NAG FL Interface
f08atf (zungqr)
1
Purpose
f08atf generates all or part of the complex unitary matrix
$Q$ from a
$QR$ factorization computed by
f08asf,
f08bsf or
f08btf.
2
Specification
Fortran Interface
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)) 

C Header Interface
#include <nag.h>
void 
f08atf_ (const Integer *m, const Integer *n, const Integer *k, Complex a[], const Integer *lda, const Complex tau[], Complex work[], const Integer *lwork, Integer *info) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f08atf_ (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 the names f08atf, nagf_lapackeig_zungqr or its LAPACK name zungqr.
3
Description
f08atf is intended to be used after a call to
f08asf,
f08bsf or
f08btf, 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 followed by
f08atf 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
$k<p$. 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
Arguments

1:
$\mathbf{m}$ – Integer
Input

On entry: $m$, the order of the unitary matrix $Q$.
Constraint:
${\mathbf{m}}\ge 0$.

2:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of columns of the matrix $Q$.
Constraint:
${\mathbf{m}}\ge {\mathbf{n}}\ge 0$.

3:
$\mathbf{k}$ – Integer
Input

On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraint:
${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.

4:
$\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (Kind=nag_wp) array
Input/Output

Note: the second dimension of the array
a
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: details of the vectors which define the elementary reflectors, as returned by
f08asf,
f08bsf or
f08btf.
On exit: the $m$ by $n$ matrix $Q$.

5:
$\mathbf{lda}$ – Integer
Input

On entry: the first dimension of the array
a as declared in the (sub)program from which
f08atf 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) array
Input

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,
f08bsf or
f08btf.

7:
$\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Complex (Kind=nag_wp) array
Workspace

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:
$\mathbf{lwork}$ – Integer
Input

On entry: the dimension of the array
work as declared in the (sub)program from which
f08atf 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, ${\mathbf{lwork}}\ge {\mathbf{n}}\times \mathit{nb}$, 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:
$\mathbf{info}$ – Integer
Output
On exit:
${\mathbf{info}}=0$ unless the routine detects an error (see
Section 6).
6
Error 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.
7
Accuracy
The computed matrix
$Q$ differs from an exactly unitary matrix by a matrix
$E$ such that
where
$\epsilon $ is the
machine precision.
8
Parallelism and Performance
f08atf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08atf 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 implementationspecific information.
The total number of real floatingpoint operations is approximately $16mnk8\left(m+n\right){k}^{2}+\frac{16}{3}{k}^{3}$; when $n=k$, the number is approximately $\frac{8}{3}{n}^{2}\left(3mn\right)$.
The real analogue of this routine is
f08aff.
10
Example
This example forms the leading
$4$ columns of the unitary matrix
$Q$ from the
$QR$ factorization of the matrix
$A$, where
The columns of
$Q$ form an orthonormal basis for the space spanned by the columns of
$A$.
10.1
Program Text
10.2
Program Data
10.3
Program Results