naginterfaces.library.lapackeig.zgeqrf

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

zgeqrf computes the factorization of a complex matrix.

For full information please refer to the NAG Library document for f08as

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/f08/f08asf.html

Parameters
acomplex, array-like, shape

The matrix .

Returns
acomplex, ndarray, shape

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

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

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

If , the factorization is given by:

where is an upper 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 columns of , and the remaining columns.

If , is trapezoidal, and the factorization can be written

where is upper triangular and is rectangular.

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 columns of the array represents a factorization of the first columns of the original matrix .

References

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