Contents

?stevd

Computes all eigenvalues and, optionally, all eigenvectors of a real symmetric tridiagonal matrix using divide and conquer algorithm.

Syntax

Include Files
• mkl.h
Description
The routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric tridiagonal matrix
T
. In other words, the routine can compute the spectral factorization of
T
as:
T
=
Z
*
Λ
*
Z
T
.
Here
Λ
is a diagonal matrix whose diagonal elements are the eigenvalues
λ
i
, and
Z
is the orthogonal matrix whose columns are the eigenvectors
z
i
. Thus,
T
*
z
i
=
λ
i
*
z
i
for
i
= 1, 2, ...,
n
.
If the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan variant of the
QL
or
QR
algorithm.
There is no complex analogue of this routine.
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.
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 off-diagonal elements of
T
.
The dimension of
e
must be at least max(1,
n
). The
n-
th element of this array is used as workspace.
ldz
The leading dimension of the output array
z
. Constraints:
ldz
1
if
job
=
'N'
;
ldz
max(1,
n
)
if
job
=
'V'
.
Output Parameters
d
On exit, if
info
= 0
, contains the eigenvalues of the matrix
T
in ascending order.
info
.
z
Array, size
max(1,
ldz
*
n
) if
jobz
=
'V'
and 1 if
jobz
=
'N'
.
If
jobz
=
'V'
, then this array is overwritten by the orthogonal matrix
Z
which contains the eigenvectors of
T
.
If
jobz
=
'N'
, then
z
is not referenced.
e
On exit, this array is overwritten with intermediate results.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
=
i
, then the algorithm failed to converge;
i
indicates the number of elements of an intermediate tridiagonal form which did not converge to zero.
If
info
=
-i
, the
i
-th parameter had an illegal value.
Application Notes
The computed eigenvalues and eigenvectors are exact for a matrix
T
+
E
such that
||
E
||
2
=
O
(
ε
)*||
T
||
2
, where
ε
is the machine precision.
If
λ
i
is an exact eigenvalue, and
μ
i
is the corresponding computed value, then
|
μ
i
-
λ
i
|
c
(
n
)*
ε
*||
T
||
2
where
c
(
n
)
is a modestly increasing function of
n
.
If
z
i
is the corresponding exact eigenvector, and
w
i
is the corresponding computed vector, then the angle
θ
(
z
i
,
w
i
)
between them is bounded as follows:
θ
(
z
i
,
w
i
)
c
(
n
)*
ε
*||
T
||
2
/ min
i
j
|
λ
i
-
λ
j
|
.
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.

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