Contents

# ?stevr

Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix using the Relatively Robust Representations.

## Syntax

Include Files
• mkl.h
Description
The routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix
T
. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
Whenever possible, the routine calls stemr to compute the eigenspectrum using Relatively Robust Representations. stegr computes eigenvalues by the
dqds
algorithm, while orthogonal eigenvectors are computed from various "good"
L*D*L
T
representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the i-th unreduced block of
T
:
1. Compute
T
-
σ
i
=
L
i
*D
i
*L
i
T
, such that
L
i
*D
i
*L
i
T
is a relatively robust representation.
2. Compute the eigenvalues,
λ
j
, of
L
i
*D
i
*L
i
T
to high relative accuracy by the
dqds
algorithm.
3. If there is a cluster of close eigenvalues, "choose"
σ
i
close to the cluster, and go to Step (a).
4. Given the approximate eigenvalue
λ
j
of
L
i
*D
i
*L
i
T
, compute the corresponding eigenvector by forming a rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the input parameter
abstol
.
The routine
?stevr
calls stemr when the full spectrum is requested on machines which conform to the IEEE-754 floating point standard.
?stevr
calls stebz and stein on non-IEEE machines and when partial spectrum requests are made.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
jobz
Must be
'N'
or
'V'
.
If
jobz
=
'N'
, then only eigenvalues are computed.
If
jobz
=
'V'
, then eigenvalues and eigenvectors are computed.
range
Must be
'A'
or
'V'
or
'I'
.
If
range
=
'A'
, the routine computes all eigenvalues.
If
range
=
'V'
, the routine computes eigenvalues
w
[
i
]
in the half-open interval:
vl
<
w
[
i
]
vu
.
If
range
=
'I'
, the routine computes eigenvalues with indices
il
to
iu
.
For
range
=
'V'
or
'I'
and
iu
-
il
<
n
-
1,
sstebz
/
dstebz
and
sstein
/
dstein
are called.
n
The order of the matrix
T
(
n
0
).
d
,
e
Arrays:
d
contains the
n
diagonal elements of the tridiagonal matrix
T
.
The dimension of
d
must be at least max(1,
n
).
e
contains the
n
-1 subdiagonal elements of
A
.
The dimension of
e
must be at least max(1,
n
-1). The
n
-th element of this array is used as workspace.
vl
,
vu
If
range
=
'V'
, the lower and upper bounds of the interval to be searched for eigenvalues.
Constraint:
vl
<
vu
.
If
range
=
'A'
or
'I'
,
vl
and
vu
are not referenced.
il
,
iu
If
range
=
'I'
, the indices in ascending order of the smallest and largest eigenvalues to be returned.
Constraint:
1
il
iu
n
, if
n
> 0
;
il
=1
and
iu
=0
if
n
= 0
.
If
range
=
'A'
or
'V'
,
il
and
iu
are not referenced.
abstol
The absolute error tolerance to which each eigenvalue/eigenvector is required.
If
jobz
=
'V'
, the eigenvalues and eigenvectors output have residual norms bounded by
abstol
, and the dot products between different eigenvectors are bounded by
abstol
. If
abstol
<
n
*eps*||
T
||
, then
n
*eps*||
T
||
will be used in its place, where
eps
is the machine precision, and
||
T
||
is the 1-norm of the matrix
T
. The eigenvalues are computed to an accuracy of
eps*||
T
||
irrespective of
abstol
.
If high relative accuracy is important, set
abstol
to
?lamch
('S').
ldz
The leading dimension of the output array
z
.
Constraints:
ldz
1
if
jobz
=
'N'
;
ldz
max(1,
n
)
for column major layout and
ldz
max(1,
m
) for row major layout
if
jobz
=
'V'
.
Output Parameters
m
The total number of eigenvalues found,
0
m
n
. If
range
=
'A'
,
m
=
n
, if
range
=
'I'
,
m
=
iu
-
il
+1
, and if
range
=
'V'
the exact value of
m
is unknown..
w
,
z
Arrays:
w
, size at least max(1,
n
).
The first
m
elements of
w
contain the selected eigenvalues of the matrix
T
in ascending order.
z
(size at least max(1,
ldz
*
m
) for column major layout and max(1,
ldz
*
n
) for row major layout)
.
If
jobz
=
'V'
, then if
info
= 0
, the first
m
columns of
z
contain the orthonormal eigenvectors of the matrix
T
corresponding to the selected eigenvalues, with the
i
-th column of
z
holding the eigenvector associated with
w
[
i
- 1]
.
If
jobz
=
'N'
, then
z
is not referenced.
d
,
e
On exit, these arrays may be multiplied by a constant factor chosen to avoid overflow or underflow in computing the eigenvalues.
isuppz
Array, size at least 2 *max(1,
m
).
The support of the eigenvectors in
z
, i.e., the indices indicating the nonzero elements in
z
. The
i
-th eigenvector is nonzero only in elements
isuppz
[2
i
- 2]
through
isuppz
[2
i
- 1]
.
Implemented only for
range
=
'A'
or
'I'
and
iu
-
il
=
n
-
1
.
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
, an internal error has occurred.
Application Notes
Normal execution of the routine
?stegr
may create
NaN
s and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the IEEE standard default manner.

#### 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