Contents

# ?hpgvx

Computes selected eigenvalues and, optionally, eigenvectors of a generalized Hermitian positive-definite eigenproblem with matrices in packed storage.

## Syntax

Include Files
• mkl.h
Description
The routine computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian positive-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 Hermitian, stored in packed format, 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
=
lambda
*
B
*
x
;
if
itype
= 2
, the problem type is
A*B*x
=
lambda
*
x
;
if
itype
= 3
, the problem type is
B
*
A*x
=
lambda
*
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
ap
and
bp
store the upper triangles of
A
and
B
;
If
uplo
=
'L'
, arrays
ap
and
bp
store the lower triangles of
A
and
B
.
n
The order of the matrices
A
and
B
(
n
0
).
ap
,
bp
Arrays:
ap
contains the packed upper or lower triangle of the Hermitian matrix
A
, as specified by
uplo
.
The dimension of
ap
must be at least max(1,
n
*(
n
+1)/2).
bp
contains the packed upper or lower triangle of the Hermitian matrix
B
, as specified by
uplo
.
The dimension of
bp
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 for the eigenvalues.
See
Application Notes
ldz
The leading dimension of the output array
z
;
ldz
1
. If
jobz
=
'V'
,
ldz
max(1,
n
)
for column major layout and
ldz
max(1,
m
) for row major layout
.
Output Parameters
ap
On exit, the contents of
ap
are overwritten.
bp
On exit, contains the triangular factor
U
or
L
from the Cholesky factorization
B
=
U
H
*
U
or
B
=
L
*
L
H
, in the same storage format as
B
.
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
).
If
info
= 0
, contains the eigenvalues in ascending order.
z
Array
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
). The eigenvectors are normalized as follows:
if
itype
= 1
or
2
,
Z
H
*
B
*
Z
= I
;
if
itype
= 3
,
Z
H
*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
,
cpptrf
/
zpptrf
and
chpevx
/
zhpevx
returned an error code:
If
info
=
i
n
,
chpevx
/
zhpevx
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
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.