naginterfaces.library.lapackeig.zgebrd

naginterfaces.library.lapackeig.zgebrd(a)[source]

zgebrd reduces a complex matrix to bidiagonal form.

For full information please refer to the NAG Library document for f08ks

https://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/f08/f08ksf.html

Parameters
acomplex, array-like, shape

The matrix .

Returns
acomplex, ndarray, shape

If , the diagonal and first superdiagonal are overwritten by the upper bidiagonal matrix , elements below the diagonal are overwritten by details of the unitary matrix and elements above the first superdiagonal are overwritten by details of the unitary matrix .

If , the diagonal and first subdiagonal are overwritten by the lower bidiagonal matrix , elements below the first subdiagonal are overwritten by details of the unitary matrix and elements above the diagonal are overwritten by details of the unitary matrix .

dfloat, ndarray, shape

The diagonal elements of the bidiagonal matrix .

efloat, ndarray, shape

The off-diagonal elements of the bidiagonal matrix .

tauqcomplex, ndarray, shape

Further details of the unitary matrix .

taupcomplex, ndarray, shape

Further details of the unitary matrix .

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

Notes

zgebrd reduces a complex matrix to real bidiagonal form by a unitary transformation: , where and are unitary matrices of order and respectively.

If , the reduction is given by:

where is a real upper bidiagonal matrix and consists of the first columns of .

If , the reduction is given by

where is a real lower bidiagonal matrix and consists of the first rows of .

The unitary matrices and are not formed explicitly but are represented as products of elementary reflectors (see the F08 Introduction for details). Functions are provided to work with and in this representation (see Further Comments).

References

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