naginterfaces.library.lapackeig.dgerqf

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

dgerqf computes an factorization of a real matrix .

For full information please refer to the NAG Library document for f08ch

https://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/f08/f08chf.html

Parameters
afloat, array-like, shape

The matrix .

Returns
afloat, ndarray, shape

If , the upper triangle of the subarray contains the upper triangular matrix .

If , the elements on and above the th subdiagonal contain the upper trapezoidal matrix ; the remaining elements, with the array , represent the orthogonal matrix as a product of elementary reflectors (see the F08 Introduction).

taufloat, ndarray, shape

The scalar factors of the elementary reflectors.

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

Notes

dgerqf forms the factorization of an arbitrary rectangular real matrix. If , the factorization is given by

where is an lower triangular matrix and is an orthogonal matrix. If the factorization is given by

where is an upper trapezoidal matrix and is again an orthogonal matrix. In the case where the factorization can be expressed as

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

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).

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