naginterfaces.library.lapackeig.zungqr

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

zungqr generates all or part of the complex unitary matrix $Q$ from a $QR$ factorization computed by zgeqrf() or zgeqp3().

For full information please refer to the NAG Library document for f08at

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08atf.html

Parameters

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

Details of the vectors which define the elementary reflectors, as returned by zgeqrf() or zgeqp3().

taucomplex, array-like, shape $(max(1,k))$

Further details of the elementary reflectors, as returned by zgeqrf() or zgeqp3().

Returns

acomplex, 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

zungqr is intended to be used after a call to zgeqrf() or zgeqp3(), which perform a $QR$ factorization of a complex matrix $A$. The unitary 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 columns.

Usually $Q$ is determined from the $QR$ factorization of an $m\times p$ matrix $A$ with $m\ge p$. The whole of $Q$ may be computed by calling zungqr with $\text{n}$ set to $m$ and $\text{k}$ set to $p$ or its leading $p$ columns by calling zungqr 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 zgeqrf() followed by zungqr can be used to orthogonalize the columns of $A$.

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

References

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