# NAG FL Interfacef06trf (zuhqr)

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## 1Purpose

f06trf performs a $QR$ or $RQ$ factorization (as a sequence of plane rotations) of a complex upper Hessenberg matrix.

## 2Specification

Fortran Interface
 Subroutine f06trf ( side, n, k1, k2, c, s, a, lda)
 Integer, Intent (In) :: n, k1, k2, lda Real (Kind=nag_wp), Intent (Inout) :: s(*) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*) Complex (Kind=nag_wp), Intent (Out) :: c(k2) Character (1), Intent (In) :: side
#include <nag.h>
 void f06trf_ (const char *side, const Integer *n, const Integer *k1, const Integer *k2, Complex c[], double s[], Complex a[], const Integer *lda, const Charlen length_side)
The routine may be called by the names f06trf or nagf_blas_zuhqr.

## 3Description

f06trf transforms an $n×n$ complex upper Hessenberg matrix $H$ to upper triangular form $R$ by applying a unitary matrix $P$ from the left or the right. $H$ is assumed to have real nonzero subdiagonal elements ${h}_{\mathit{k}+1,\mathit{k}}$, for $\mathit{k}={\mathit{k}}_{1},\dots ,{\mathit{k}}_{2}-1$, only; $R$ has real diagonal elements. $P$ 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 $P=D{P}_{{k}_{2}-1}\cdots {P}_{{k}_{1}+1}{P}_{{k}_{1}}$ and $D=\mathrm{diag}\left(1,\dots ,1,{d}_{{k}_{2}},1,\dots ,1\right)$ with $|{d}_{{k}_{2}}|=1$.
If ${\mathbf{side}}=\text{'R'}$, the rotations are applied from the right:
 $HPH=R ,$
where $P=D{P}_{{k}_{1}}{P}_{{k}_{1}+1}\cdots {P}_{{k}_{2}-1}$ and $D=\mathrm{diag}\left(1,\dots ,1,{d}_{{k}_{1}},1,\dots ,1\right)$ with $|{d}_{{k}_{1}}|=1$.
In either case, ${P}_{k}$ is a rotation in the $\left(k,k+1\right)$ plane, chosen to annihilate ${h}_{k+1,k}$.
The $2×2$ plane rotation part of ${P}_{k}$ has the form
 $( c¯k sk -sk ck )$
with ${s}_{k}$ real.

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 dimension of the array c as declared in the (sub)program from which f06trf is called. 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: $\mathbf{c}\left({\mathbf{k2}}\right)$Complex (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 $\mathit{k}={\mathit{k}}_{1},\dots ,{\mathit{k}}_{2}-1$; ${\mathbf{c}}\left({k}_{2}\right)$ holds ${d}_{{k}_{2}}$, the ${k}_{2}$th diagonal element of $D$, if ${\mathbf{side}}=\text{'L'}$, or ${d}_{{k}_{1}}$, the ${k}_{1}$th diagonal element of $D$, if ${\mathbf{side}}=\text{'R'}$.
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 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: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (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 $n×n$ upper Hessenberg matrix $H$.
On exit: the upper triangular matrix $R$. The imaginary parts of the diagonal elements are set to zero.
8: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f06trf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

None.

Not applicable.