f08atf generates all or part of the complex unitary matrix
$Q$ from a
$QR$ factorization computed by
f08asf,
f08bsf or
f08btf.
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.
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)

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).
The computed matrix
$Q$ differs from an exactly unitary matrix by a matrix
$E$ such that
where
$\epsilon $ is the
machine precision.
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 real analogue of this routine is
f08aff.
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
$A$.