NAG FL Interface
f08awf (zunglq)
1
Purpose
f08awf generates all or part of the complex unitary matrix
$Q$ from an
$LQ$ factorization computed by
f08avf.
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 
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) 

C++ Header Interface
#include <nag.h> extern "C" {
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 the names f08awf, nagf_lapackeig_zunglq or its LAPACK name zunglq.
3
Description
f08awf is intended to be used after a call to
f08avf, 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 followed by
f08awf 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
$k<p$. 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)
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 number of rows of the matrix $Q$.
Constraint:
${\mathbf{m}}\ge 0$.

2:
$\mathbf{n}$ – Integer
Input

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

3:
$\mathbf{k}$ – Integer
Input

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) 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
f08avf.
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
f08awf 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
f08avf.

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
f08awf 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, ${\mathbf{lwork}}\ge {\mathbf{m}}\times \mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint:
${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ 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
f08awf 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 $m=k$, the number is approximately $\frac{8}{3}{m}^{2}\left(3nm\right)$.
The real analogue of this routine is
f08ajf.
10
Example
This example forms the leading
$4$ rows of the unitary matrix
$Q$ from the
$LQ$ factorization of the matrix
$A$, where
The rows of
$Q$ form an orthonormal basis for the space spanned by the rows of
$A$.
10.1
Program Text
10.2
Program Data
10.3
Program Results