The routine may be called by the names f08tef, nagf_lapackeig_dspgst or its LAPACK name dspgst.
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.
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
1: – IntegerInput
On entry: indicates how the standard form is computed.
if , ;
if , .
if , ;
if , .
, or .
2: – Character(1)Input
On entry: indicates whether the upper or lower triangular part of is stored and how has been factorized.
The upper triangular part of is stored and .
The lower triangular part of is stored and .
3: – IntegerInput
On entry: , the order of the matrices and .
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array ap
must be at least
On entry: the upper or lower triangle of the symmetric matrix , packed by columns.
if , the upper triangle of must be stored with element in for ;
if , the lower triangle of must be stored with element in for .
On exit: the upper or lower triangle of ap is overwritten by the corresponding upper or lower triangle of as specified by itype and uplo, using the same packed storage format as described above.
5: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array bp
must be at least
On entry: the Cholesky factor of as specified by uplo and returned by f07gdf.
6: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
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.
8Parallelism and Performance
f08tef makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
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 total number of floating-point operations is approximately .
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.