naginterfaces.library.lapackeig.dormbr

naginterfaces.library.lapackeig.dormbr(vect, side, trans, k, a, tau, c)

dormbr multiplies an arbitrary real $m\times n$ matrix $C$ by one of the real orthogonal matrices $Q$ or $P$ which were determined by dgebrd() when reducing a real matrix to bidiagonal form.

For full information please refer to the NAG Library document for f08kg

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08kgf.html

Parameters

vectstr, length 1

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

$vect=\text{'Q'}$

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

$vect=\text{'P'}$

$P$ or $P^T$ is applied to $C$.

sidestr, length 1

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

$side=\text{'L'}$

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

$side=\text{'R'}$

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

transstr, length 1

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

$trans=\text{'N'}$

$Q$ or $P$ is applied to $C$.

$trans=\text{'T'}$

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

kint

If $vect=\text{'Q'}$, the number of columns in the original matrix $A$.

If $vect=\text{'P'}$, the number of rows in the original matrix $A$.

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

Note: the required extent for this argument in dimension 1 is determined as follows: if $vect=\text{'Q'}$: $\text{r}$; if $vect=\text{'P'}$: $min(\text{r},k)$; otherwise: $0$.

Note: the required extent for this argument in dimension 2 is determined as follows: if $vect=\text{'Q'}$: $min(\text{r},k)$; if $vect=\text{'P'}$: $\text{r}$; otherwise: $0$.

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

taufloat, array-like, shape $(min(\text{r},k))$

Further details of the elementary reflectors, as returned by dgebrd() in its argument $\text{tauq}$ if $vect=\text{'Q'}$, or in its argument $\text{taup}$ if $vect=\text{'P'}$.

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

The matrix $C$.

Returns

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

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

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $vect$.

Constraint: $vect=\text{'Q'}$ or $\text{'P'}$.

(errno $-2$)

On entry, error in parameter $side$.

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

(errno $-3$)

On entry, error in parameter $trans$.

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

(errno $-4$)

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

Constraint: $m\ge 0$.

(errno $-5$)

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

Constraint: $n\ge 0$.

(errno $-6$)

On entry, error in parameter $k$.

Constraint: $k\ge 0$.

Notes

dormbr is intended to be used after a call to dgebrd(), which reduces a real rectangular matrix $A$ to bidiagonal form $B$ by an orthogonal transformation: $A=QB{P}^T$. dgebrd() represents the matrices $Q$ and $P^T$ as products of elementary reflectors.

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

$$QC,Q^{T}C,CQ,C{Q}^T,PC,P^{T}C,CP\text{or}C{P}^T\text{,}$$

overwriting the result on $C$ (which may be any real rectangular matrix).

References

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