Contents

# ?heevr

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

## Syntax

Include Files
• mkl.h
Description
The routine computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix
A
. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
The routine first reduces the matrix
A
to tridiagonal form
T
with a call to hetrd. Then, whenever possible,
?heevr
calls stegr 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 each unreduced block (submatrix) of
T
:
1. Compute
T
-
σ
*
I
=
L
*
D
*
L
T
, so that
L
and
D
define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of
D
and
L
cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix
T
does not have this property in general.
2. Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see Steps c) and d).
3. For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy.
4. For each eigenvalue with a large enough relative separation, compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to Step c) for any clusters that remain.
The desired accuracy of the output can be specified by the input parameter
abstol
.
The routine
?heevr
calls stemr when the full spectrum is requested on machines which conform to the IEEE-754 floating point standard, or stebz and stein on non-IEEE machines and when partial spectrum requests are made.
Note that the routine
?heevr
is preferable for most cases of complex Hermitian eigenvalue problems as its underlying algorithm is fast and uses less workspace.
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
job
=
'N'
, then only eigenvalues are computed.
If
job
=
'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
lambda
(
i
)
in the half-open interval:
vl
<
lambda
(
i
)
vu
.
If
range
=
'I'
, the routine computes eigenvalues with indices
il
to
iu
.
For
range
=
'V'
or
'I'
,
sstebz
/
dstebz
and
cstein
/
zstein
are called.
uplo
Must be
'U'
or
'L'
.
If
uplo
=
'U'
,
a
stores the upper triangular part of
A
.
If
uplo
=
'L'
,
a
stores the lower triangular part of
A
.
n
The order of the matrix
A
(
n
0
).
a
a
(size max(1,
lda
*
n
))
is an array containing either upper or lower triangular part of the Hermitian matrix
A
, as specified by
uplo
.
lda
The leading dimension of the array
a
.
Must be at least max(1,
n
).
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
||
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
a
On exit, the lower triangle (if
uplo
=
'L'
) or the upper triangle (if
uplo
=
'U'
) of
A
, including the diagonal, is overwritten.
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
w
Array, size at least max(1,
n
), contains the selected eigenvalues in ascending order, stored in
w

to
w
[
m
- 1]
.
z
Array
z
(size 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
A
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.
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]
. Referenced only if eigenvectors are needed (
jobz
=
'V'
) and all eigenvalues are needed, that is,
range
=
'A'
or
range
=
'I'
and
il
= 1 and
iu
=
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.
If
info
=
i
, an internal error has occurred.
Application Notes
Normal execution of
?stemr
may create
NaN
s and infinities and hence may abort due to a floating point exception in environments which do not handle
NaN
s and infinities in the IEEE standard default manner.
For more details, see
?stemr
and these references:
• Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices," Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
• Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.
• Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.

#### Product and Performance Information

1

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