Contents

# ?hpevx

Computes selected eigenvalues and, optionally, eigenvectors of a Hermitian matrix in packed storage.

## Syntax

Include Files
• mkl.h
Description
The routine computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix
A
in packed storage. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
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
w
[
i
]
in the half-open interval:
vl
<
w
[
i
]
vu
.
If
range
=
'I'
, the routine computes eigenvalues with indices
il
to
iu
.
uplo
Must be
'U'
or
'L'
.
If
uplo
=
'U'
,
ap
stores the packed upper triangular part of
A
.
If
uplo
=
'L'
,
ap
stores the packed lower triangular part of
A
.
n
The order of the matrix
A
(
n
0
).
ap
Array
ap
contains the packed upper or lower triangle of the Hermitian matrix
A
, as specified by
uplo
.
The size of
ap
must be at least max(1,
n
*(
n
+1)/2).
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 is required.
See
Application notes
for details on error tolerance.
ldz
The leading dimension of the output array
z
.
Constraints:
if
jobz
=
'N'
, then
ldz
1
;
if
jobz
=
'V'
, then
ldz
max(1,
n
)
for column major layout and
ldz
max(1,
m
) for row major layout
.
Output Parameters
ap
On exit, this array is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of
A
.
m
The total number of eigenvalues found,
0
m
n
.
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
).
If
info
= 0
, contains the selected eigenvalues of the matrix
A
in ascending order.
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
).
If an eigenvector fails to converge, then that column of
z
contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in
ifail
.
If
jobz
=
'N'
, then
z
is not referenced.
ifail
Array, size at least max(1,
n
).
If
jobz
=
'V'
, then if
info
= 0
, the first
m
elements of
ifail
are zero; if
info
> 0
, the
ifail
contains the indices the eigenvectors that failed to converge.
If
jobz
=
'N'
, then
ifail
is not referenced.
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
, then
i
eigenvectors failed to converge; their indices are stored in the array
ifail
.
Application Notes
An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to
abstol
+
ε
*max(|a|,|b|)
, where
ε
is the machine precision.
If
abstol
is less than or equal to zero, then
ε
*||
T
||
1
will be used in its place, where
T
is the tridiagonal matrix obtained by reducing
A
to tridiagonal form. Eigenvalues will be computed most accurately when
abstol
is set to twice the underflow threshold 2*
?lamch
('S'), not zero.
If this routine returns with
info
> 0
, indicating that some eigenvectors did not converge, try setting
abstol
to 2*
?lamch
('S').

#### Product and Performance Information

1

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