# NAG FL Interfacef06qsf (dusqr)

## 1Purpose

f06qsf performs a $QR$ or $RQ$ factorization (as a sequence of plane rotations) of a real upper spiked matrix.

## 2Specification

Fortran Interface
 Subroutine f06qsf ( side, n, k1, k2, c, s, a, lda)
 Integer, Intent (In) :: n, k1, k2, lda Real (Kind=nag_wp), Intent (Inout) :: s(*), a(lda,*) Real (Kind=nag_wp), Intent (Out) :: c(k2-1) Character (1), Intent (In) :: side
#include <nag.h>
 void f06qsf_ (const char *side, const Integer *n, const Integer *k1, const Integer *k2, double c[], double s[], double a[], const Integer *lda, const Charlen length_side)
The routine may be called by the names f06qsf or nagf_blas_dusqr.

## 3Description

f06qsf transforms an $n$ by $n$ real upper spiked matrix $H$ to upper triangular form $R$ by applying a real orthogonal matrix $P$ from the left or the right. $P$ is formed as a sequence of plane rotations in planes ${k}_{1}$ to ${k}_{2}$.
If ${\mathbf{side}}=\text{'L'}$, $H$ is assumed to have a row spike, with nonzero elements ${h}_{{\mathit{k}}_{2},\mathit{k}}$, for $\mathit{k}={\mathit{k}}_{1},\dots ,{\mathit{k}}_{2}-1$. The rotations are applied from the left:
 $PH=R ,$
where $P={P}_{{k}_{2}-1}\cdots {P}_{{k}_{1}+1}{P}_{{k}_{1}}$ and ${P}_{k}$ is a rotation in the $\left(k,{k}_{2}\right)$ plane.
If ${\mathbf{side}}=\text{'R'}$, $H$ is assumed to have a column spike, with nonzero elements ${h}_{\mathit{k}+1,{\mathit{k}}_{1}}$, for $\mathit{k}={\mathit{k}}_{1},\dots ,{\mathit{k}}_{2}-1$. The rotations are applied from the right:
 $HPT=R ,$
where $P={P}_{{k}_{1}}{P}_{{k}_{1}+1}\cdots {P}_{{k}_{2}-1}$ and ${P}_{k}$ is a rotation in the $\left({k}_{1},k+1\right)$ plane.
The $2$ by $2$ plane rotation part of ${P}_{k}$ has the form
 $ck sk -sk ck .$

None.

## 5Arguments

1: $\mathbf{side}$Character(1) Input
On entry: specifies whether $H$ is operated on from the left or the right.
${\mathbf{side}}=\text{'L'}$
$H$ is pre-multiplied from the left.
${\mathbf{side}}=\text{'R'}$
$H$ is post-multiplied from the right.
Constraint: ${\mathbf{side}}=\text{'L'}$ or $\text{'R'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $H$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{k1}$Integer Input
4: $\mathbf{k2}$Integer Input
On entry: the values ${k}_{1}$ and ${k}_{2}$.
If ${\mathbf{k1}}<1$ or ${\mathbf{k2}}\le {\mathbf{k1}}$ or ${\mathbf{k2}}>{\mathbf{n}}$, an immediate return is effected.
5: $\mathbf{c}\left({\mathbf{k2}}-1\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{c}}\left(\mathit{k}\right)$ holds ${c}_{\mathit{k}}$, the cosine of the rotation ${P}_{\mathit{k}}$, for $\mathit{k}={\mathit{k}}_{1},\dots ,{\mathit{k}}_{2}-1$.
6: $\mathbf{s}\left(*\right)$Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array s must be at least ${\mathbf{k2}}-1$.
On entry: the nonzero elements of the spike of $H$: ${\mathbf{s}}\left(\mathit{k}\right)$ must hold ${h}_{{\mathit{k}}_{2},\mathit{k}}$ if ${\mathbf{side}}=\text{'L'}$, and ${h}_{\mathit{k}+1,{\mathit{k}}_{1}}$ if ${\mathbf{side}}=\text{'R'}$, for $\mathit{k}={\mathit{k}}_{1},\dots ,{\mathit{k}}_{2}-1$.
On exit: ${\mathbf{s}}\left(\mathit{k}\right)$ holds ${s}_{\mathit{k}}$, the sine of the rotation ${P}_{\mathit{k}}$, for $\mathit{k}={\mathit{k}}_{1},\dots ,{\mathit{k}}_{2}-1$.
7: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least ${\mathbf{n}}$.
On entry: the upper triangular part of the $n$ by $n$ upper spiked matrix $H$.
On exit: the upper triangular matrix $R$.
8: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f06qsf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

None.

Not applicable.

## 8Parallelism and Performance

f06qsf is not threaded in any implementation.