# NAG Library Routine Document

## 1Purpose

f06fuf applies a LINPACK (as opposed to NAG) style real elementary reflection to a real vector.

## 2Specification

Fortran Interface
 Subroutine f06fuf ( n, z, incz, z1, x, incx)
 Integer, Intent (In) :: n, incz, incx Real (Kind=nag_wp), Intent (In) :: z(*), z1 Real (Kind=nag_wp), Intent (Inout) :: alpha, x(*)
#include nagmk26.h
 void f06fuf_ (const Integer *n, const double z[], const Integer *incz, const double *z1, double *alpha, double x[], const Integer *incx)

## 3Description

f06fuf applies a real elementary reflection (Householder matrix) $P$, as generated by f06fsf, to a given real vector:
 $α x ←P α x$
where $x$ is an $n$-element real vector and $\alpha$ a real scalar.

None.

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of elements in $x$ and $z$.
2:     $\mathbf{z}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array z must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)×\left|{\mathbf{incz}}\right|\right)$.
On entry: the vector $z$, as returned by f06fsf.
If ${\mathbf{incz}}>0$, ${z}_{\mathit{i}}$ must be stored in ${\mathbf{z}}\left(1+\left(\mathit{i}-1\right)×{\mathbf{incz}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
If ${\mathbf{incz}}<0$, ${z}_{\mathit{i}}$ must be stored in ${\mathbf{z}}\left(1-\left({\mathbf{n}}-\mathit{i}\right)×{\mathbf{incz}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:     $\mathbf{incz}$ – IntegerInput
On entry: the increment in the subscripts of z between successive elements of $z$.
4:     $\mathbf{z1}$ – Real (Kind=nag_wp)Input
On entry: the scalar $\zeta$, as returned by f06fsf.
If $\zeta =0$, $P$ is assumed to be the unit matrix and the transformation is skipped.
5:     $\mathbf{alpha}$ – Real (Kind=nag_wp)Input/Output
On entry: the original scalar $\alpha$.
On exit: the transformed scalar $\alpha$.
6:     $\mathbf{x}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)×\left|{\mathbf{incx}}\right|\right)$.
On entry: the original vector $x$.
If ${\mathbf{incx}}>0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(1+\left(\mathit{i}-1\right)×{\mathbf{incx}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
If ${\mathbf{incx}}<0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(1-\left({\mathbf{n}}-\mathit{i}\right)×{\mathbf{incx}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
On exit: the transformed vector $x$ stored in the same array elements used to supply the original vector $x$.
7:     $\mathbf{incx}$ – IntegerInput
On entry: the increment in the subscripts of x between successive elements of $x$.

None.

Not applicable.

## 8Parallelism and Performance

f06fuf 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.