naginterfaces.library.lapackeig.dorgrq

naginterfaces.library.lapackeig.dorgrq(a, tau)

dorgrq generates all or part of the real $n\times n$ orthogonal matrix $Q$ from an $RQ$ factorization computed by dgerqf().

For full information please refer to the NAG Library document for f08cj

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08cjf.html

Parameters

afloat, array-like, shape $(m,n)$

Details of the vectors which define the elementary reflectors, as returned by dgerqf().

taufloat, array-like, shape $\left(k\right)$

$tau[i-1]$ must contain the scalar factor of the elementary reflector $H_i$, as returned by dgerqf().

Returns

afloat, ndarray, shape $(m,n)$

The $m\times n$ matrix $Q$.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $\text{m}$.

Constraint: $m\ge 0$.

(errno $-2$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge m$.

(errno $-3$)

On entry, error in parameter $\text{k}$.

Constraint: $m\ge k\ge 0$.

Notes

dorgrq is intended to be used following a call to dgerqf(), which performs an $RQ$ factorization of a real matrix $A$ and represents the orthogonal matrix $Q$ as a product of $k$ elementary reflectors of order $n$.

This function may be used to generate $Q$ explicitly as a square matrix, or to form only its trailing rows.

Usually $Q$ is determined from the $RQ$ factorization of a $p\times n$ matrix $A$ with $p\le n$. The whole of $Q$ may be computed by calling dorgrq with $\text{m}$ set to $n$ and $\text{k}$ set to $p$ or its trailing $p$ rows by calling dorgrq with $\text{m}$ and $\text{k}$ set to $p$.

The rows of $Q$ returned by the last call form an orthonormal basis for the space spanned by the rows of $A$; thus dgerqf() followed by dorgrq can be used to orthogonalize the rows of $A$.

The information returned by dgerqf() also yields the $RQ$ factorization of the trailing $k$ rows of $A$, where $k<p$. The orthogonal matrix arising from this factorization can be computed by calling dorgrq with $\text{m}$ set to $n$ or its leading $k$ columns by calling dorgrq with $\text{m}$ set to $k$.

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