Contents

# ?hbgvx

Computes selected eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian positive-definite eigenproblem with banded matrices.

## Syntax

Include Files
• mkl.h
Description
The routine computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian positive-definite banded eigenproblem, of the form
A
*
x
=
λ
*
B
*
x
. Here
A
and
B
are assumed to be Hermitian and banded, and
B
is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, 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
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
ab
and
bb
store the upper triangles of
A
and
B
;
If
uplo
=
'L'
, arrays
ab
and
bb
store the lower triangles of
A
and
B
.
n
The order of the matrices
A
and
B
(
n
0
).
ka
The number of super- or sub-diagonals in
A
(
ka
0
).
kb
The number of super- or sub-diagonals in
B
(
kb
0).
ab
,
bb
Arrays:
ab
(size at least max(1,
ldab
*
n
) for column major layout and max(1,
ldab
*(
ka
+ 1)) for row major layout)
is an array containing either upper or lower triangular part of the Hermitian matrix
A
(as specified by
uplo
) in band storage format.
bb
(size at least max(1,
ldbb
*
n
) for column major layout and max(1,
ldbb
*(
kb
+ 1)) for row major layout)
is an array containing either upper or lower triangular part of the Hermitian matrix
B
(as specified by
uplo
) in band storage format.
ldab
The leading dimension of the array
ab
; must be at least
ka
+1
for column major layout and at least max(1,
n
) for row major layout
.
ldbb
The leading dimension of the array
bb
; must be at least
kb
+1
for column major layout and at least max(1,
n
) for row major layout
.
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
for more information.
ldz
The leading dimension of the output array
z
;
ldz
1
. If
jobz
=
'V'
,
ldz
max(1,
n
)
for column major layout and at least max(1,
m
) for row major layout
.
ldq
The leading dimension of the output array
q
;
ldq
1
. If
jobz
=
'V'
,
ldq
max(1,
n
)
.
Output Parameters
ab
On exit, the contents of
ab
are overwritten.
bb
On exit, contains the factor
S
from the split Cholesky factorization
B
=
S
H
*S
, as returned by pbstf/pbstf.
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
w
, size at least max(1,
n
).
If
info
= 0
, contains the eigenvalues in ascending order.
z
,
q
Arrays:
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
,
z
contains the matrix
Z
of eigenvectors, with the
i
-th column of
z
holding the eigenvector associated with
w
[
i
- 1]
. The eigenvectors are normalized so that
Z
H
*
B
*
Z
= I
.
If
jobz
=
'N'
, then
z
is not referenced.
q
(size max(1,
ldq
*
n
))
.
If
jobz
=
'V'
, then
q
contains the
n
-by-
n
matrix used in the reduction of
Ax
=
λ
Bx
to standard form, that is,
Cx
=
λ
x
and consequently
C
to tridiagonal form.
If
jobz
=
'N'
, then
q
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 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
, and
if
i
n
, the algorithm failed to converge, and
i
off-diagonal elements of an intermediate tridiagonal did not converge to zero;
if
info
=
n
+
i
, for
1
i
n
, then pbstf/pbstf returned
info
=
i
and
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
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.