Contents

# ?syevx

Computes selected eigenvalues and, optionally, eigenvectors of a symmetric matrix.

## Syntax

Include Files
• mkl.h
Description
The routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix
A
. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
Note that for most cases of real symmetric eigenvalue problems the default choice should be syevr function as its underlying algorithm is faster and uses less workspace.
?syevx
is faster for a few selected 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
jobz
=
'N'
, then only eigenvalues are computed.
If
jobz
=
'V'
, then eigenvalues and eigenvectors are computed.
range
Must be
'A'
,
'V'
, or
'I'
.
If
range
=
'A'
, all eigenvalues will be found.
If
range
=
'V'
, all eigenvalues in the half-open interval (
vl
,
vu
] will be found.
If
range
=
'I'
, the eigenvalues with indices
il
through
iu
will be found.
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 symmetric 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;
vl
vu
. Not referenced if
range
=
'A'
or
'I'
.
il
,
iu
If
range
=
'I'
, the indices of the smallest and largest eigenvalues to be returned.
Constraints:
1
il
iu
n
, if
n
> 0
;
il
= 1
and
iu
= 0
, if
n
= 0
.
Not referenced if
range
=
'A'
or
'V'
.
abstol
The absolute error tolerance for the eigenvalues.
See
Application Notes
ldz
The leading dimension of the output array
z
; .
If
jobz
=
'V'
, then
ldz
max(1,
n
)
for column major layout and
lda
max(1,
m
) for row major layout
.
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
, and if
range
=
'I'
,
m
=
iu
-
il
+
1
.
w
Array, size at least max(1,
n
). The first
m
elements contain 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)
contains eigenvectors.
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.
Note: you must ensure that at least max(1,
m
) columns are supplied in the array
z
; if
range
=
'V'
, the exact value of
m
is not known in advance and an upper bound must be used.
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
, then
ifail
contains the indices of the eigenvectors that failed to converge.
If
jobz
=
'V'
, 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
|}
is used as tolerance, where ||
T
|| is the 1-norm of the tridiagonal matrix obtained by reducing
A
to tridiagonal form. Eigenvalues are 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.