F08TEF (DSPGST) reduces a real symmetric-definite generalized eigenproblem
Az=λBz,
ABz=λz or
BAz=λz to the standard form
Cy=λy, where
A is a real symmetric matrix and
B has been factorized by
F07GDF (DPPTRF), using packed storage.
To reduce the real symmetric-definite generalized eigenproblem
Az=λBz,
ABz=λz or
BAz=λz to the standard form
Cy=λy using packed storage, F08TEF (DSPGST) must be preceded by a call to
F07GDF (DPPTRF) which computes the Cholesky factorization of
B;
B must be positive definite.
The different problem types are specified by the parameter
ITYPE, as indicated in the table below. The table shows how
C is computed by the routine, and also how the eigenvectors
z of the original problem can be recovered from the eigenvectors of the standard form.
ITYPE |
Problem |
UPLO |
B |
C |
z |
1 |
Az=λBz |
'U'
'L' |
UTU
LLT |
U-TAU-1
L-1AL-T |
U-1y
L-Ty |
2 |
ABz=λz |
'U'
'L' |
UTU
LLT |
UAUT
LTAL |
U-1y
L-Ty |
3 |
BAz=λz |
'U'
'L' |
UTU
LLT |
UAUT
LTAL |
UTy
Ly |
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
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 F08TEF (DSPGST) 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.
See the document for
F08SAF (DSYGV) for further details.
The complex analogue of this routine is
F08TSF (ZHPGST).
This example computes all the eigenvalues of
Az=λBz, where
using packed storage. Here
B is symmetric positive definite and must first be factorized by
F07GDF (DPPTRF). The program calls F08TEF (DSPGST) to reduce the problem to the standard form
Cy=λy; then
F08GEF (DSPTRD) to reduce
C to tridiagonal form, and
F08JFF (DSTERF) to compute the eigenvalues.