The routine may be called by the names f08taf, nagf_lapackeig_dspgv or its LAPACK name dspgv.
f08taf first performs a Cholesky factorization of the matrix as , when or , when . The generalized problem is then reduced to a standard symmetric eigenvalue problem
which is solved for the eigenvalues and, optionally, the eigenvectors; the eigenvectors are then backtransformed to give the eigenvectors of the original problem.
For the problem , the eigenvectors are normalized so that the matrix of eigenvectors, , satisfies
where is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem we correspondingly have
and for we have
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
1: – IntegerInput
On entry: specifies the problem type to be solved.
, or .
2: – Character(1)Input
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
3: – Character(1)Input
On entry: if , the upper triangles of and are stored.
If , the lower triangles of and are stored.
4: – IntegerInput
On entry: , the order of the matrices and .
5: – 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 entry: the first dimension of the array z as declared in the (sub)program from which f08taf is called.
if , ;
10: – Real (Kind=nag_wp) arrayWorkspace
11: – 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.
The algorithm failed to converge; off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
If , for , then the leading minor of order of is not positive definite. The factorization of could not be completed and no eigenvalues or eigenvectors were computed.
If is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.
The example program below illustrates the computation of approximate error bounds.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08taf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08taf 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 proportional to .