naginterfaces.library.lapackeig.dsbgvd¶
- naginterfaces.library.lapackeig.dsbgvd(jobz, uplo, ka, kb, ab, bb)[source]¶
dsbgvd
computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the formwhere and are symmetric and banded, and is also positive definite. If eigenvectors are desired, it uses a divide-and-conquer algorithm.
For full information please refer to the NAG Library document for f08uc
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08ucf.html
- Parameters
- jobzstr, length 1
Indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
- uplostr, length 1
If , the upper triangles of and are stored.
If , the lower triangles of and are stored.
- kaint
If , the number of superdiagonals, , of the matrix .
If , the number of subdiagonals, , of the matrix .
- kbint
If , the number of superdiagonals, , of the matrix .
If , the number of subdiagonals, , of the matrix .
- abfloat, array-like, shape
The upper or lower triangle of the symmetric band matrix .
- bbfloat, array-like, shape
The upper or lower triangle of the symmetric band matrix .
- Returns
- abfloat, ndarray, shape
The contents of are overwritten.
- bbfloat, ndarray, shape
The factor from the split Cholesky factorization , as returned by
dpbstf()
.- wfloat, ndarray, shape
The eigenvalues in ascending order.
- zfloat, ndarray, shape
If , contains the matrix of eigenvectors, with the th column of holding the eigenvector associated with . The eigenvectors are normalized so that .
If , is not referenced.
- Raises
- NagValueError
- (errno )
On entry, error in parameter .
Constraint: or .
- (errno )
On entry, error in parameter .
Constraint: or .
- (errno )
On entry, error in parameter .
Constraint: .
- (errno )
On entry, error in parameter .
Constraint: .
- (errno )
On entry, error in parameter .
Constraint: .
- (errno )
The algorithm failed to converge; off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
- (errno )
If , for ,
dpbstf()
returned : is not positive definite. The factorization of could not be completed and no eigenvalues or eigenvectors were computed.
- Notes
The generalized symmetric-definite band problem
is first reduced to a standard band symmetric problem
where is a symmetric band matrix, using Wilkinson’s modification to Crawford’s algorithm (see Crawford (1973) and Wilkinson (1977)). The symmetric eigenvalue problem is then solved for the eigenvalues and the eigenvectors, if required, which are then backtransformed to the eigenvectors of the original problem.
The eigenvectors are normalized so that the matrix of eigenvectors, , satisfies
where is the diagonal matrix whose diagonal elements are the eigenvalues.
- References
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
Crawford, C R, 1973, Reduction of a band-symmetric generalized eigenvalue problem, Comm. ACM (16), 41–44
Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore
Wilkinson, J H, 1977, Some recent advances in numerical linear algebra, The State of the Art in Numerical Analysis, (ed D A H Jacobs), Academic Press