Contents

# ?geesx

Computes the eigenvalues and Schur factorization of a general matrix, orders the factorization and computes reciprocal condition numbers.

## Syntax

Include Files
• mkl.h
Description
The routine computes for an
n
-by-
n
real/complex nonsymmetric matrix
A
, the eigenvalues, the real-Schur/Schur form
T
, and, optionally, the matrix of Schur vectors
Z
. This gives the Schur factorization
A
=
Z
*
T
*
Z
H
.
Optionally, it also orders the eigenvalues on the diagonal of the real-Schur/Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues (
rconde
); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues (
rcondv
Z
form an orthonormal basis for this invariant subspace.
For further explanation of the reciprocal condition numbers
rconde
and
rcondv
, see [LUG], Section 4.10 (where these quantities are called
s
and
sep
respectively).
A real matrix is in real-Schur form if it is upper quasi-triangular with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the form where
b
*
c
< 0
. The eigenvalues of such a block are A complex matrix is in Schur form if it is upper triangular.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
jobvs
Must be
'N'
or
'V'
.
If
jobvs
=
'N'
, then Schur vectors are not computed.
If
jobvs
=
'V'
, then Schur vectors are computed.
sort
Must be
'N'
or
'S'
. Specifies whether or not to order the eigenvalues on the diagonal of the Schur form.
If
sort
=
'N'
, then eigenvalues are not ordered.
If
sort
=
'S'
, eigenvalues are ordered (see
select
).
select
If
sort
=
'S'
,
select
is used to select eigenvalues to sort to the top left of the Schur form.
If
sort
=
'N'
,
select
is not referenced.
For real flavors
:
An eigenvalue
wr
[
j
]
+sqrt(-1)*
wi
[
j
] is selected if
select
(
wr
[
j
],
wi
[
j
])
is true; that is, if either one of a complex conjugate pair of eigenvalues is selected, then both complex eigenvalues are selected.
For complex flavors:
An eigenvalue
w
[
j
] is selected if
select
(
w
[
j
])
is true.
Note that a selected complex eigenvalue may no longer satisfy
select
(
wr
[
j
],
wi
[
j
])= 1
after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned); in this case
info
may be set to
n
+2 (see
info
below).
sense
Must be
'N'
,
'E'
,
'V'
, or
'B'
. Determines which reciprocal condition number are computed.
If
sense
=
'N'
, none are computed;
If
sense
=
'E'
, computed for average of selected eigenvalues only;
If
=
'V'
, computed for selected right invariant subspace only;
If
sense
=
'B'
, computed for both.
If
sense
is
'E'
,
'V'
, or
'B'
, then
sort
must equal
'S'
.
n
The order of the matrix
A
(
n
0
).
a
Arrays:
a
(size at least max(1,
lda
*
n
))
is an array containing the
n
-by-
n
matrix
A
.
lda
The leading dimension of the array
a
. Must be at least max(1,
n
).
ldvs
The leading dimension of the output array
vs
. Constraints:
ldvs
1
;
ldvs
max(1,
n
)
if
jobvs
=
'V'
.
Output Parameters
a
On exit, this array is overwritten by the real-Schur/Schur form
T
.
sdim
If
sort
=
'N'
,
sdim
= 0.
If
sort
=
'S'
,
sdim
is equal to the number of eigenvalues (after sorting) for which
select
is true.
Note that for real flavors complex conjugate pairs for which
select
is true for either eigenvalue count as 2.
wr
,
wi
Arrays, size at least max (1,
n
) each. Contain the real and imaginary parts, respectively, of the computed eigenvalues, in the same order that they appear on the diagonal of the output real-Schur form
T
. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having positive imaginary part first.
w
Array, size at least max(1,
n
). Contains the computed eigenvalues. The eigenvalues are stored in the same order as they appear on the diagonal of the output Schur form
T
.
vs
Array
vs
(size at least max(1,
ldvs
*
n
))
If
jobvs
=
'V'
,
vs
contains the orthogonal/unitary matrix
Z
of Schur vectors.
If
jobvs
=
'N'
,
vs
is not referenced.
rconde
,
rcondv
If
sense
=
'E'
or
'B'
,
rconde
contains the reciprocal condition number for the average of the selected eigenvalues.
If
sense
=
'N'
or
'V'
,
rconde
is not referenced.
If
sense
=
'V'
or
'B'
,
rcondv
contains the reciprocal condition number for the selected right invariant subspace.
If
sense
=
'N'
or
'E'
,
rcondv
is not referenced.
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
=
i
, and
i
n
:
the
QR
algorithm failed to compute all the eigenvalues; elements 1:
ilo
-1 and
i
+1:
n
of
wr
and
wi
(for real flavors) or
w
(for complex flavors) contain those eigenvalues which have converged; if
jobvs
=
'V'
,
vs
contains the transformation which reduces
A
to its partially converged Schur form;
i
=
n
+1
:
the eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned);
i
=
n
+2
:
after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy
select
= 1
. This could also be caused by underflow due to scaling.

#### Product and Performance Information

1

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