naginterfaces.library.lapackeig.dgemqrt

naginterfaces.library.lapackeig.dgemqrt(side, trans, v, t, c)

dgemqrt multiplies an arbitrary real matrix $C$ by the real orthogonal matrix $Q$ from a $QR$ factorization computed by dgeqrt().

For full information please refer to the NAG Library document for f08ac

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08acf.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 $C$.

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

Note: the required extent for this argument in dimension 1 is determined as follows: if $side=\text{'L'}$: $m$; if $side=\text{'R'}$: $n$; otherwise: $0$.

Details of the vectors which define the elementary reflectors, as returned by dgeqrt() in the first $k$ columns of its array argument $\text{a}$.

tfloat, array-like, shape $(\text{nb},k)$

Further details of the orthogonal matrix $Q$ as returned by dgeqrt(). The number of blocks is $b=\lceil \frac{k}{\text{nb}}\rceil$, where $k=min(m,n)$ and each block is of order $\text{nb}$ except for the last block, which is of order $k-(b-1)\times \text{nb}$. For the $b$ blocks the upper triangular block reflector factors $T_1,T_2,\dots ,T_b$ are stored in the $\text{nb}\times n$ matrix $T$ as $T=[T_1|T_2|\cdots |T_b]$.

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

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

Returns

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: $n\ge k\ge 0$.

(errno $-6$)

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

Constraint: $\text{nb}\ge 1$.

(errno $-6$)

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

Constraint: $\text{nb}\le k$.

Notes

dgemqrt is intended to be used after a call to dgeqrt() which performs a $QR$ factorization of a real matrix $A$. The orthogonal matrix $Q$ is represented as a product of elementary reflectors.

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

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

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

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

References

Golub, G H and Van Loan, C F, 2012, Matrix Computations, (4th Edition), Johns Hopkins University Press, Baltimore