NAG FL Interface
f08avf (zgelqf)
1
Purpose
f08avf computes the $LQ$ factorization of a complex $m$ by $n$ matrix.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
m, n, lda, lwork 
Integer, Intent (Out) 
:: 
info 
Complex (Kind=nag_wp), Intent (Inout) 
:: 
a(lda,*), tau(*) 
Complex (Kind=nag_wp), Intent (Out) 
:: 
work(max(1,lwork)) 

C Header Interface
#include <nag.h>
void 
f08avf_ (const Integer *m, const Integer *n, Complex a[], const Integer *lda, Complex tau[], Complex work[], const Integer *lwork, Integer *info) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f08avf_ (const Integer &m, const Integer &n, Complex a[], const Integer &lda, Complex tau[], Complex work[], const Integer &lwork, Integer &info) 
}

The routine may be called by the names f08avf, nagf_lapackeig_zgelqf or its LAPACK name zgelqf.
3
Description
f08avf forms the $LQ$ factorization of an arbitrary rectangular complex $m$ 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 (with real diagonal elements) and
$Q$ is an
$n$ by
$n$ unitary 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
${A}^{\mathrm{H}}$, 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:
$\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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: the $m$ by $n$ matrix $A$.
On exit: if
$m\le n$, the elements above the diagonal are overwritten by details of the unitary 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 unitary matrix $Q$ and the remaining elements are overwritten by the corresponding elements of the $m$ by $n$ lower trapezoidal matrix $L$.
The diagonal elements of $L$ are real.

4:
$\mathbf{lda}$ – Integer
Input

On entry: the first dimension of the array
a as declared in the (sub)program from which
f08avf is called.
Constraint:
${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.

5:
$\mathbf{tau}\left(*\right)$ – Complex (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 unitary matrix $Q$.

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

7:
$\mathbf{lwork}$ – Integer
Input

On entry: the dimension of the array
work as declared in the (sub)program from which
f08avf 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
f08avf 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 $\frac{8}{3}{m}^{2}\left(3nm\right)$ if $m\le n$ or $\frac{8}{3}{n}^{2}\left(3mn\right)$ if $m>n$.
To form the unitary matrix
$Q$ f08avf may be followed by a call to
f08awf:
Call zunglq(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
f08avf.
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 zunglq(m,n,m,a,lda,tau,work,lwork,info)
To apply
$Q$ to an arbitrary complex rectangular matrix
$C$,
f08avf may be followed by a call to
f08axf. For example,
Call zunmlq('Left','Conjugate Transpose',m,p,min(m,n),a,lda,tau, &
c,ldc,work,lwork,info)
forms the matrix product $C={Q}^{\mathrm{H}}C$, where $C$ is $m$ by $p$.
The real analogue of this routine is
f08ahf.
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
$B$,
and
10.1
Program Text
10.2
Program Data
10.3
Program Results