Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgelqf (f08av)

## Purpose

nag_lapack_zgelqf (f08av) computes the $LQ$ factorization of a complex $m$ by $n$ matrix.

## Syntax

[a, tau, info] = f08av(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_zgelqf(a, 'm', m, 'n', n)

## Description

nag_lapack_zgelqf (f08av) 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:
 $A = L 0 Q$
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
 $A = L 0 Q1 Q2$
which reduces to
 $A = LQ1 ,$
where ${Q}_{1}$ consists of the first $m$ rows of $Q$, and ${Q}_{2}$ the remaining $n-m$ rows.
If $m>n$, $L$ is trapezoidal, and the factorization can be written
 $A = L1 L2 Q$
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
 $A = L 0 Q⇔AH= QH LH 0 .$
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). Functions are provided to work with $Q$ in this representation (see Further Comments).
Note also that for any $k, 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 $A$.

None.

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $n$ matrix $A$.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the first dimension of the array a.
$m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array a.
$n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If $m\le n$, the elements above the diagonal store details of the unitary matrix $Q$ and the lower triangle stores the corresponding elements of the $m$ by $m$ lower triangular matrix $L$.
If $m>n$, the strictly upper triangular part stores details of the unitary matrix $Q$ and the remaining elements store the corresponding elements of the $m$ by $n$ lower trapezoidal matrix $L$.
The diagonal elements of $L$ are real.
2:     $\mathrm{tau}\left(:\right)$ – complex array
The dimension of the array tau will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$
Further details of the unitary matrix $Q$.
3:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

${\mathbf{info}}=-i$
If ${\mathbf{info}}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: m, 2: n, 3: a, 4: lda, 5: tau, 6: work, 7: lwork, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

## Accuracy

The computed factorization is the exact factorization of a nearby matrix $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision.

The total number of real floating-point operations is approximately $\frac{8}{3}{m}^{2}\left(3n-m\right)$ if $m\le n$ or $\frac{8}{3}{n}^{2}\left(3m-n\right)$ if $m>n$.
To form the unitary matrix $Q$ nag_lapack_zgelqf (f08av) may be followed by a call to nag_lapack_zunglq (f08aw):
```[a, info] = f08aw(a(1:n,:), tau);
```
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 nag_lapack_zgelqf (f08av).
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:
```[a, info] = f08aw(a, tau, 'k', m);
```
To apply $Q$ to an arbitrary complex rectangular matrix $C$, nag_lapack_zgelqf (f08av) may be followed by a call to nag_lapack_zunmlq (f08ax). For example,
```[c, info] = f08ax('Left', 'Conjugate Transpose', a(:,1:p), tau, c);
```
forms the matrix product $C={Q}^{\mathrm{H}}C$, where $C$ is $m$ by $p$.
The real analogue of this function is nag_lapack_dgelqf (f08ah).

## Example

This example finds the minimum norm solutions of the under-determined systems of linear equations
 $Ax1= b1 and Ax2= b2$
where ${b}_{1}$ and ${b}_{2}$ are the columns of the matrix $B$,
 $A = 0.28-0.36i 0.50-0.86i -0.77-0.48i 1.58+0.66i -0.50-1.10i -1.21+0.76i -0.32-0.24i -0.27-1.15i 0.36-0.51i -0.07+1.33i -0.75+0.47i -0.08+1.01i$
and
 $B = -1.35+0.19i 4.83-2.67i 9.41-3.56i -7.28+3.34i -7.57+6.93i 0.62+4.53i .$
```function f08av_example

fprintf('f08av example results\n\n');

a = [ 0.28 - 0.36i,  0.50 - 0.86i, -0.77 - 0.48i,  1.58 + 0.66i;
-0.50 - 1.10i, -1.21 + 0.76i, -0.32 - 0.24i, -0.27 - 1.15i;
0.36 - 0.51i, -0.07 + 1.33i, -0.75 + 0.47i, -0.08 + 1.01i];
b = [-1.35 + 0.19i,  4.83 - 2.67i;
9.41 - 3.56i, -7.28 + 3.34i;
-7.57 + 6.93i,  0.62 + 4.53i;
0,             0];
[m,n] = size(a);

% Compute the LQ factorization of a
[lq, tau, info] = f08av(a);

% Solve l*y = b
l = tril(lq(:, 1:m));
y = [inv(l)*b(1:m,:); b(m+1:n,:)];

% Compute minimum-norm solution x = (q^h)*y
[x, info] = f08ax( ...
'Left', 'Conjugate Transpose', lq, tau, y);

disp('Minimum-norm solution(s)');
disp(x);

```
```f08av example results

Minimum-norm solution(s)
-2.8501 + 6.4683i  -1.1682 - 1.8886i
1.6264 - 0.7799i   2.8377 + 0.7654i
6.9290 + 4.6481i  -1.7610 - 0.7041i
1.4048 + 3.2400i   1.0518 - 1.6365i

```