Contents

# ?trsna

Estimates condition numbers for specified eigenvalues and right eigenvectors of an upper (quasi-) triangular matrix.

## Syntax

Include Files
• mkl.h
Description
The routine estimates condition numbers for specified eigenvalues and/or right eigenvectors of an upper triangular matrix
T
(or, for real flavors, upper quasi-triangular matrix
T
in canonical Schur form). These are the same as the condition numbers of the eigenvalues and right eigenvectors of an original matrix
A
=
Z*T*Z
H
(with unitary or, for real flavors, orthogonal
Z
), from which
T
may have been derived.
The routine computes the reciprocal of the condition number of an eigenvalue
λ
i
as
s
i
= |
v
T
*
u
|/(||
u
||
E
||
v
||
E
)
for real flavors and
s
i
= |
v
H
*
u
|/(||
u
||
E
||
v
||
E
)
for complex flavors,
where:
• u
and
v
are the right and left eigenvectors of
T
, respectively, corresponding to
λ
i
.
• v
T
/
v
H
denote transpose/conjugate transpose of
v
, respectively.
This reciprocal condition number always lies between zero (ill-conditioned) and one (well-conditioned).
An approximate error estimate for a computed eigenvalue
λ
i
is then given by
ε
*||
T
||/
s
i
, where
ε
is the machine precision .
To estimate the reciprocal of the condition number of the right eigenvector corresponding to
λ
i
, the routine first calls trexc to reorder the diagonal elements of matrix
T
so that
λ
i The reciprocal condition number of the eigenvector is then estimated as
sep
i
, the smallest singular value of the matrix
T
22
-
λ
i
*
I
.
An approximate error estimate for a computed right eigenvector u corresponding to
λ
i
is then given by
ε
*||
T
||/sep
i
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
job
Must be
'E'
or
'V'
or
'B'
.
If
job
=
'E'
, then condition numbers for eigenvalues only are computed.
If
job
=
'V'
, then condition numbers for eigenvectors only are computed.
If
job
=
'B'
, then condition numbers for both eigenvalues and eigenvectors are computed.
howmny
Must be
'A'
or
'S'
.
If
howmny
=
'A'
, then the condition numbers for all eigenpairs are computed.
If
howmny
=
'S'
, then condition numbers for selected eigenpairs (as specified by
select
) are computed.
select
Array, size at least max (1,
n
) if
howmny
=
'S'
and at least 1 otherwise.
Specifies the eigenpairs for which condition numbers are to be computed if
howmny
=
'S'
.
For real flavors
:
To select condition numbers for the eigenpair corresponding to the real eigenvalue
λ
j
,
select
[
j
]
must be set
1
;
to select condition numbers for the eigenpair corresponding to a complex conjugate pair of eigenvalues
λ
j
and
λ
j
+ 1
),
select
[
j
- 1]
and/or
select
[
j
]
must be set
1
For complex flavors
To select condition numbers for the eigenpair corresponding to the eigenvalue
λ
j
,
select
[
j
]
must be set
1
select
is not referenced if
howmny
=
'A'
.
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
.
vl
(size max(1,
ldvl
*
mm
) for column major layout and max(1,
ldvl
*
n
) for row major layout)
If
job
=
'E'
or
'B'
, then
vl
must contain the left eigenvectors of
T
(or of any matrix
Q*T*Q
H
with
Q
unitary or orthogonal) corresponding to the eigenpairs specified by
howmny
and
select
. The eigenvectors must be stored in consecutive columns of
vl
, as returned by trevc or hsein .
The array
vl
is not referenced if
job
=
'V'
.
vr
(size max(1,
ldvr
*
mm
) for column major layout and max(1,
ldvr
*
n
) for row major layout)
If
job
=
'E'
or
'B'
, then
vr
must contain the right eigenvectors of
T
(or of any matrix
Q*T*Q
H
with
Q
unitary or orthogonal) corresponding to the eigenpairs specified by
howmny
and
select
. The eigenvectors must be stored in consecutive columns of
vr
, as returned by trevc or hsein .
The array
vr
is not referenced if
job
=
'V'
.
ldt
t
; at least max(1,
n
).
ldvl
vl
.
If
job
=
'E'
or
'B'
,
ldvl
max(1,
n
)
for column major layout and
ldvl
max(1,
mm
) for row major layout
.
If
job
=
'V'
,
ldvl
1
.
ldvr
vr
.
If
job
=
'E'
or
'B'
,
ldvr
max(1,
n
)
for column major layout and
ldvr
max(1,
mm
) for row major layout
.
If
job
=
'R'
,
ldvr
1
.
mm
The number of elements in the arrays
s
and
sep
, and the number of columns in
vl
and
vr
(if used). Must be at least
m
(the precise number required).
If
howmny
=
'A'
,
mm
=
n
;
if
howmny
=
'S'
,
for real flavors
mm
is obtained by counting 1 for each selected real eigenvalue and 2 for each selected complex conjugate pair of eigenvalues.
for complex flavors
mm
is the number of selected eigenpairs (see
select
). Constraint:
0
mm
n
.
Output Parameters
s
Array, size at least max(1,
mm
) if
job
=
'E'
or
'B'
and at least 1 if
job
=
'V'
.
Contains the reciprocal condition numbers of the selected eigenvalues if
job
=
'E'
or
'B'
, stored in consecutive elements of the array. Thus
s
[
j
- 1]
,
sep
[
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 have been selected).
For real flavors
: for a complex conjugate pair of eigenvalues, two consecutive elements of
s
are set to the same value. The array
s
is not referenced if
job
=
'V'
.
sep
Array, size at least max(1,
mm
) if
job
=
'V'
or
'B'
and at least 1 if
job
=
'E'
. Contains the estimated reciprocal condition numbers of the selected right eigenvectors if
job
=
'V'
or
'B'
, stored in consecutive elements of the array.
For real flavors
: for a complex eigenvector, two consecutive elements of
sep
are set to the same value; if the eigenvalues cannot be reordered to compute
sep
[
j
- 1]
, then
sep
[
j
- 1]
is set to zero; this can only occur when the true value would be very small anyway. The array
sep
is not referenced if
job
=
'E'
.
m
For complex flavors
: the number of selected eigenpairs.
If
howmny
=
'A'
,
m
is set to
n
.
For real flavors
: the number of elements of
s
and/or
sep
actually used to store the estimated condition numbers.
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.
Application Notes
The computed values
sep
i
may overestimate the true value, but seldom by a factor of more than 3.

#### Product and Performance Information

1

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804