The routine may be called by the names f08sbf, nagf_lapackeig_dsygvx or its LAPACK name dsygvx.
f08sbf 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).
15: – IntegerOutput
On exit: the total number of eigenvalues found. .
If , .
If , .
16: – Real (Kind=nag_wp) arrayOutput
On exit: the first m elements contain the selected eigenvalues in ascending order.
17: – 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.
Note: you must ensure that at least columns are supplied in the array z; if , the exact value of m is not known in advance and an upper bound of at least n must be used.
18: – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f08sbf is called.
if , ;
19: – Real (Kind=nag_wp) arrayWorkspace
On exit: if , contains the minimum value of lwork required for optimal performance.
20: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08sbf is called.
If , a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
for optimal performance, , where is the optimal block size for f08fef.
21: – Integer arrayWorkspace
22: – Integer arrayOutput
Note: the dimension of the array jfail
must be at least
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 failed to converge.
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
f08sbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08sbf 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 .