naginterfaces.library.lapackeig.dgeqlf

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

dgeqlf computes a factorization of a real matrix .

For full information please refer to the NAG Library document for f08ce

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/f08/f08cef.html

Parameters
afloat, array-like, shape

The matrix .

Returns
afloat, ndarray, shape

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

If , the elements on and below the th superdiagonal contain the lower 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 (see Further Comments).

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

Notes

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

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

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

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