naginterfaces.library.lapackeig.zgelqf

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

zgelqf computes the factorization of a complex matrix.

For full information please refer to the NAG Library document for f08av

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/f08/f08avf.html

Parameters
acomplex, array-like, shape

The matrix .

Returns
acomplex, ndarray, shape

If , the elements above the diagonal are overwritten by details of the unitary matrix and the lower triangle is overwritten by the corresponding elements of the lower triangular matrix .

If , the strictly upper triangular part is overwritten by details of the unitary matrix and the remaining elements are overwritten by the corresponding elements of the lower trapezoidal matrix .

The diagonal elements of are real.

taucomplex, 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

zgelqf forms the factorization of an arbitrary rectangular complex matrix. No pivoting is performed.

If , the factorization is given by:

where is an lower triangular matrix (with real diagonal elements) and is an unitary matrix. It is sometimes more convenient to write the factorization as

which reduces to

where consists of the first rows of , and the remaining rows.

If , is trapezoidal, and the factorization can be written

where is lower triangular and is rectangular.

The factorization of is essentially the same as the factorization of , since

The matrix is not formed explicitly but is represented as a product of elementary reflectors (see the F08 Introduction for details). Functions are provided to work with in this representation (see Further Comments).

Note also that for any , the information returned in the first rows of the array represents an factorization of the first rows of the original matrix .