# NAG Library Routine Document

## 1Purpose

f08awf (zunglq) generates all or part of the complex unitary matrix $Q$ from an $LQ$ factorization computed by f08avf (zgelqf).

## 2Specification

Fortran Interface
 Subroutine f08awf ( m, n, k, a, lda, tau, work, info)
 Integer, Intent (In) :: m, n, k, 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 f08awf_ (const Integer *m, const Integer *n, const Integer *k, Complex a[], const Integer *lda, const Complex tau[], Complex work[], const Integer *lwork, Integer *info)
The routine may be called by its LAPACK name zunglq.

## 3Description

f08awf (zunglq) is intended to be used after a call to f08avf (zgelqf), which performs an $LQ$ factorization of a complex matrix $A$. The unitary matrix $Q$ is represented as a product of elementary reflectors.
This routine may be used to generate $Q$ explicitly as a square matrix, or to form only its leading rows.
Usually $Q$ is determined from the $LQ$ factorization of a $p$ by $n$ matrix $A$ with $p\le n$. The whole of $Q$ may be computed by:
```Call zunglq(n,n,p,a,lda,tau,work,lwork,info)
```
(note that the array a must have at least $n$ rows) or its leading $p$ rows by:
```Call zunglq(p,n,p,a,lda,tau,work,lwork,info)
```
The rows of $Q$ returned by the last call form an orthonormal basis for the space spanned by the rows of $A$; thus f08avf (zgelqf) followed by f08awf (zunglq) can be used to orthogonalize the rows of $A$.
The information returned by the $LQ$ factorization routines also yields the $LQ$ factorization of the leading $k$ rows of $A$, where $k. The unitary matrix arising from this factorization can be computed by:
```Call zunglq(n,n,k,a,lda,tau,work,lwork,info)
```
or its leading $k$ rows by:
```Call zunglq(k,n,k,a,lda,tau,work,lwork,info)
```

## 4References

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

## 5Arguments

1:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $Q$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge {\mathbf{m}}$.
3:     $\mathbf{k}$ – IntegerInput
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraint: ${\mathbf{m}}\ge {\mathbf{k}}\ge 0$.
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 f08avf (zgelqf).
On exit: the $m$ by $n$ matrix $Q$.
5:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08awf (zunglq) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
6:     $\mathbf{tau}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry: further details of the elementary reflectors, as returned by f08avf (zgelqf).
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 f08awf (zunglq) is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, ${\mathbf{lwork}}\ge {\mathbf{m}}×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ 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

f08awf (zunglq) 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 $16mnk-8\left(m+n\right){k}^{2}+\frac{16}{3}{k}^{3}$; when $m=k$, the number is approximately $\frac{8}{3}{m}^{2}\left(3n-m\right)$.
The real analogue of this routine is f08ajf (dorglq).

## 10Example

This example forms the leading $4$ rows of the unitary matrix $Q$ from the $LQ$ factorization of the matrix $A$, where
 $A = 0.28-0.36i 0.50-0.86i -0.77-0.48i 1.58+0.66i -0.50-1.10i -1.21+0.76i -0.32-0.24i -0.27-1.15i 0.36-0.51i -0.07+1.33i -0.75+0.47i -0.08+1.01i .$
The rows of $Q$ form an orthonormal basis for the space spanned by the rows of $A$.

### 10.1Program Text

Program Text (f08awfe.f90)

### 10.2Program Data

Program Data (f08awfe.d)

### 10.3Program Results

Program Results (f08awfe.r)