Generalized Symmetric-Definite Eigenvalue
Problems: LAPACK Computational Routines
Generalized
symmetric-definite eigenvalue problems are as follows: find the
eigenvalues
λ
and the
corresponding eigenvectors
z
that satisfy
one of these equations:
Az
=
λ
Bz
ABz
=
λ
zB
A
z
=
λ
z
where
A
is an
n
-by-n
symmetric or Hermitian
matrix, and
B
is an
n
-by-n
symmetric positive-definite
or Hermitian positive-definite matrix.
In these problems, there exist
n
real
eigenvectors corresponding to real eigenvalues (even for complex Hermitian
matrices
A
and
B
).
Routines described in this
, which you can solve
by calling LAPACK routines described earlier in this chapter (see
Symmetric Eigenvalue Problems).
topic
allow you to reduce the above generalized problems to standard
symmetric eigenvalue problem
Cy
=
λ
y
The reduction routine for the banded matrices
A
and
B
uses a split
Cholesky factorization for which a specific routine
pbstf is provided. This refinement halves the
amount of work required to form matrix
C
.
Table
"Computational Routines for Reducing
Generalized Eigenproblems to Standard Problems"
lists LAPACK
routines that can be used to solve generalized symmetric-definite eigenvalue
problems.