naginterfaces.library.lapackeig.dorglq

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

dorglq generates all or part of the real orthogonal matrix $Q$ from an $LQ$ factorization computed by dgelqf().

For full information please refer to the NAG Library document for f08aj

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08ajf.html

Parameters

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

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

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

Further details of the elementary reflectors, as returned by dgelqf().

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

dorglq is intended to be used after a call to dgelqf(), which performs an $LQ$ factorization of a real matrix $A$. The orthogonal matrix $Q$ is represented as a product of elementary reflectors.

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

Usually $Q$ is determined from the $LQ$ factorization of a $p\times n$ matrix $A$ with $p\le n$. The whole of $Q$ may be computed by calling dorglq with $\text{m}$ set to $n$ and $\text{k}$ set to $p$ or its leading $p$ rows by calling dorglq 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 dgelqf() followed by dorglq can be used to orthogonalize the rows of $A$.

The information returned by the $LQ$ factorization functions also yields the $LQ$ factorization of the leading $k$ rows of $A$, where $k<p$. The orthogonal matrix arising from this factorization can be computed by calling dorglq with $\text{m}$ set to $n$ or its leading $k$ rows by calling dorglq with $\text{m}$ set to $k$.

References

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