NAG FL Interface
f08ahf (dgelqf)
1
Purpose
f08ahf computes the $LQ$ 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 
f08ahf_ (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 
f08ahf_ (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 f08ahf, nagf_lapackeig_dgelqf or its LAPACK name dgelqf.
3
Description
f08ahf forms the $LQ$ factorization of an arbitrary rectangular real $m$ by $n$ matrix. No pivoting is performed.
If
$m\le n$, the factorization is given by:
where
$L$ is an
$m$ by
$m$ lower triangular matrix and
$Q$ is an
$n$ by
$n$ orthogonal matrix. It is sometimes more convenient to write the factorization as
which reduces to
where
${Q}_{1}$ consists of the first
$m$ rows of
$Q$, and
${Q}_{2}$ the remaining
$nm$ rows.
If
$m>n$,
$L$ is trapezoidal, and the factorization can be written
where
${L}_{1}$ is lower triangular and
${L}_{2}$ is rectangular.
The
$LQ$ factorization of
$A$ is essentially the same as the
$QR$ factorization of
${A}^{\mathrm{T}}$, since
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
$k<m$, the information returned in the first
$k$ rows of the array
a represents an
$LQ$ factorization of the first
$k$ rows of the original matrix
$A$.
4
References
None.
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
$m\le n$, the elements above the diagonal are overwritten by details of the orthogonal matrix
$Q$ and the lower triangle is overwritten by the corresponding elements of the
$m$ by
$m$ lower triangular matrix
$L$.
If $m>n$, the strictly upper 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 $n$ lower trapezoidal matrix $L$.

4:
$\mathbf{lda}$ – Integer
Input

On entry: the first dimension of the array
a as declared in the (sub)program from which
f08ahf 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
f08ahf 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 ${\mathbf{lwork}}=1$.

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
f08ahf 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 $\frac{2}{3}{m}^{2}\left(3nm\right)$ if $m\le n$ or $\frac{2}{3}{n}^{2}\left(3mn\right)$ if $m>n$.
To form the orthogonal matrix
$Q$ f08ahf may be followed by a call to
f08ajf:
Call dorglq(n,n,min(m,n),a,lda,tau,work,lwork,info)
but note that the first dimension of the array
a, specified by the argument
lda, must be at least
n, which may be larger than was required by
f08ahf.
When
$m\le n$, it is often only the first
$m$ rows of
$Q$ that are required, and they may be formed by the call:
Call dorglq(m,n,m,a,lda,tau,work,lwork,info)
To apply
$Q$ to an arbitrary real rectangular matrix
$C$,
f08ahf may be followed by a call to
f08akf. For example,
Call dormlq('Left','Transpose',m,p,min(m,n),a,lda,tau,c,ldc, &
work,lwork,info)
forms the matrix product $C={Q}^{\mathrm{T}}C$, where $C$ is $m$ by $p$.
The complex analogue of this routine is
f08avf.
10
Example
This example finds the minimum norm solutions of the underdetermined systems of linear equations
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