F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08ATF (ZUNGQR)

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

F08ATF (ZUNGQR) generates all or part of the complex unitary matrix $Q$ from a $QR$ factorization computed by F08ASF (ZGEQRF), F08BSF (ZGEQPF) or F08BTF (ZGEQP3).

## 2  Specification

 SUBROUTINE F08ATF ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
 INTEGER M, N, K, LDA, LWORK, INFO COMPLEX (KIND=nag_wp) A(LDA,*), TAU(*), WORK(max(1,LWORK))
The routine may be called by its LAPACK name zungqr.

## 3  Description

F08ATF (ZUNGQR) is intended to be used after a call to F08ASF (ZGEQRF), F08BSF (ZGEQPF) or F08BTF (ZGEQP3), which perform a $QR$ 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 columns.
Usually $Q$ is determined from the $QR$ factorization of an $m$ by $p$ matrix $A$ with $m\ge p$. The whole of $Q$ may be computed by:
```CALL ZUNGQR(M,M,P,A,LDA,TAU,WORK,LWORK,INFO)
```
(note that the array A must have at least $m$ columns) or its leading $p$ columns by:
```CALL ZUNGQR(M,P,P,A,LDA,TAU,WORK,LWORK,INFO)
```
The columns of $Q$ returned by the last call form an orthonormal basis for the space spanned by the columns of $A$; thus F08ASF (ZGEQRF) followed by F08ATF (ZUNGQR) can be used to orthogonalize the columns of $A$.
The information returned by the $QR$ factorization routines also yields the $QR$ factorization of the leading $k$ columns of $A$, where $k. The unitary matrix arising from this factorization can be computed by:
```CALL ZUNGQR(M,M,K,A,LDA,TAU,WORK,LWORK,INFO)
```
or its leading $k$ columns by:
```CALL ZUNGQR(M,K,K,A,LDA,TAU,WORK,LWORK,INFO)
```

## 4  References

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

## 5  Parameters

1:     M – INTEGERInput
On entry: $m$, the order of the unitary matrix $Q$.
Constraint: ${\mathbf{M}}\ge 0$.
2:     N – INTEGERInput
On entry: $n$, the number of columns of the matrix $Q$.
Constraint: ${\mathbf{M}}\ge {\mathbf{N}}\ge 0$.
3:     K – INTEGERInput
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraint: ${\mathbf{N}}\ge {\mathbf{K}}\ge 0$.
4:     A(LDA,$*$) – 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 F08ASF (ZGEQRF), F08BSF (ZGEQPF) or F08BTF (ZGEQP3).
On exit: the $m$ by $n$ matrix $Q$.
5:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08ATF (ZUNGQR) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
6:     TAU($*$) – 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 F08ASF (ZGEQRF), F08BSF (ZGEQPF) or F08BTF (ZGEQP3).
7:     WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – 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:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08ATF (ZUNGQR) 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, ${\mathbf{LWORK}}\ge {\mathbf{N}}×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ or ${\mathbf{LWORK}}=-1$.
9:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7  Accuracy

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

The total number of real floating point operations is approximately $16mnk-8\left(m+n\right){k}^{2}+\frac{16}{3}{k}^{3}$; when $n=k$, the number is approximately $\frac{8}{3}{n}^{2}\left(3m-n\right)$.
The real analogue of this routine is F08AFF (DORGQR).

## 9  Example

This example forms the leading $4$ columns of the unitary matrix $Q$ from the $QR$ factorization of the matrix $A$, where
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i .$
The columns of $Q$ form an orthonormal basis for the space spanned by the columns of $A$.

### 9.1  Program Text

Program Text (f08atfe.f90)

### 9.2  Program Data

Program Data (f08atfe.d)

### 9.3  Program Results

Program Results (f08atfe.r)