# NAG Library Routine Document

## 1Purpose

f06qwf transforms a real upper triangular matrix to an upper spiked matrix by applying a given sequence of plane rotations.

## 2Specification

Fortran Interface
 Subroutine f06qwf ( side, n, k1, k2, c, s, a, lda)
 Integer, Intent (In) :: n, k1, k2, lda Real (Kind=nag_wp), Intent (In) :: c(*) Real (Kind=nag_wp), Intent (Inout) :: s(*), a(lda,*) Character (1), Intent (In) :: side
#include nagmk26.h
 void f06qwf_ (const char *side, const Integer *n, const Integer *k1, const Integer *k2, const double c[], double s[], double a[], const Integer *lda, const Charlen length_side)

## 3Description

f06qwf transforms an $n$ by $n$ real upper triangular matrix $U$ to an upper spiked matrix $H$, by applying a given sequence of plane rotations from either the left or the right, in planes ${k}_{1}$ to ${k}_{2}$.
If ${\mathbf{side}}=\text{'L'}$, $H$ has a row spike, with nonzero elements ${h}_{{k}_{2},k}$, for $k={k}_{1},{k}_{1}+1,\dots ,{k}_{2}-1$. The rotations are applied from the left:
 $H=PU ,$
where $P={P}_{{k}_{1}}{P}_{{k}_{1}+1}\cdots {P}_{{k}_{2}-1}$ and ${P}_{k}$ is a rotation in the $\left(k,{k}_{2}\right)$ plane.
If ${\mathbf{side}}=\text{'R'}$, $H$ has a column spike, with nonzero elements ${h}_{k+1,{k}_{1}}$, for $k={k}_{1},{k}_{1}+1,\dots ,{k}_{2}-1$. The rotations are applied from the right:
 $HPT = R ,$
where $P={P}_{{k}_{2}-1}\cdots {P}_{{k}_{1}+1}{P}_{{k}_{1}}$ and ${P}_{k}$ is a rotation in the $\left({k}_{1},k+1\right)$ 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 $U$ is operated on from the left or the right.
${\mathbf{side}}=\text{'L'}$
$U$ is pre-multiplied from the left.
${\mathbf{side}}=\text{'R'}$
$U$ is post-multiplied from the right.
Constraint: ${\mathbf{side}}=\text{'L'}$ or $\text{'R'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrices $U$ and $H$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{k1}$ – IntegerInput
4:     $\mathbf{k2}$ – IntegerInput
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:     $\mathbf{c}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array c must be at least ${\mathbf{k2}}-1$.
On entry: ${\mathbf{c}}\left(\mathit{k}\right)$ must hold ${c}_{\mathit{k}}$, 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) arrayInput/Output
Note: the dimension of the array s must be at least ${\mathbf{k2}}-1$.
On entry: ${\mathbf{s}}\left(\mathit{k}\right)$ must hold ${s}_{\mathit{k}}$, the sine of the rotation ${P}_{\mathit{k}}$, for $\mathit{k}={\mathit{k}}_{1},\dots ,{\mathit{k}}_{2}-1$.
On exit: ${\mathbf{s}}\left(\mathit{k}\right)$ holds a nonzero element of the spike of $H$: ${h}_{{\mathit{k}}_{2},\mathit{k}}$ if ${\mathbf{side}}=\text{'L'}$, or ${h}_{\mathit{k}+1,{\mathit{k}}_{1}}$ if ${\mathbf{side}}=\text{'R'}$, for $\mathit{k}={\mathit{k}}_{1},\dots ,{\mathit{k}}_{2}-1$.
7:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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 part of the upper spiked matrix $H$.
8:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f06qwf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

None.

Not applicable.

## 8Parallelism and Performance

f06qwf is not threaded in any implementation.