naginterfaces.library.lapackeig.dorgbr

naginterfaces.library.lapackeig.dorgbr(vect, k, a, tau)

dorgbr generates one of the real orthogonal matrices $Q$ or $P^T$ which were determined by dgebrd() when reducing a real matrix to bidiagonal form.

For full information please refer to the NAG Library document for f08kf

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08kff.html

Parameters

vectstr, length 1

Indicates whether the orthogonal matrix $Q$ or $P^T$ is generated.

$vect=\text{'Q'}$

$Q$ is generated.

$vect=\text{'P'}$

$P^T$ is generated.

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 $(m,n)$

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

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

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

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 $vect=\text{'P'}$.

Returns

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

The orthogonal matrix $Q$ or $P^T$, or the leading rows or columns thereof, as specified by $vect$, $m$ and $n$.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $vect$.

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

(errno $-2$)

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

Constraint: $m\ge 0$.

(errno $-3$)

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

Constraint: $n\ge 0$.

(errno $-3$)

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

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

(errno $-3$)

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

Constraint: $m=n$.

(errno $-3$)

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

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

(errno $-3$)

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

Constraint: $n=m$.

(errno $-4$)

On entry, error in parameter $k$.

Constraint: $k\ge 0$.

Notes

dorgbr 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 generate $Q$ or $P^T$ explicitly as square matrices, or in some cases just the leading columns of $Q$ or the leading rows of $P^T$.

The various possibilities are specified by the arguments $vect$, $\text{m}$, $\text{n}$ and $k$. The appropriate values to cover the most likely cases are as follows (assuming that $A$ was an $m\times n$ matrix):

1. To form the full $m\times m$ matrix $Q$: set $vect=\text{'Q'}$, $n=m$ and $k=n$ (note that the array $a$ must have at least $m$ columns).

2. If $m>n$, to form the $n$ leading columns of $Q$: set $vect=\text{'Q'}$, and $k=n$

3. To form the full $n\times n$ matrix $P^T$: set $vect=\text{'P'}$, $m=n$ and $k=m$ (note that the array $a$ must have at least $n$ rows).

4. If $m<n$, to form the $m$ leading rows of $P^T$: set $vect=\text{'P'}$ and $k=m$

References

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