naginterfaces.library.lapackeig.dormrq

naginterfaces.library.lapackeig.dormrq(side, trans, a, tau, c)

dormrq multiplies a general real $m\times n$ matrix $C$ by the real orthogonal matrix $Q$ from an $RQ$ factorization computed by dgerqf().

For full information please refer to the NAG Library document for f08ck

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08ckf.html

Parameters

sidestr, length 1

Indicates how $Q$ or $Q^T$ is to be applied to $C$.

$side=\text{'L'}$

$Q$ or $Q^T$ is applied to $C$ from the left.

$side=\text{'R'}$

$Q$ or $Q^T$ is applied to $C$ from the right.

transstr, length 1

Indicates whether $Q$ or $Q^T$ is to be applied to $C$.

$trans=\text{'N'}$

$Q$ is applied to $C$.

$trans=\text{'T'}$

$Q^T$ is applied to $C$.

afloat, array-like, shape $(k,:)$

Note: the required extent for this argument in dimension 2 is determined as follows: if $side=\text{'L'}$: $m$; if $side=\text{'R'}$: $n$; otherwise: $0$.

The $\text{i}$th row of $a$ must contain the vector which defines the elementary reflector $H_{\text{i}}$, for $\text{i}=1,2,\dots ,k$, 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().

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

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

Returns

afloat, ndarray, shape $(k,:)$

Is modified by dormrq but restored on exit.

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

$c$ is overwritten by $QC$ or $Q^{T}C$ or $CQ$ or $C{Q}^T$ as specified by $side$ and $trans$.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $side$.

Constraint: $side=\text{'L'}$ or $\text{'R'}$.

(errno $-2$)

On entry, error in parameter $trans$.

Constraint: $trans=\text{'N'}$ or $\text{'T'}$.

(errno $-3$)

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

Constraint: $m\ge 0$.

(errno $-4$)

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

Constraint: $n\ge 0$.

(errno $-5$)

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

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

(errno $-5$)

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

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

Notes

dormrq 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 elementary reflectors.

This function may be used to form one of the matrix products

$$QC\text{,}{Q}^{T}C\text{,}CQ\text{,}C{Q}^T\text{,}$$

overwriting the result on $C$, which may be any real rectangular $m\times n$ matrix.

A common application of this function is in solving underdetermined linear least squares problems, as described in the F08 Introduction.

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