naginterfaces.library.lapackeig.zgehrd

naginterfaces.library.lapackeig.zgehrd(ilo, ihi, a)

zgehrd reduces a complex general matrix to Hessenberg form.

For full information please refer to the NAG Library document for f08ns

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/f08/f08nsf.html

Parameters

iloint

If $A$ has been output by zgebal(), $ilo$ and $ihi$ must contain the values returned by that function. Otherwise, $ilo$ must be set to $1$ and $ihi$ to $\text{n}$.

ihiint

If $A$ has been output by zgebal(), $ilo$ and $ihi$ must contain the values returned by that function. Otherwise, $ilo$ must be set to $1$ and $ihi$ to $\text{n}$.

acomplex, array-like, shape $(n,n)$

The $n\times n$ general matrix $A$.

Returns

acomplex, ndarray, shape $(n,n)$

$a$ is overwritten by the upper Hessenberg matrix $H$ and details of the unitary matrix $Q$. The subdiagonal elements of $H$ are real.

taucomplex, ndarray, shape $(max(1,n-1))$

Further details of the unitary 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

zgehrd reduces a complex general matrix $A$ to upper Hessenberg form $H$ by a unitary similarity transformation: $A=QH{Q}^H$. $H$ has real subdiagonal elements.

The matrix $Q$ is not formed explicitly, but is represented as a product of elementary reflectors (see the F08 Introduction for details). Functions are provided to work with $Q$ in this representation (see Further Comments).

The function can take advantage of a previous call to zgebal(), which may produce a matrix with the structure:

$$\begin{array}{c}\left(\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ & A_{22}& A_{23}\\ & & A_{33}\end{array}\right)\end{array}$$

where $A_{11}$ and $A_{33}$ are upper triangular. If so, only the central diagonal block $A_{22}$, in rows and columns $i_{lo}$ to $i_{hi}$, needs to be reduced to Hessenberg form (the blocks $A_{12}$ and $A_{23}$ will also be affected by the reduction). Therefore, the values of $i_{lo}$ and $i_{hi}$ determined by zgebal() can be supplied to the function directly. If zgebal() has not previously been called however, then $i_{lo}$ must be set to $1$ and $i_{hi}$ to $n$.

References

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