NAG FL Interface
f08aef (dgeqrf)
1
Purpose
f08aef computes the $QR$ factorization of a real $m$ by $n$ matrix.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
m, n, lda, lwork 
Integer, Intent (Out) 
:: 
info 
Real (Kind=nag_wp), Intent (Inout) 
:: 
a(lda,*), tau(*) 
Real (Kind=nag_wp), Intent (Out) 
:: 
work(max(1,lwork)) 

C Header Interface
#include <nag.h>
void 
f08aef_ (const Integer *m, const Integer *n, double a[], const Integer *lda, double tau[], double work[], const Integer *lwork, Integer *info) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f08aef_ (const Integer &m, const Integer &n, double a[], const Integer &lda, double tau[], double work[], const Integer &lwork, Integer &info) 
}

The routine may be called by the names f08aef, nagf_lapackeig_dgeqrf or its LAPACK name dgeqrf.
3
Description
f08aef forms the $QR$ factorization of an arbitrary rectangular real $m$ by $n$ matrix. No pivoting is performed.
If
$m\ge n$, the factorization is given by:
where
$R$ is an
$n$ by
$n$ upper triangular matrix and
$Q$ is an
$m$ by
$m$ orthogonal matrix. It is sometimes more convenient to write the factorization as
which reduces to
where
${Q}_{1}$ consists of the first
$n$ columns of
$Q$, and
${Q}_{2}$ the remaining
$mn$ columns.
If
$m<n$,
$R$ is trapezoidal, and the factorization can be written
where
${R}_{1}$ is upper triangular and
${R}_{2}$ is rectangular.
The matrix
$Q$ is not formed explicitly but is represented as a product of
$\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see the
F08 Chapter Introduction for details). Routines are provided to work with
$Q$ in this representation (see
Section 9).
Note also that for any
$k<n$, the information returned in the first
$k$ columns of the array
a represents a
$QR$ factorization of the first
$k$ columns of the original matrix
$A$.
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 $A$.
Constraint:
${\mathbf{m}}\ge 0$.

2:
$\mathbf{n}$ – Integer
Input

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

3:
$\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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: the $m$ by $n$ matrix $A$.
On exit: if
$m\ge n$, the elements below the diagonal are overwritten by details of the orthogonal matrix
$Q$ and the upper triangle is overwritten by the corresponding elements of the
$n$ by
$n$ upper triangular matrix
$R$.
If $m<n$, the strictly lower triangular part is overwritten by details of the orthogonal matrix $Q$ and the remaining elements are overwritten by the corresponding elements of the $m$ by $n$ upper trapezoidal matrix $R$.

4:
$\mathbf{lda}$ – Integer
Input

On entry: the first dimension of the array
a as declared in the (sub)program from which
f08aef is called.
Constraint:
${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.

5:
$\mathbf{tau}\left(*\right)$ – Real (Kind=nag_wp) array
Output

Note: the dimension of the array
tau
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: further details of the orthogonal matrix $Q$.

6:
$\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – 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.

7:
$\mathbf{lwork}$ – Integer
Input

On entry: the dimension of the array
work as declared in the (sub)program from which
f08aef 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 ${\mathbf{lwork}}=1$.

8:
$\mathbf{info}$ – 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 factorization is the exact factorization of a nearby matrix
$\left(A+E\right)$, where
and
$\epsilon $ is the
machine precision.
8
Parallelism and Performance
f08aef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08aef 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 $\frac{2}{3}{n}^{2}\left(3mn\right)$ if $m\ge n$ or $\frac{2}{3}{m}^{2}\left(3nm\right)$ if $m<n$.
To form the orthogonal matrix
$Q$ f08aef may be followed by a call to
f08aff
:
Call dorgqr(m,m,min(m,n),a,lda,tau,work,lwork,info)
but note that the second dimension of the array
a must be at least
m, which may be larger than was required by
f08aef.
When
$m\ge n$, it is often only the first
$n$ columns of
$Q$ that are required, and they may be formed by
the call:
Call dorgqr(m,n,n,a,lda,tau,work,lwork,info)
To apply
$Q$ to an arbitrary
$m$ by
$p$ real rectangular matrix
$C$,
f08aef may be followed by a call to
f08agf
. For example,
Call dormqr('Left','Transpose',m,p,min(m,n),a,lda,tau,c,ldc,work, &
lwork,info)
forms the matrix product
$C={Q}^{\mathrm{T}}C$.
To compute a
$QR$ factorization with column pivoting, use
f08bbf or
f08bef.
The complex analogue of this routine is
f08asf.
10
Example
This example solves the linear least squares problems
where
${b}_{1}$ and
${b}_{2}$ are the columns of the matrix
$B$,
10.1
Program Text
10.2
Program Data
10.3
Program Results