NAG FL Interface
f08aff (dorgqr)
1
Purpose
f08aff generates all or part of the real orthogonal matrix
$Q$ from a
$QR$ factorization computed by
f08aef,
f08bef or
f08bff.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
m, n, k, lda, lwork 
Integer, Intent (Out) 
:: 
info 
Real (Kind=nag_wp), Intent (In) 
:: 
tau(*) 
Real (Kind=nag_wp), Intent (Inout) 
:: 
a(lda,*) 
Real (Kind=nag_wp), Intent (Out) 
:: 
work(max(1,lwork)) 

C Header Interface
#include <nag.h>
void 
f08aff_ (const Integer *m, const Integer *n, const Integer *k, double a[], const Integer *lda, const double tau[], double work[], const Integer *lwork, Integer *info) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f08aff_ (const Integer &m, const Integer &n, const Integer &k, double a[], const Integer &lda, const double tau[], double work[], const Integer &lwork, Integer &info) 
}

The routine may be called by the names f08aff, nagf_lapackeig_dorgqr or its LAPACK name dorgqr.
3
Description
f08aff is intended to be used after a call to
f08aef,
f08bef or
f08bff.
which perform a
$QR$ factorization of a real matrix
$A$. The orthogonal 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 dorgqr(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 dorgqr(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
f08aef followed by
f08aff 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 orthogonal matrix arising from this factorization can be computed by:
Call dorgqr(m,m,k,a,lda,tau,work,lwork,info)
or its leading
$k$ columns by:
Call dorgqr(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 orthogonal 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:
$\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (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
f08aef,
f08bef or
f08bff.
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
f08aff is called.
Constraint:
${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.

6:
$\mathbf{tau}\left(*\right)$ – Real (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
f08aef,
f08bef or
f08bff.

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

On exit: if
${\mathbf{info}}={\mathbf{0}}$,
${\mathbf{work}}\left(1\right)$ 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
f08aff 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 orthogonal matrix by a matrix
$E$ such that
where
$\epsilon $ is the
machine precision.
8
Parallelism and Performance
f08aff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08aff 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 floatingpoint operations is approximately $4mnk2\left(m+n\right){k}^{2}+\frac{4}{3}{k}^{3}$; when $n=k$, the number is approximately $\frac{2}{3}{n}^{2}\left(3mn\right)$.
The complex analogue of this routine is
f08atf.
10
Example
This example forms the leading
$4$ columns of the orthogonal 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