# NAG FL Interfacef08nff (dorghr)

## 1Purpose

f08nff generates the real orthogonal matrix $Q$ which was determined by f08nef when reducing a real general matrix $A$ to Hessenberg form.

## 2Specification

Fortran Interface
 Subroutine f08nff ( n, ilo, ihi, a, lda, tau, work, info)
 Integer, Intent (In) :: n, ilo, ihi, lda, lwork Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: tau(*) Real (Kind=nag_wp), Intent (Inout) :: a(lda,*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
#include <nag.h>
 void f08nff_ (const Integer *n, const Integer *ilo, const Integer *ihi, double a[], const Integer *lda, const double tau[], double work[], const Integer *lwork, Integer *info)
The routine may be called by the names f08nff, nagf_lapackeig_dorghr or its LAPACK name dorghr.

## 3Description

f08nff is intended to be used following a call to f08nef, which reduces a real general matrix $A$ to upper Hessenberg form $H$ by an orthogonal similarity transformation: $A=QH{Q}^{\mathrm{T}}$. f08nef represents the matrix $Q$ as a product of ${i}_{\mathrm{hi}}-{i}_{\mathrm{lo}}$ elementary reflectors. Here ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$ are values determined by f08nhf when balancing the matrix; if the matrix has not been balanced, ${i}_{\mathrm{lo}}=1$ and ${i}_{\mathrm{hi}}=n$.
This routine may be used to generate $Q$ explicitly as a square matrix. $Q$ has the structure:
 $Q = I 0 0 0 Q22 0 0 0 I$
where ${Q}_{22}$ occupies rows and columns ${i}_{\mathrm{lo}}$ to ${i}_{\mathrm{hi}}$.

## 4References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{ilo}$Integer Input
3: $\mathbf{ihi}$Integer Input
On entry: these must be the same arguments ilo and ihi, respectively, as supplied to f08nef.
Constraints:
• if ${\mathbf{n}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{n}}$;
• if ${\mathbf{n}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$.
4: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: details of the vectors which define the elementary reflectors, as returned by f08nef.
On exit: the $n$ by $n$ orthogonal matrix $Q$.
5: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08nff is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6: $\mathbf{tau}\left(*\right)$Real (Kind=nag_wp) array Input
Note: the dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: further details of the elementary reflectors, as returned by f08nef.
7: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Real (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
8: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08nff is called, unless ${\mathbf{lwork}}=-1$, in which case a workspace query is assumed and the routine only calculates the optimal dimension of work (using the formula given below).
Suggested value: for optimal performance lwork should be at least $\left({\mathbf{ihi}}-{\mathbf{ilo}}\right)×nb$, where $nb$ is the block size.
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{ihi}}-{\mathbf{ilo}}\right)$ or ${\mathbf{lwork}}=-1$.
9: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7Accuracy

The computed matrix $Q$ differs from an exactly orthogonal matrix by a matrix $E$ such that
 $E2 = Oε ,$
where $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08nff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nff makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is approximately $\frac{4}{3}{q}^{3}$, where $q={i}_{\mathrm{hi}}-{i}_{\mathrm{lo}}$.
The complex analogue of this routine is f08ntf.

## 10Example

This example computes the Schur factorization of the matrix $A$, where
 $A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 .$
Here $A$ is general and must first be reduced to Hessenberg form by f08nef. The program then calls f08nff to form $Q$, and passes this matrix to f08pef which computes the Schur factorization of $A$.

### 10.1Program Text

Program Text (f08nffe.f90)

### 10.2Program Data

Program Data (f08nffe.d)

### 10.3Program Results

Program Results (f08nffe.r)