Generalized symmetric-definite eigenvalue problems are as follows: find the eigenvalues
λand the corresponding eigenvectors
zthat satisfy one of these equations:
nsymmetric or Hermitian matrix, and
nsymmetric positive-definite or Hermitian positive-definite matrix.
In these problems, there exist
nreal eigenvectors corresponding to real eigenvalues (even for complex Hermitian matrices
Routines described in this
topicallow you to reduce the above generalized problems to standard symmetric eigenvalue problem
, which you can solve by calling LAPACK routines described earlier in this chapter (see Symmetric Eigenvalue Problems).
The reduction routine for the banded matrices
Buses a split Cholesky factorization for which a specific routine pbstf is provided. This refinement halves the amount of work required to form matrix
"Computational Routines for Reducing Generalized Eigenproblems to Standard Problems"lists LAPACK routines that can be used to solve generalized symmetric-definite eigenvalue problems.