Contents

# ?hegst

Reduces a complex Hermitian positive-definite generalized eigenvalue problem to the standard form.

## Syntax

Include Files
• mkl.h
Description
The routine reduces a complex Hermitian positive-definite generalized eigenvalue problem to standard form.
itype
Problem
Result
1
A
*
x
=
λ
*
B
*
x
A
overwritten by
inv(
U
H
)*
A
*inv(
U
)
or
inv(
L
)*
A
*inv(
L
H
)
2
A
*
B
*
x
=
λ
*
x
A
overwritten by
U
*
A
*
U
H
or
L
H
*
A
*
L
3
B
*
A
*
x
=
λ
*
x
Before calling this routine, you must call
?potrf
to compute the Cholesky factorization:
B
=
U
H
*U
or
B
=
L*L
H
.
Input Parameters
itype
Must be 1 or 2 or 3.
If
itype
= 1
, the generalized eigenproblem is
A*z
=
lambda
*
B
*
z
for
uplo
=
'U'
:
C
= (
U
H
)
-1
*
A
*
U
-1
;
for
uplo
=
'L'
:
C
=
L
-1
*
A
*(
L
H
)
-1
.
If
itype
= 2
, the generalized eigenproblem is
A
*
B
*
z
=
lambda
*
z
for
uplo
=
'U'
:
C
=
U
*
A
*
U
H
;
for
uplo
=
'L'
:
C
=
L
H
*
A
*
L
.
If
itype
= 3
, the generalized eigenproblem is
B*A
*
z
=
lambda
*
z
for
uplo
=
'U'
:
C
=
U
*
A
*
U
H
;
for
uplo
=
'L'
:
C
=
L
H
*
A
*
L
.
uplo
Must be
'U'
or
'L'
.
If
uplo
=
'U'
, the array
a
stores the upper triangle of
A
; you must supply
B
in the factored form
B
=
U
H
*U
.
If
uplo
=
'L'
, the array
a
stores the lower triangle of
A
; you must supply
B
in the factored form
B
=
L*L
H
.
n
The order of the matrices
A
and
B
(
n
0
).
a
,
b
Arrays:
a
(size max(1,
lda
*
n
))
contains the upper or lower triangle of
A
.
b
(size max(1,
ldb
*
n
))
contains the Cholesky-factored matrix
B
:
B
=
U
H
*U
or
B
=
L*L
H
(as returned by
?potrf
).
lda
The leading dimension of
a
; at least max(1,
n
).
ldb
The leading dimension of
b
; at least max(1,
n
).
Output Parameters
a
The upper or lower triangle of
A
is overwritten by the upper or lower triangle of
C
, as specified by the arguments
itype
and
uplo
.
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.
Application Notes
Forming the reduced matrix
C
is a stable procedure. However, it involves implicit multiplication by
B
-1
(if
itype
= 1
) or
B
(if
itype
= 2
or 3). When the routine is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if
B
is ill-conditioned with respect to inversion.
The approximate number of floating-point operations is
n
3
.

#### Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.