*
P
The eigenvalues are not input to this routine, but are computed directly from the diagonal blocks or diagonal elements of
S
and
P
.
This routine returns the matrices
X
and/or
Y
of right and left eigenvectors of (
S
,
P
), or the products
Z
*
X
and/or
Q
*
Y
, where
Z
and
Q
are input matrices.
If
Q
and
Z
are the orthogonal/unitary factors from the generalized Schur factorization of a matrix pair (
A
,
B
), then
Z
*
X
and
Q
*
Y
are the matrices of right and left eigenvectors of (
A
,
B
).
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
side
Must be
'R'
,
'L'
, or
'B'
.
If
side
=
'R'
, compute right eigenvectors only.
If
side
=
'L'
, compute left eigenvectors only.
If
side
=
'B'
, compute both right and left eigenvectors.
howmny
Must be
'A'
,
'B'
, or
'S'
.
If
howmny
=
'A'
, compute all right and/or left eigenvectors.
If
howmny
=
'B'
, compute all right and/or left eigenvectors, backtransformed by the matrices in
vr
and/or
vl
.
If
howmny
=
'S'
, compute selected right and/or left eigenvectors, specified by the logical array
select
.
select
Array, size at least max (1,
n
).
If
howmny
=
'S'
,
select
specifies the eigenvectors to be computed.
If
howmny
=
'A'
or
'B'
,
select
is not referenced.
For real flavors
:
If
w
[
j
]
is a real eigenvalue, the corresponding real eigenvector is computed if
select
[
j
]
is
1
.
If
w
[
j
]
and
omega
[
j
+ 1]
are the real and imaginary parts of a complex eigenvalue, the corresponding complex eigenvector is computed if either
select
[
j
]
or
select
[
j
+ 1]
is
1
, and on exit
select
[
j
]
is set to
1
and
select
[
j
+ 1]
is set to
0
.
For complex flavors:
The eigenvector corresponding to the
j
-th eigenvalue is computed if
select
[
j
]
is
1
.
n
The order of the matrices
S
and
P
(
n
0
).
s
,
p
,
vl
,
vr
Arrays:
s
(size max(1,
lds
*
n
))
contains the matrix
S
from a generalized Schur factorization as computed by
?hgeqz
. This matrix is upper quasi-triangular for real flavors, and upper triangular for complex flavors.
p
(size max(1,
ldp
*
n
))
contains the upper triangular matrix
P
from a generalized Schur factorization as computed by
?hgeqz
.
For real flavors, 2-by-2 diagonal blocks of
P
corresponding to 2-by-2 blocks of
S
must be in positive diagonal form.
For complex flavors,
P
must have real diagonal elements.
If
side
=
'L'
or
'B'
and
howmny
=
'B'
,
vl
(size max(1,
ldvl
*
mm
) for column major layout and max(1,
ldvl
*
n
) for row major layout)
must contain an
n
-by-
n
matrix
Q
(usually the orthogonal/unitary matrix
Q
of left Schur vectors returned by
?hgeqz
).
If
side
=
'R'
, vl
is not referenced.
If
side
=
'R'
or
'B'
and
howmny
=
'B'
,
vr
(size max(1,
ldvr
*
mm
) for column major layout and max(1,
ldvr
*
n
) for row major layout)
must contain an
n
-by-
n
matrix
Z
(usually the orthogonal/unitary matrix
Z
of right Schur vectors returned by
?hgeqz
).
If
side
=
'L'
,
vr
is not referenced.
lds
s
; at least
max(1,
n
)
.
ldp
p
; at least
max(1,
n
)
.
ldvl
vl
;
If
side
=
'L'
or
'B'
, then
ldvl
n
for column major layout and
ldvl
max(1,
mm
) for row major layout
.
If
side
=
'R'
, then
ldvl
1 .
ldvr
vr
;
If
side
=
'R'
or
'B'
, then
ldvr
n
for column major layout and
ldvr
max(1,
mm
) for row major layout
.
If
side
=
'L'
, then
ldvr
1.
mm
The number of columns in the arrays
vl
and/or
vr
(
mm
m
).
Output Parameters
vl
On exit, if
side
=
'L'
or
'B'
,
vl
contains:
if
howmny
=
'A'
, the matrix
Y
of left eigenvectors of (
S
,
P
);
if
howmny
=
'B'
, the matrix
Q
*
Y
;
if
howmny
=
'S'
, the left eigenvectors of (
S
,
P
) specified by
select
, stored consecutively in the columns of
vl
, in the same order as their eigenvalues.
For real flavors
:
A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part.
vr
On exit, if
side
=
'R'
or
'B'
,
vr
contains:
if
howmny
=
'A'
, the matrix
X
of right eigenvectors of (
S
,
P
);
if
howmny
=
'B'
, the matrix
Z
*
X
;
if
howmny
=
'S'
, the right eigenvectors of (
S
,
P
) specified by
select
, stored consecutively in the columns of
vr
, in the same order as their eigenvalues.
For real flavors
:
A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part.
m
The number of columns in the arrays
vl
and/or
vr
actually used to store the eigenvectors.
If
howmny
=
'A'
or
'B'
,
m
is set to
n
.
For real flavors
:
Each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns.
For complex flavors
:
Each selected eigenvector occupies one column.
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.
For real flavors
:
if
info
=
i
>0
, the 2-by-2 block (
i
:
i
+1) does not have a complex eigenvalue.

#### Product and Performance Information

1

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Notice revision #20110804