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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgelqf (f08ah)

## Purpose

nag_lapack_dgelqf (f08ah) computes the LQ$LQ$ factorization of a real m$m$ by n$n$ 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$LQ$ factorization of an arbitrary rectangular real m$m$ by n$n$ matrix. No pivoting is performed.
If mn$m\le n$, the factorization is given by:
A =
 ( L 0 )
Q
$A = L 0 Q$
where L$L$ is an m$m$ by m$m$ lower triangular matrix and Q$Q$ is an n$n$ by n$n$ orthogonal matrix. It is sometimes more convenient to write the factorization as
A =
 ( L 0 )
 ( Q1 ) Q2
$A = L 0 Q1 Q2$
which reduces to
 A = LQ1 , $A = LQ1 ,$
where Q1${Q}_{1}$ consists of the first m$m$ rows of Q$Q$, and Q2${Q}_{2}$ the remaining nm$n-m$ rows.
If m > n$m>n$, L$L$ is trapezoidal, and the factorization can be written
A =
 ( L1 ) L2
Q
$A = L1 L2 Q$
where L1${L}_{1}$ is lower triangular and L2${L}_{2}$ is rectangular.
The LQ$LQ$ factorization of A$A$ is essentially the same as the QR$QR$ factorization of AT${A}^{\mathrm{T}}$, since
A =
 ( L 0 )
QAT = QT
 ( LT ) 0
.
$A = L 0 Q⇔AT= QT LT 0 .$
The matrix Q$Q$ is not formed explicitly but is represented as a product of min (m,n)$\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with Q$Q$ in this representation (see Section [Further Comments]).
Note also that for any k < m$k, the information returned in the first k$k$ rows of the array a represents an LQ$LQ$ factorization of the first k$k$ rows of the original matrix A$A$.

None.

## Parameters

### Compulsory Input Parameters

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

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
m$m$, the number of rows of the matrix A$A$.
Constraint: m0${\mathbf{m}}\ge 0$.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array a.
n$n$, the number of columns of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda work lwork

### Output Parameters

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

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ 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 (A + E)$\left(A+E\right)$, where
 ‖E‖2 = O(ε) ‖A‖2 , $‖E‖2 = O(ε) ‖A‖2 ,$
and ε$\epsilon$ is the machine precision.

The total number of floating point operations is approximately (2/3) m2 (3nm) $\frac{2}{3}{m}^{2}\left(3n-m\right)$ if mn$m\le n$ or (2/3) n2 (3mn) $\frac{2}{3}{n}^{2}\left(3m-n\right)$ if m > n$m>n$.
To form the orthogonal matrix Q$Q$ 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 parameter lda, must be at least n, which may be larger than was required by nag_lapack_dgelqf (f08ah).
When mn$m\le n$, it is often only the first m$m$ rows of Q$Q$ that are required, and they may be formed by the call:
```[a, info] = f08aj(a, tau, 'k', m);
```
To apply Q$Q$ to an arbitrary real rectangular matrix C$C$, 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 = QTC$C={Q}^{\mathrm{T}}C$, where C$C$ is m$m$ by p$p$.
The complex analogue of this function is nag_lapack_zgelqf (f08av).

## Example

```function nag_lapack_dgelqf_example
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] = nag_lapack_dgelqf(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 in b
[b, info] = nag_lapack_dormlq('Left', 'Transpose', a, tau, b)
```
```

b =

0.2371    0.7383
-0.4575    0.0158
-0.0085   -0.0161
-0.5192    1.0768
0.0239   -0.6436
-0.0543   -0.6613

info =

0

```
```function f08ah_example
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 in b
[b, info] = f08ak('Left', 'Transpose', a, tau, b)
```
```

b =

0.2371    0.7383
-0.4575    0.0158
-0.0085   -0.0161
-0.5192    1.0768
0.0239   -0.6436
-0.0543   -0.6613

info =

0

```