Contents

# ?tgsna

Estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a pair of matrices in generalized real Schur canonical form.

## Syntax

Include Files
• mkl.h
Description
The real flavors
stgsna
/
dtgsna
of this routine estimate reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (
A
,
B
) in generalized real Schur canonical form (or of any matrix pair (
Q
*A*
Z
T
,
Q*
B*
Z
T
) with orthogonal matrices
Q
and
Z
.
(
A
,
B
) must be in generalized real Schur form (as returned by gges/gges), that is,
A
is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular.
The complex flavors
ctgsna
/
ztgsna
estimate reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (
A
,
B
). (
A
,
B
) must be in generalized Schur canonical form, that is,
A
and
B
are both upper triangular.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
job
Specifies whether condition numbers are required for eigenvalues or eigenvectors. Must be
'E'
or
'V'
or
'B'
.
If
job
=
'E'
, for eigenvalues only (compute
s
).
If
job
=
'V'
, for eigenvectors only (compute
dif
).
If
job
=
'B'
, for both eigenvalues and eigenvectors (compute both
s
and
dif
).
howmny
Must be
'A'
or
'S'
.
If
howmny
=
'A'
, compute condition numbers for all eigenpairs.
If
howmny
=
'S'
, compute condition numbers for selected eigenpairs specified by the logical array
select
.
select
Array, size at least max (1,
n
).
If
howmny
=
'S'
,
select
specifies the eigenpairs for which condition numbers are required.
If
howmny
=
'A'
,
select
is not referenced.
For real flavors
:
To select condition numbers for the eigenpair corresponding to a real eigenvalue
ω
j
,
select
[
j
- 1]
must be set to
1
; to select condition numbers corresponding to a complex conjugate pair of eigenvalues
ω
j
and
ω
j
+ 1
, either
select
[
j
- 1]
or
select
[
j
]
must be set to
1.
For complex flavors:
To select condition numbers for the corresponding
j
-th eigenvalue and/or eigenvector,
select
[
j
- 1]
must be set to
1
.
n
The order of the square matrix pair (
A
,
B
)
(
n
0).
a
,
b
,
vl
,
vr
Arrays:
a
(size max(1,
lda
*
n
))
contains the upper quasi-triangular (for real flavors) or upper triangular (for complex flavors) matrix
A
in the pair (
A
,
B
).
b
(size max(1,
ldb
*
n
))
contains the upper triangular matrix
B
in the pair (
A
,
B
).
If
job
=
'E'
or
'B'
,
vl
(size max(1,
ldvl
*
m
) for column major layout and max(1,
ldvl
*
n
) for row major layout)
must contain left eigenvectors of (
A
,
B
), corresponding to the eigenpairs specified by
howmny
and
select
. The eigenvectors must be stored in consecutive columns of
vl
, as returned by
?tgevc
.
If
job
=
'V'
,
vl
is not referenced.
If
job
=
'E'
or
'B'
,
vr
(size max(1,
ldvr
*
m
) for column major layout and max(1,
ldvr
*
n
) for row major layout)
must contain right eigenvectors of (
A
,
B
), corresponding to the eigenpairs specified by
howmny
and
select
. The eigenvectors must be stored in consecutive columns of
vr
, as returned by
?tgevc
.
If
job
=
'V'
,
vr
is not referenced.
lda
a
; at least max(1,
n
).
ldb
b
; at least max(1,
n
).
ldvl
vl
;
ldvl
1
.
If
job
=
'E'
or
'B'
, then
ldvl
max(1,
n
)
for column major layout and
ldvl
max(1,
m
) for row major layout
.
ldvr
vr
;
ldvr
1
.
If
job
=
'E'
or
'B'
, then
ldvr
max(1,
n
)
for column major layout and
ldvr
max(1,
m
) for row major layout
.
mm
The number of elements in the arrays
s
and
dif
(
mm
m
).
Output Parameters
s
Array,
size
mm
.
If
job
=
'E'
or
'B'
, contains the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array.
If
job
=
'V'
,
s
is not referenced.
For real flavors
:
For a complex conjugate pair of eigenvalues two consecutive elements of
s
are set to the same value. Thus,
s
[
j
- 1]
,
dif
[
j
- 1]
, and the
j
-th columns of
vl
and
vr
all correspond to the same eigenpair (but not in general the
j
-th eigenpair, unless all eigenpairs are selected).
dif
Array,
size
mm
.
If
job
=
'V'
or
'B'
, contains the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array.
If the eigenvalues cannot be reordered to compute
dif
[
j
]
,
dif
[
j
]
is set to 0; this can only occur when the true value would be very small anyway.
If
job
=
'E'
,
dif
is not referenced.
For real flavors
:
For a complex eigenvector, two consecutive elements of
dif
are set to the same value.
For complex flavors
:
For each eigenvalue/vector specified by
select
,
dif
stores a Frobenius norm-based estimate of Difl.
m
The number of elements in the arrays
s
and
dif
used to store the specified condition numbers; for each selected eigenvalue one element is used.
If
howmny
=
'A'
,
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.

#### Product and Performance Information

1

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