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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgelqf (f08ah)

## Purpose

nag_lapack_dgelqf (f08ah) computes the $LQ$ factorization of a real $m$ by $n$ matrix.

## Syntax

[a, tau, info] = f08ah(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_dgelqf(a, 'm', m, 'n', n)

## Description

nag_lapack_dgelqf (f08ah) 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:
 $A = L 0 Q$
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
 $A = L 0 Q1 Q2$
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 ${A}^{\mathrm{T}}$, since
 $A = L 0 Q⇔AT= QT LT 0 .$
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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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 orthogonal 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 orthogonal matrix $Q$ and the remaining elements store the corresponding elements of the $m$ by $n$ lower trapezoidal matrix $L$.
2:     $\mathrm{tau}\left(:\right)$ – double 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 orthogonal 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 floating-point operations is approximately $\frac{2}{3}{m}^{2}\left(3n-m\right)$ if $m\le n$ or $\frac{2}{3}{n}^{2}\left(3m-n\right)$ if $m>n$.
To form the orthogonal matrix $Q$ nag_lapack_dgelqf (f08ah) may be followed by a call to nag_lapack_dorglq (f08aj):
```[a, info] = f08aj(a, tau, 'k', min(m,n));
```
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_dgelqf (f08ah).
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] = f08aj(a, tau, 'k', m);
```
To apply $Q$ to an arbitrary real rectangular matrix $C$, nag_lapack_dgelqf (f08ah) may be followed by a call to nag_lapack_dormlq (f08ak). For example,
```[c, info] = f08ak('Left', 'Transpose', a, tau, c, 'k', min(m, n));
```
forms the matrix product $C={Q}^{\mathrm{T}}C$, where $C$ is $m$ by $p$.
The complex analogue of this function is nag_lapack_zgelqf (f08av).

## 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 $B$,
 $A = -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50 and B= -2.87 -5.23 1.63 0.29 -3.52 4.76 0.45 -8.41 .$
```function f08ah_example

fprintf('f08ah example results\n\n');

a = [-5.42, 3.28, -3.68, 0.27, 2.06, 0.46;
-1.65, -3.4, -3.2, -1.03, -4.06, -0.01;
-0.37, 2.35, 1.9, 4.31, -1.76, 1.13;
-3.15, -0.11, 1.99, -2.7, 0.26, 4.5];
b = [-2.87, -5.23;
1.63,  0.29;
-3.52,  4.76;
0.45, -8.41;
0,     0;
0,     0];
% Compute the LQ factorization of a
[a, tau, info] = f08ah(a);

% solve l*y=b
l = tril(a(:, 1:4));
b(1:4,:) = inv(l)*b(1:4,:);

% Compute minimum-norm solution x = (q^t)*b
[x, info] = f08ak( ...
'Left', 'Transpose', a, tau, b);

mtitle = 'Minimum-norm solution(s)';
[ifail] = x04ca( ...
'General', ' ', x, mtitle);

```
```f08ah example results

Minimum-norm solution(s)
1          2
1      0.2371     0.7383
2     -0.4575     0.0158
3     -0.0085    -0.0161
4     -0.5192     1.0768
5      0.0239    -0.6436
6     -0.0543    -0.6613
```