naginterfaces.library.lapackeig.dormhr

naginterfaces.library.lapackeig.dormhr(side, trans, ilo, ihi, a, tau, c)

dormhr multiplies an arbitrary real matrix $C$ by the real orthogonal matrix $Q$ which was determined by dgehrd() when reducing a real general matrix to Hessenberg form.

For full information please refer to the NAG Library document for f08ng

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08ngf.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 $C$.

iloint

These must be the same arguments $ilo$ and $ihi$, respectively, as supplied to dgehrd().

ihiint

These must be the same arguments $ilo$ and $ihi$, respectively, as supplied to dgehrd().

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

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: $0$.

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: $0$.

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

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

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

Further details of the elementary reflectors, as returned by dgehrd().

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 $ilo$.

(errno $-6$)

On entry, error in parameter $ihi$.

Notes

dormhr is intended to be used following a call to dgehrd(), which reduces a real general matrix $A$ to upper Hessenberg form $H$ by an orthogonal similarity transformation: $A=QH{Q}^T$. dgehrd() represents the matrix $Q$ as a product of $i_{hi}-i_{lo}$ elementary reflectors. Here $i_{lo}$ and $i_{hi}$ are values determined by dgebal() when balancing the matrix; if the matrix has not been balanced, $i_{lo}=1$ and $i_{hi}=n$.

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 to transform a matrix $V$ of eigenvectors of $H$ to the matrix $\text{QV}$ of eigenvectors of $A$.

References

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