naginterfaces.library.lapackeig.dspevd

naginterfaces.library.lapackeig.dspevd(job, uplo, n, ap)[source]

dspevd computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix held in packed storage. 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 f08gc

https://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/f08/f08gcf.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.

nint

, the order of the matrix .

apfloat, array-like, shape

The upper or lower triangle of the symmetric matrix , packed by columns.

Returns
apfloat, ndarray, shape

is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of .

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 .

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 )

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

dspevd computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix (held in packed storage). 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