Contents

# ?trevc

Computes selected eigenvectors of an upper (quasi-) triangular matrix computed by
?hseqr
.

## Syntax

Include Files
• mkl.h
Description
The routine computes some or all of the right and/or left eigenvectors of an upper triangular matrix
T
(or, for real flavors, an upper quasi-triangular matrix
T
). Matrices of this type are produced by the Schur factorization of a general matrix:
A
=
Q
*
T
*
Q
H
, as computed by hseqr.
The right eigenvector
x
and the left eigenvector
y
of
T
corresponding to an eigenvalue
w
, are defined by:
T
*
x
=
w
*
x
,
y
H
*
T
=
w
*
y
H
, where
y
H
denotes the conjugate transpose of
y
.
The eigenvalues are not input to this routine, but are read directly from the diagonal blocks of
T
.
This routine returns the matrices
X
and/or
Y
of right and left eigenvectors of
T
, or the products
Q
*
X
and/or
Q*
Y
, where
Q
is an input matrix.
If
Q
is the orthogonal/unitary factor that reduces a matrix
A
to Schur form
T
, then
Q
*
X
and
Q
*
Y
are the matrices of right and left eigenvectors of
A
.
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'
or
'L'
or
'B'
.
If
side
=
'R'
, then only right eigenvectors are computed.
If
side
=
'L'
, then only left eigenvectors are computed.
If
side
=
'B'
, then all eigenvectors are computed.
howmny
Must be
'A'
or
'B'
or
'S'
.
If
howmny
=
'A'
, then all eigenvectors (as specified by
side
) are computed.
If
howmny
=
'B'
, then all eigenvectors (as specified by
side
) are computed and backtransformed by the matrices supplied in
vl
and
vr
.
If
howmny
=
'S'
, then selected eigenvectors (as specified by
side
and
select
) are computed.
select
Array, size at least max (1,
n
).
If
howmny
=
'S'
,
select
specifies which eigenvectors are to be computed.
If
howmny
=
'A'
or
'B'
,
select
is not referenced.
For real flavors
:
If
omega
[
j
]
is a real eigenvalue, the corresponding real eigenvector is computed if
select
[
j
]
is
1
.
If
omega
[
j
- 1]
and
omega
[
j
]
are the real and imaginary parts of a complex eigenvalue, the corresponding complex eigenvector is computed if either
select
[
j
- 1]
or
select
[
j
]
is
1
, and on exit
select
[
j
- 1]
is set to
1
and
select
[
j
]
is set to
0
.
For complex flavors:
The eigenvector corresponding to the
j
-th eigenvalue is computed if
select
[
j
- 1]
is
1
.
n
The order of the matrix
T
(
n
0
).
t
,
vl
,
vr
Arrays:
t
(size max(1,
ldt
*
n
))
contains the
n
-by-
n
matrix
T
in Schur canonical form. For complex flavors
ctrevc
and
ztrevc
, contains the upper triangular matrix
T
.
vl
(size max(1,
ldvl
*
mm
) for column major layout and max(1,
ldvl
*
n
) for row major layout)
If
howmny
=
'B'
and
side
=
'L'
or
'B'
, then
vl
must contain an
n
-by-
n
matrix
Q
(usually the matrix of Schur vectors returned by
?hseqr
).
If
howmny
=
'A'
or
'S'
, then
vl
need not be set.
The array
vl
is not referenced if
side
=
'R'
.
vr
(size max(1,
ldvr
*
mm
) for column major layout and max(1,
ldvr
*
n
) for row major layout)
If
howmny
=
'B'
and
side
=
'R'
or
'B'
, then
vr
must contain an
n
-by-
n
matrix
Q
(usually the matrix of Schur vectors returned by
?hseqr
). .
If
howmny
=
'A'
or
'S'
, then
vr
need not be set.
The array
vr
is not referenced if
side
=
'L'
.
ldt
t
; at least max(1,
n
).
ldvl
vl
.
If
side
=
'L'
or
'B'
,
ldvl
n
.
If
side
=
'R'
,
ldvl
1
.
ldvr
vr
.
If
side
=
'R'
or
'B'
,
ldvr
n
.
If
side
=
'L'
,
ldvr
1
.
mm
The number of columns in the arrays
vl
and/or
vr
. Must be at least
m
(the precise number of columns required).
If
howmny
=
'A'
or
'B'
,
mm
=
n
.
If
howmny
=
'S'
:
for real flavors
,
mm
is obtained by counting 1 for each selected real eigenvector and 2 for each selected complex eigenvector;
for complex flavors
,
mm
is the number of selected eigenvectors (see
select
).
Constraint:
0
mm
n
.
Output Parameters
select
If a complex eigenvector of a real matrix was selected as specified above, then
select
[
j
]
is set to
1
and
select
[
j
+ 1]
to
0
t
ctrevc
/
ztrevc
modify the
t
array, which is restored on exit.
vl
,
vr
If
side
=
'L'
or
'B'
,
vl
contains the computed left eigenvectors (as specified by
howmny
and
select
).
If
side
=
'R'
or
'B'
,
vr
contains the computed right eigenvectors (as specified by
howmny
and
select
).
The eigenvectors treated column-wise form a rectangular
n
-by-
mm
matrix.
For real flavors
: a real eigenvector corresponding to a real eigenvalue occupies one column of the matrix; a complex eigenvector corresponding to a complex eigenvalue occupies two columns: the first column holds the real part of the eigenvector and the second column holds the imaginary part of the eigenvector. The matrix is stored in a one-dimensional array as described by
matrix_layout
(using either column major or row major layout).
m
For complex flavors
: the number of selected eigenvectors.
If
howmny
=
'A'
or
'B'
,
m
is set to
n
.
For real flavors
: the number of columns of
vl
and/or
vr
actually used to store the selected eigenvectors.
If
howmny
=
'A'
or
'B'
,
m
is set to
n
.
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.
Application Notes
If
x
i
is an exact right eigenvector and
y
i
is the corresponding computed eigenvector, then the angle
θ
(
y
i
,
x
i
)
between them is bounded as follows:
θ
(
y
i
,
x
i
)
(
c
(
n
)
ε
||
T
||
2
)/sep
i
where sep
i
is the reciprocal condition number of
x
i
. The condition number sep
i
may be computed by calling
?trsna
.

#### Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.