naginterfaces.library.lapackeig.dgeqrt

naginterfaces.library.lapackeig.dgeqrt(nb, a)[source]

dgeqrt recursively computes, with explicit blocking, the factorization of a real matrix.

For full information please refer to the NAG Library document for f08ab

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/f08/f08abf.html

Parameters
nbint

The explicitly chosen block size to be used in computing the factorization. See Further Comments for details.

afloat, array-like, shape

The matrix .

Returns
afloat, ndarray, shape

If , the elements below the diagonal are overwritten by details of the orthogonal 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 orthogonal matrix and the remaining elements are overwritten by the corresponding elements of the upper trapezoidal matrix .

tfloat, ndarray, shape

Further details of the orthogonal matrix . The number of blocks is , where and each block is of order except for the last block, which is of order . For each of the blocks, an upper triangular block reflector factor is computed: . These are stored in the matrix as .

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

Notes

dgeqrt forms the factorization of an arbitrary rectangular real matrix. No pivoting is performed.

It differs from dgeqrf() in that it: requires an explicit block size; stores reflector factors that are upper triangular matrices of the chosen block size (rather than scalars); and recursively computes the factorization based on the algorithm of Elmroth and Gustavson (2000).

If , the factorization is given by:

where is an upper triangular matrix and is an orthogonal 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 upper 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 represents a factorization of the first columns of the original matrix .

References

Elmroth, E and Gustavson, F, 2000, Applying Recursion to Serial and Parallel Factorization Leads to Better Performance, IBM Journal of Research and Development. (Volume 44) (4), 605–624

Golub, G H and Van Loan, C F, 2012, Matrix Computations, (4th Edition), Johns Hopkins University Press, Baltimore