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
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

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