naginterfaces.library.lapackeig.zheevd

naginterfaces.library.lapackeig.zheevd(job, uplo, a)[source]

zheevd computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian 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 f08fq

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/f08/f08fqf.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.

acomplex, array-like, shape

The Hermitian matrix .

Returns
acomplex, ndarray, shape

If , is overwritten by the unitary matrix which contains the eigenvectors of .

wfloat, ndarray, shape

The eigenvalues of the matrix in ascending order.

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

zheevd computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix . In other words, it can compute the spectral factorization of as

where is a real diagonal matrix whose diagonal elements are the eigenvalues , and is the (complex) unitary 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