F06 Chapter Contents
F06 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF06HRF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F06HRF generates a complex elementary reflection.

## 2  Specification

 SUBROUTINE F06HRF ( N, ALPHA, X, INCX, TOL, THETA)
 INTEGER N, INCX REAL (KIND=nag_wp) TOL COMPLEX (KIND=nag_wp) ALPHA, X(*), THETA

## 3  Description

F06HRF generates details of a complex elementary reflection (Householder matrix), $P$, such that
 $P α x = β 0$
where $P$ is unitary, $\alpha$ is a complex scalar, $\beta$ is a real scalar, and $x$ is an $n$-element complex vector.
$P$ is given in the form
 $P=I-γ ζ z ζ zH ,$
where $z$ is an $n$-element complex vector, $\gamma$ is a complex scalar such that $\mathrm{Re}\left(\gamma \right)=1$, and $\zeta$ is a real scalar. $\gamma$ and $\zeta$ are returned in a single complex value $\theta =\left(\zeta ,\mathrm{Im}\left(\gamma \right)\right)$. Thus $\zeta =\mathrm{Re}\left(\theta \right)$ and $\gamma =\left(1,\mathrm{Im}\left(\theta \right)\right)$.
If $x$ is such that
 $maxRexi,Imxi≤maxtol,εmaxReα,Imα,$
where $\epsilon$ is the machine precision and $\mathit{tol}$ is a user-supplied tolerance, then:
• either $\theta$ is set to $0$, in which case $P$ can be taken to be the unit matrix;
• or $\theta$ is set so that $\mathrm{Re}\left(\theta \right)\le 0$ and $\theta \ne 0$, in which case
 $P= θ 0 0 I .$
Otherwise $1\le \mathrm{Re}\left(\theta \right)\le \sqrt{2}$.

None.

## 5  Parameters

1:     N – INTEGERInput
On entry: $n$, the number of elements in $x$ and $z$.
2:     ALPHA – COMPLEX (KIND=nag_wp)Input/Output
On entry: the scalar $\alpha$.
On exit: the scalar $\beta$.
3:     X($*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array X must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{N}}-1\right)×{\mathbf{INCX}}\right)$.
On entry: the $n$-element vector $x$. ${x}_{\mathit{i}}$ must be stored in ${\mathbf{X}}\left(1+\left(\mathit{i}-1\right)×{\mathbf{INCX}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{N}}$.
Intermediate elements of X are not referenced.
On exit: the referenced elements are overwritten by details of the complex elementary reflection.
4:     INCX – INTEGERInput
On entry: the increment in the subscripts of X between successive elements of $x$.
Constraint: ${\mathbf{INCX}}>0$.
5:     TOL – REAL (KIND=nag_wp)Input
On entry: the value $\mathit{tol}$.
6:     THETA – COMPLEX (KIND=nag_wp)Output
On exit: the scalar $\theta$.

None.

Not applicable.

None.

None.