## Developer Reference

• 0.9
• 09/09/2020
• Public Content
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.