# NAG Library Routine Document

## 1Purpose

f08ntf (zunghr) generates the complex unitary matrix $Q$ which was determined by f08nsf (zgehrd) when reducing a complex general matrix $A$ to Hessenberg form.

## 2Specification

Fortran Interface
 Subroutine f08ntf ( n, ilo, ihi, a, lda, tau, work, info)
 Integer, Intent (In) :: n, ilo, ihi, lda, lwork Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (In) :: tau(*) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
#include nagmk26.h
 void f08ntf_ (const Integer *n, const Integer *ilo, const Integer *ihi, Complex a[], const Integer *lda, const Complex tau[], Complex work[], const Integer *lwork, Integer *info)
The routine may be called by its LAPACK name zunghr.

## 3Description

f08ntf (zunghr) is intended to be used following a call to f08nsf (zgehrd), which reduces a complex general matrix $A$ to upper Hessenberg form $H$ by a unitary similarity transformation: $A=QH{Q}^{\mathrm{H}}$. f08nsf (zgehrd) 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 f08nvf (zgebal) 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{ilo}$ – IntegerInput
3:     $\mathbf{ihi}$ – IntegerInput
On entry: these must be the same arguments ilo and ihi, respectively, as supplied to f08nsf (zgehrd).
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:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (Kind=nag_wp) arrayInput/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 f08nsf (zgehrd).
On exit: the $n$ by $n$ unitary matrix $Q$.
5:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08ntf (zunghr) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6:     $\mathbf{tau}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
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 f08nsf (zgehrd).
7:     $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Complex (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the real part of ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
8:     $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08ntf (zunghr) 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:     $\mathbf{info}$ – IntegerOutput
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 unitary matrix by a matrix $E$ such that
 $E2 = Oε ,$
where $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08ntf (zunghr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ntf (zunghr) 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 real floating-point operations is approximately $\frac{16}{3}{q}^{3}$, where $q={i}_{\mathrm{hi}}-{i}_{\mathrm{lo}}$.
The real analogue of this routine is f08nff (dorghr).

## 10Example

This example computes the Schur factorization of the matrix $A$, where
 $A = -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i .$
Here $A$ is general and must first be reduced to Hessenberg form by f08nsf (zgehrd). The program then calls f08ntf (zunghr) to form $Q$, and passes this matrix to f08psf (zhseqr) which computes the Schur factorization of $A$.

### 10.1Program Text

Program Text (f08ntfe.f90)

### 10.2Program Data

Program Data (f08ntfe.d)

### 10.3Program Results

Program Results (f08ntfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017