naginterfaces.library.lapackeig.dgerqf

naginterfaces.library.lapackeig.dgerqf(a)

dgerqf computes an $RQ$ factorization of a real $m\times n$ matrix $A$.

For full information please refer to the NAG Library document for f08ch

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08chf.html

Parameters

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

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

Returns

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

If $m\le n$, the upper triangle of the subarray $a[0:m,n-m:n]$ contains the $m\times m$ upper triangular matrix $R$.

If $m\ge n$, the elements on and above the $(m-n)$th subdiagonal contain the $m\times n$ upper trapezoidal matrix $R$; the remaining elements, with the array $tau$, represent the orthogonal matrix $Q$ as a product of $min(m,n)$ elementary reflectors (see the F08 Introduction).

taufloat, ndarray, shape $(min(m,n))$

The scalar factors of the elementary reflectors.

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: $n\ge 0$.

Notes

dgerqf forms the $RQ$ factorization of an arbitrary rectangular real $m\times n$ matrix. If $m\le n$, the factorization is given by

$$A=\left(\begin{array}{cc}0& R\end{array}\right)Q\text{,}$$

where $R$ is an $m\times m$ lower triangular matrix and $Q$ is an $n\times n$ orthogonal matrix. If $m>n$ the factorization is given by

$$A=RQ\text{,}$$

where $R$ is an $m\times n$ upper trapezoidal matrix and $Q$ is again an $n\times n$ orthogonal matrix. In the case where $m<n$ the factorization can be expressed as

$$\begin{array}{c}A=\left(\begin{array}{cc}0& R\end{array}\right)\left(\begin{array}{c}{Q}_1\\ Q_2\end{array}\right)=R{Q}_2\text{,}\end{array}$$

where $Q_1$ consists of the first $(n-m)$ rows of $Q$ and $Q_2$ the remaining $m$ rows.

The matrix $Q$ is not formed explicitly, but is represented as a product of $min(m,n)$ elementary reflectors (see the F08 Introduction for details). Functions are provided to work with $Q$ in this representation (see Further Comments).

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