naginterfaces.library.lapackeig.dorgql

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

dorgql generates all or part of the real $m\times m$ orthogonal matrix $Q$ from a $QL$ factorization computed by dgeqlf().

For full information please refer to the NAG Library document for f08cf

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08cff.html

Parameters

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

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

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

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

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: $m\ge n\ge 0$.

(errno $-3$)

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

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

Notes

dorgql is intended to be used after a call to dgeqlf(), which performs a $QL$ 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 trailing columns.

Usually $Q$ is determined from the $QL$ factorization of an $m\times p$ matrix $A$ with $m\ge p$. The whole of $Q$ may be computed by calling dorgql with $\text{n}$ set to $m$ and $\text{k}$ set to $p$ or its trailing $p$ columns by calling dorgql with $\text{n}$ and $\text{k}$ set to $p$.

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

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