Contents

# ?sygvx

Computes selected eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem.

## Syntax

Include Files
• mkl.h
Description
The routine computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form
A
*
x
=
λ
*
B
*
x
,
A
*
B
*
x
=
λ
*
x
, or
B
*
A
*
x
=
λ
*
x
.
Here
A
and
B
are assumed to be symmetric and
B
is also positive definite. 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
).
itype
Must be 1 or 2 or 3. Specifies the problem type to be solved:
if
itype
= 1
, the problem type is
A*x
=
λ
*B*x
;
if
itype
= 2
, the problem type is
A
*
B
*
x
=
λ
*
x
;
if
itype
= 3
, the problem type is
B*A
*x =
λ
*
x
.
jobz
Must be
'N'
or
'V'
.
If
jobz
=
'N'
, then compute eigenvalues only.
If
jobz
=
'V'
, then compute eigenvalues and eigenvectors.
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'
, arrays
a
and
b
store the upper triangles of
A
and
B
;
If
uplo
=
'L'
, arrays
a
and
b
store the lower triangles of
A
and
B
.
n
The order of the matrices
A
and
B
(
n
0
).
a
,
b
Arrays:
a
(size at least max(1,
lda
*
n
))
contains the upper or lower triangle of the symmetric matrix
A
, as specified by
uplo
.
b
(size at least max(1,
ldb
*
n
))
contains the upper or lower triangle of the symmetric positive definite matrix
B
, as specified by
uplo
.
lda
a
; at least max(1,
n
).
ldb
b
; 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
ldz
The leading dimension of the output array
z
. Constraints:
ldz
1
; if
jobz
=
'V'
,
ldz
max(1,
n
)
for column major layout and
ldz
max(1,
m
) for row major layout
.
Output Parameters
a
On exit, the upper triangle (if
uplo
=
'U'
) or the lower triangle (if
uplo
=
'L'
) of
A
, including the diagonal, is overwritten.
b
On exit, if
info
n
, the part of
b
containing the matrix is overwritten by the triangular factor
U
or
L
from the Cholesky factorization
B
=
U
T
*U
or
B
=
L*
L
T
.
m
The total number of eigenvalues found,
0
m
n
. If
range
=
'A'
,
m
=
n
, and if
range
=
'I'
,
m
=
iu
-
il
+1
.
w
,
z
Arrays:
w
, size at least max(1,
n
).
The first
m
elements of
w
contain the selected eigenvalues 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
A
corresponding to the selected eigenvalues, with the
i
-th column of
z
holding the eigenvector associated with
w
[
i
- 1]
. The eigenvectors are normalized as follows:
if
itype
= 1
or
2
,
Z
T
*
B
*
Z
= I
;
if
itype
= 3
,
Z
T
*inv(
B
)*
Z
= I
;
If
jobz
=
'N'
, then
z
is not referenced.
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
.
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
, the
ifail
contains the indices of 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
> 0
,
spotrf
/
dpotrf
and
ssyevx
/
dsyevx
returned an error code:
If
info
=
i
n
,
ssyevx
/
dsyevx
failed to converge, and
i
eigenvectors failed to converge. Their indices are stored in the array
ifail
;
If
info
=
n
+
i
, for
1
i
n
, then the leading minor of order
i
of
B
is not positive-definite. The factorization of
B
could not be completed and no eigenvalues or eigenvectors were computed.
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
is used as tolerance, where
T
is the tridiagonal matrix obtained by reducing
C
to tridiagonal form, where
C
is the symmetric matrix of the standard symmetric problem to which the generalized problem is transformed. 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, set
abstol
to 2*
?lamch
('S').

#### 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