# NAG Library Routine Document

## 1Purpose

f06qqf performs a $QR$ factorization (as a sequence of plane rotations) of a real upper triangular matrix that has been augmented by a full row.

## 2Specification

Fortran Interface
 Subroutine f06qqf ( n, x, incx, a, lda, c, s)
 Integer, Intent (In) :: n, incx, lda Real (Kind=nag_wp), Intent (In) :: alpha Real (Kind=nag_wp), Intent (Inout) :: x(*), a(lda,*) Real (Kind=nag_wp), Intent (Out) :: c(n), s(n)
#include nagmk26.h
 void f06qqf_ ( const Integer *n, const double *alpha, double x[], const Integer *incx, double a[], const Integer *lda, double c[], double s[])

## 3Description

f06qqf performs the factorization
 $U αxT =Q R 0$
where $U$ and $R$ are $n$ by $n$ real upper triangular matrices, $x$ is an $n$-element real vector, $\alpha$ is a real scalar, and $Q$ is a real orthogonal matrix.
$Q$ is formed as a sequence of plane rotations
 $QT = Qn ⋯ Q2 Q1$
where ${Q}_{k}$ is a rotation in the $\left(k,n+1\right)$ plane, chosen to annihilate ${x}_{k}$.
The $2$ by $2$ plane rotation part of ${Q}_{k}$ has the form
 $ck sk -sk ck .$

None.

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrices $U$ and $R$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{alpha}$ – Real (Kind=nag_wp)Input
On entry: the scalar $\alpha$.
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 vector $x$. ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(1+\left(\mathit{i}–1\right)×{\mathbf{incx}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
On exit: the referenced elements are overwritten by the tangents of the rotations ${Q}_{\mathit{k}}$, for $\mathit{k}=1,2,\dots ,n$.
4:     $\mathbf{incx}$ – IntegerInput
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}>0$.
5:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least ${\mathbf{n}}$.
On entry: the $n$ by $n$ upper triangular matrix $U$.
On exit: the upper triangular matrix $R$.
6:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f06qqf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:     $\mathbf{c}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the values ${c}_{\mathit{k}}$, the cosines of the rotations ${Q}_{\mathit{k}}$, for $\mathit{k}=1,2,\dots ,n$.
8:     $\mathbf{s}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the values ${s}_{\mathit{k}}$, the sines of the rotations ${Q}_{\mathit{k}}$, for $\mathit{k}=1,2,\dots ,n$.

None.

Not applicable.

## 8Parallelism and Performance

f06qqf is not threaded in any implementation.