F06 Chapter Contents
F06 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF06QRF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F06QRF performs a $QR$ or $RQ$ factorization (as a sequence of plane rotations) of a real upper Hessenberg matrix.

## 2  Specification

 SUBROUTINE F06QRF ( SIDE, N, K1, K2, C, S, A, LDA)
 INTEGER N, K1, K2, LDA REAL (KIND=nag_wp) C(K2-1), S(K2-1), A(LDA,*) CHARACTER(1) SIDE

## 3  Description

F06QRF transforms an $n$ by $n$ real upper Hessenberg matrix $H$ to upper triangular form $R$ by applying an orthogonal matrix $P$ from the left or the right. $H$ is assumed to have nonzero subdiagonal elements ${h}_{\mathit{k}+1,\mathit{k}}$, for $\mathit{k}={\mathit{k}}_{1},\dots ,{\mathit{k}}_{2}-1$, only. $P$ is formed as a sequence of plane rotations in planes ${k}_{1}$ to ${k}_{2}$.
If ${\mathbf{SIDE}}=\text{'L'}$, the rotations are applied from the left:
 $PH=R ,$
where $P={P}_{{k}_{2}-1}\cdots {P}_{{k}_{1}+1}{P}_{{k}_{1}}$.
If ${\mathbf{SIDE}}=\text{'R'}$, the rotations are applied from the right:
 $HPT=R ,$
where $P={P}_{{k}_{1}}{P}_{{k}_{1}+1}\cdots {P}_{{k}_{2}-1}$.
In either case, ${P}_{k}$ is a rotation in the $\left(k,k+1\right)$ plane, chosen to annihilate ${h}_{k+1,k}$.
The $2$ by $2$ plane rotation part of ${P}_{k}$ has the form
 $ck sk -sk ck .$

None.

## 5  Parameters

1:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $H$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     K1 – INTEGERInput
4:     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:     C(${\mathbf{K2}}-1$) – REAL (KIND=nag_wp) arrayOutput
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:     S(${\mathbf{K2}}-1$) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the nonzero subdiagonal elements of $H$: ${\mathbf{S}}\left(\mathit{k}\right)$ must hold ${h}_{\mathit{k}+1,\mathit{k}}$, 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:     A(LDA,$*$) – REAL (KIND=nag_wp) arrayInput/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 Hessenberg matrix $H$.
On exit: the upper triangular matrix $R$.
8:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F06QRF is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.

None.

Not applicable.