naginterfaces.library.matop.real_​gen_​rq

naginterfaces.library.matop.real_gen_rq(m, a)[source]

real_gen_rq finds the factorization of the real () matrix , so that is reduced to upper triangular form by means of orthogonal transformations from the right.

For full information please refer to the NAG Library document for f01qj

https://www.nag.com/numeric/nl/nagdoc_28.4/flhtml/f01/f01qjf.html

Parameters
mint

, the number of rows of the matrix .

When then an immediate return is effected.

afloat, array-like, shape

The leading part of the array must contain the matrix to be factorized.

Returns
afloat, ndarray, shape

The upper triangular part of will contain the upper triangular matrix , and the strictly lower triangular part of and the rectangular part of to the right of the upper triangular part will contain details of the factorization as described in Notes.

zetafloat, ndarray, shape

contains the scalar for the th transformation. If then , otherwise contains as described in Notes and is always in the range .

Raises
NagValueError
(errno )

On entry, and .

Constraint: .

(errno )

On entry, .

Constraint: .

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

The matrix is factorized as

where is an orthogonal matrix and is an upper triangular matrix.

is given as a sequence of Householder transformation matrices

the ()th transformation matrix, , being used to introduce zeros into the th row of . has the form

where

is a scalar, is an element vector and is an element vector. is chosen to annihilate the elements in the th row of .

The vector is returned in the th element of and in the th row of , such that is in , the elements of are in and the elements of are in . The elements of are returned in the upper triangular part of .

References

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

Wilkinson, J H, 1965, The Algebraic Eigenvalue Problem, Oxford University Press, Oxford