f08tef reduces a real symmetric-definite generalized eigenproblem
,
or
to the standard form
, where
is a real symmetric matrix and
has been factorized by
f07gdf, using packed storage.
To reduce the real symmetric-definite generalized eigenproblem
,
or
to the standard form
using packed storage,
f08tef must be preceded by a call to
f07gdf which computes the Cholesky factorization of
;
must be positive definite.
The different problem types are specified by the argument
itype, as indicated in the table below. The table shows how
is computed by the routine, and also how the eigenvectors
of the original problem can be recovered from the eigenvectors of the standard form.
itype |
Problem |
uplo |
|
|
|
|
|
'U'
'L' |
|
|
|
|
|
'U'
'L' |
|
|
|
|
|
'U'
'L' |
|
|
|
Forming the reduced matrix
is a stable procedure. However it involves implicit multiplication by
if (
) or
(if
or
). When
f08tef 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
is ill-conditioned with respect to inversion.
See the document for
f08saf for further details.
Background information to multithreading can be found in the
Multithreading documentation.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The complex analogue of this routine is
f08tsf.
This example computes all the eigenvalues of
, where
using packed storage. Here
is symmetric positive definite and must first be factorized by
f07gdf. The program calls
f08tef to reduce the problem to the standard form
; then
f08gef to reduce
to tridiagonal form, and
f08jff to compute the eigenvalues.