naginterfaces.library.lapackeig.dorghr

naginterfaces.library.lapackeig.dorghr(ilo, ihi, a, tau)

dorghr generates the real orthogonal matrix $Q$ which was determined by dgehrd() when reducing a real general matrix $A$ to Hessenberg form.

For full information please refer to the NAG Library document for f08nf

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08nff.html

Parameters

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 $(n,n)$

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

taufloat, array-like, shape $(max(1,n-1))$

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

Returns

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

The $n\times n$ orthogonal matrix $Q$.

Raises

NagValueError

(errno $-1$)

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

Constraint: $n\ge 0$.

(errno $-2$)

On entry, error in parameter $ilo$.

(errno $-3$)

On entry, error in parameter $ihi$.

Notes

dorghr 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 generate $Q$ explicitly as a square matrix. $Q$ has the structure:

$$\begin{array}{c}Q=\left(\begin{array}{ccc}I& 0& 0\\ 0& Q_{22}& 0\\ 0& 0& I\end{array}\right)\end{array}$$

where $Q_{22}$ occupies rows and columns $i_{lo}$ to $i_{hi}$.

References

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