# naginterfaces.library.lapackeig.dsbevd¶

naginterfaces.library.lapackeig.dsbevd(job, uplo, kd, ab)[source]

dsbevd computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric band matrix. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the or algorithm.

For full information please refer to the NAG Library document for f08hc

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/f08/f08hcf.html

Parameters
jobstr, length 1

Indicates whether eigenvectors are computed.

Only eigenvalues are computed.

Eigenvalues and eigenvectors are computed.

uplostr, length 1

Indicates whether the upper or lower triangular part of is stored.

The upper triangular part of is stored.

The lower triangular part of is stored.

kdint

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 .

Returns
abfloat, ndarray, shape

is overwritten by values generated during the reduction to tridiagonal form.

The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix are returned in using the same storage format as described above.

wfloat, ndarray, shape

The eigenvalues of the matrix in ascending order.

zfloat, ndarray, shape

If , is overwritten by the orthogonal matrix which contains the eigenvectors of . The th column of contains the eigenvector which corresponds to the eigenvalue .

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 )

If and , the algorithm failed to converge; elements of an intermediate tridiagonal form did not converge to zero; if and , then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column through .

Notes

dsbevd computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric band matrix . In other words, it can compute the spectral factorization of as

where is a diagonal matrix whose diagonal elements are the eigenvalues , and is the orthogonal matrix whose columns are the eigenvectors . Thus

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

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore