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# NAG Toolbox: nag_lapack_dgeqrf (f08ae)

## Purpose

nag_lapack_dgeqrf (f08ae) computes the $QR$ factorization of a real $m$ by $n$ matrix.

## Syntax

[a, tau, info] = f08ae(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_dgeqrf(a, 'm', m, 'n', n)

## Description

nag_lapack_dgeqrf (f08ae) forms the $QR$ factorization of an arbitrary rectangular real $m$ by $n$ matrix. No pivoting is performed.
If $m\ge n$, the factorization is given by:
 $A = Q R 0 ,$
where $R$ is an $n$ by $n$ upper triangular matrix and $Q$ is an $m$ by $m$ orthogonal matrix. It is sometimes more convenient to write the factorization as
 $A = Q1 Q2 R 0 ,$
which reduces to
 $A = Q1R ,$
where ${Q}_{1}$ consists of the first $n$ columns of $Q$, and ${Q}_{2}$ the remaining $m-n$ columns.
If $m, $R$ is trapezoidal, and the factorization can be written
 $A = Q R1 R2 ,$
where ${R}_{1}$ is upper triangular and ${R}_{2}$ is rectangular.
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 $k$ columns of the array a represents a $QR$ factorization of the first $k$ columns of the original matrix $A$.

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 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 $m\ge n$, the elements below the diagonal store details of the orthogonal matrix $Q$ and the upper triangle stores the corresponding elements of the $n$ by $n$ upper triangular matrix $R$.
If $m, the strictly lower triangular part stores details of the orthogonal matrix $Q$ and the remaining elements store the corresponding elements of the $m$ by $n$ upper trapezoidal matrix $R$.
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 $\frac{2}{3}{n}^{2}\left(3m-n\right)$ if $m\ge n$ or $\frac{2}{3}{m}^{2}\left(3n-m\right)$ if $m.
To form the orthogonal matrix $Q$ nag_lapack_dgeqrf (f08ae) may be followed by a call to nag_lapack_dorgqr (f08af):
```[a, info] = f08af(a, tau, 'k', min(m,n));
```
but note that the second dimension of the array a must be at least m, which may be larger than was required by nag_lapack_dgeqrf (f08ae).
When $m\ge n$, it is often only the first $n$ columns of $Q$ that are required, and they may be formed by the call:
```[a, info] = f08af(a, tau);
```
To apply $Q$ to an arbitrary real rectangular matrix $C$, nag_lapack_dgeqrf (f08ae) may be followed by a call to nag_lapack_dormqr (f08ag). For example,
```[c, info] = f08ag('Left', 'Transpose', a, tau, c, 'k', min(m,n));
```
forms $C={Q}^{\mathrm{T}}C$, where $C$ is $m$ by $p$.
To compute a $QR$ factorization with column pivoting, use nag_lapack_dtpqrt (f08bb) or nag_lapack_dgeqpf (f08be).
The complex analogue of this function is nag_lapack_zgeqrf (f08as).

## Example

This example solves the linear least squares problems
 $minimize⁡ Axi - bi 2 , i=1,2$
where ${b}_{1}$ and ${b}_{2}$ are the columns of the matrix $B$,
 $A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 and B= -3.15 2.19 -0.11 -3.64 1.99 0.57 -2.70 8.23 0.26 -6.35 4.50 -1.48 .$
```function f08ae_example

fprintf('f08ae example results\n\n');

a = [-0.57, -1.28, -0.39,  0.25;
-1.93,  1.08, -0.31, -2.14;
2.30,  0.24,  0.40, -0.35;
-1.93,  0.64, -0.66,  0.08;
0.15,  0.30,  0.15, -2.13;
-0.02,  1.03, -1.43,  0.50];
b = [-2.67,  0.41;
-0.55, -3.10;
3.34, -4.01;
-0.77,  2.76;
0.48, -6.17;
4.10,  0.21];
% Compute the QR Factorisation of A
[a, tau, info] = f08ae(a);

% Compute C = (C1) = (Q^T)*B, storing the result in B (C2)
[b, info] = f08ag(...
'Left', 'Transpose', a, tau, b);

% Compute least-squares solutions by backsubstitution in R*X = C1
[b, info] = f07te(...
'Upper', 'No Transpose', 'Non-Unit', a, b, 'n', int64(4));
if (info >0)
fprintf('The upper triangular factor, R, of A is singular,\n');
fprintf('the least squares solution could not be computed.\n');
else
% Print least-squares solutions
[ifail] = x04ca('General', ' ', b(1:4,:), 'Least-squares solution(s)');
% Compute and print estimates of the square roots of the residual
% sums of squares
rnorm = zeros(2,1);
for j=1:2
rnorm(j) = norm(b(5:6,j));
end
fprintf('\nSquare root(s) of the residual sum(s) of squares\n');
fprintf('\t%11.2e    %11.2e\n', rnorm(1), rnorm(2));
end

```
```f08ae example results

Least-squares solution(s)
1          2
1      1.5339    -1.5753
2      1.8707     0.5559
3     -1.5241     1.3119
4      0.0392     2.9585

Square root(s) of the residual sum(s) of squares
2.22e-02       1.38e-02
```

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