## Developer Reference

• 2020.2
• 07/15/2020
• Public Content
Contents

# ?hbgst

Reduces a complex Hermitian positive-definite generalized eigenproblem for banded matrices to the standard form using the factorization performed by
?pbstf
.

## Syntax

Include Files
• mkl.fi
,
lapack.f90
Description
To reduce the complex Hermitian positive-definite generalized eigenproblem
A
*
z
=
λ
*
B
*
z
to the standard form
C
*
x
=
λ
*
y
, where
A
,
B
and
C
are banded, this routine must be preceded by a call to pbstf/pbstf, which computes the split Cholesky factorization of the positive-definite matrix
B
:
B
=
S
H
*S
. The split Cholesky factorization, compared with the ordinary Cholesky factorization, allows the work to be approximately halved.
This routine overwrites
A
with
C
=
X
H
*A*X
, where
X
=
inv(
S
)
*
Q
, and
Q
is a unitary matrix chosen (implicitly) to preserve the bandwidth of
A
. The routine also has an option to allow the accumulation of
X
, and then, if
z
is an eigenvector of
C
,
X*z
is an eigenvector of the original system.
Input Parameters
vect
CHARACTER*1
.
Must be
'N'
or
'V'
.
If
vect
=
'N'
, then matrix
X
is not returned;
If
vect
=
'V'
, then matrix
X
is returned.
uplo
CHARACTER*1
.
Must be
'U'
or
'L'
.
If
uplo
=
'U'
,
ab
stores the upper triangular part of
A
.
If
uplo
=
'L'
,
ab
stores the lower triangular part of
A
.
n
INTEGER
.
The order of the matrices
A
and
B
(
n
0
).
ka
INTEGER
.
The number of super- or sub-diagonals in
A
(
ka
0
).
kb
INTEGER
.
The number of super- or sub-diagonals in
B
(
ka
kb
0
).
ab
,
bb