f08tbf computes selected eigenvalues and, optionally, eigenvectors of a real generalized symmetric-definite eigenproblem, of the form
where and are symmetric, stored in packed storage, and is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
The routine may be called by the names f08tbf, nagf_lapackeig_dspgvx or its LAPACK name dspgvx.
f08tbf 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 desired eigenvalues and 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
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput.11 873–912
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 , all eigenvalues will be found.
If , all eigenvalues in the half-open interval will be found.
On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval of width less than or equal to
where is the machine precision. If abstol is less than or equal to zero, then will be used in its place, where is the tridiagonal matrix obtained by reducing to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold , not zero. If this routine returns with , indicating that some eigenvectors did not converge, try setting abstol to . See Demmel and Kahan (1990).
13: – IntegerOutput
On exit: the total number of eigenvalues found. .
If , .
If , .
14: – Real (Kind=nag_wp) arrayOutput
On exit: the first m elements contain the selected eigenvalues in ascending order.
15: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array z
must be at least
if , and at least otherwise.
On exit: if , then
if , the first m columns of contain the orthonormal eigenvectors of the matrix corresponding to the selected eigenvalues, with the th column of holding the eigenvector associated with . The eigenvectors are normalized as follows:
if or , ;
if , ;
if an eigenvector fails to converge (), then that column of contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
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; eigenvectors did not converge. Their indices are stored in array jfail.
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.
8Parallelism and Performance
f08tbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08tbf 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 .