# NAG Library Routine Document

## 1Purpose

f06fsf generates a real elementary reflection in the LINPACK (as opposed to NAG) style.

## 2Specification

Fortran Interface
 Subroutine f06fsf ( n, x, incx, tol, z1)
 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) :: z1
#include nagmk26.h
 void f06fsf_ ( const Integer *n, double *alpha, double x[], const Integer *incx, const double *tol, double *z1)

## 3Description

f06fsf 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
 $P=I-1ζ ζ z ζ zT ,$
where $z$ is an $n$-element real vector and $\zeta$ is a real scalar. (This form is compatible with that used by LINPACK.)
If the elements of $x$ are all zero, or if the elements of $x$ are all less than $\mathit{tol}×\left|\alpha \right|$ in absolute value, then $\zeta$ is set to $0$ and $P$ can be taken to be the unit matrix. Otherwise $\zeta$ always lies in the range $\left(1,2\right)$.

None.

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
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) 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 real elementary reflection.
4:     $\mathbf{incx}$ – IntegerInput
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 $\mathit{tol}$.
If tol is not in the range $\left(0,1\right)$, the value $0$ is used for $\mathit{tol}$.
6:     $\mathbf{z1}$ – Real (Kind=nag_wp)Output
On exit: the scalar $\zeta$.

None.

Not applicable.

## 8Parallelism and Performance

f06fsf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.