# NAG FL Interfacef06frf (dnhousg)

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## 1Purpose

f06frf generates a real elementary reflection in the NAG (as opposed to LINPACK) style.

## 2Specification

Fortran Interface
 Subroutine f06frf ( n, x, incx, tol, zeta)
 Integer, Intent (In) :: n, incx Real (Kind=nag_wp), Intent (In) :: tol Real (Kind=nag_wp), Intent (Inout) :: alpha, x(*) Real (Kind=nag_wp), Intent (Out) :: zeta
#include <nag.h>
 void f06frf_ (const Integer *n, double *alpha, double x[], const Integer *incx, const double *tol, double *zeta)
The routine may be called by the names f06frf or nagf_blas_dnhousg.

## 3Description

f06frf generates details of a real elementary reflection (Householder matrix), $P$, such that
 $P ( α x )=( β 0 )$
where $P$ is orthogonal, $\alpha$ and $\beta$ are real scalars, and $x$ is an $n$-element real vector.
$P$ is given in the form
 $P=I-( ζ z ) ( ζ zT ) ,$
where $z$ is an $n$-element real vector and $\zeta$ is a real scalar.
If $x$ is such that
 $max|xi|≤max(tol,ε|α|)$
where $\epsilon$ is the machine precision and $\mathit{tol}$ is a user-supplied tolerance, then $\zeta$ is set to $0$, and $P$ can be taken to be the unit matrix. Otherwise $1\le \zeta \le \sqrt{2}$.
None.

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of elements in $x$ and $z$.
2: $\mathbf{alpha}$Real (Kind=nag_wp) Input/Output
On entry: the scalar $\alpha$.
On exit: the scalar $\beta$.
3: $\mathbf{x}\left(*\right)$Real (Kind=nag_wp) array Input/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 real elementary reflection.
4: $\mathbf{incx}$Integer Input
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}>0$.
5: $\mathbf{tol}$Real (Kind=nag_wp) Input
On entry: the value $\mathit{tol}$.
6: $\mathbf{zeta}$Real (Kind=nag_wp) Output
On exit: the scalar $\zeta$.

None.

Not applicable.