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# NAG Toolbox: nag_lapack_dorgqr (f08af)

## Purpose

nag_lapack_dorgqr (f08af) generates all or part of the real orthogonal matrix $Q$ from a $QR$ factorization computed by nag_lapack_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf).

## Syntax

[a, info] = f08af(a, tau, 'm', m, 'n', n, 'k', k)
[a, info] = nag_lapack_dorgqr(a, tau, 'm', m, 'n', n, 'k', k)

## Description

nag_lapack_dorgqr (f08af) is intended to be used after a call to nag_lapack_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf). which perform a $QR$ factorization of a real matrix $A$. The orthogonal matrix $Q$ is represented as a product of elementary reflectors.
This function may be used to generate $Q$ explicitly as a square matrix, or to form only its leading columns.
Usually $Q$ is determined from the $QR$ factorization of an $m$ by $p$ matrix $A$ with $m\ge p$. The whole of $Q$ may be computed by:
```[a, info] = f08af(a, tau, 'k', p);
```
(note that the array a must have $m$ columns) or its leading $p$ columns by:
```[a, info] = f08af(a(:,1:p), tau, 'k', p);
```
The columns of $Q$ returned by the last call form an orthonormal basis for the space spanned by the columns of $A$; thus nag_lapack_dgeqrf (f08ae) followed by nag_lapack_dorgqr (f08af) can be used to orthogonalize the columns of $A$.
The information returned by the $QR$ factorization functions also yields the $QR$ factorization of the leading $k$ columns of $A$, where $k. The orthogonal matrix arising from this factorization can be computed by:
```[a, info] = f08af(a, tau, 'k', k);
```
or its leading $k$ columns by:
```[a, info] = f08af(a(:,1:p), tau, 'k', k);
```

## 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf).
2:     $\mathrm{tau}\left(:\right)$ – double array
The dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$
Further details of the elementary reflectors, as returned by nag_lapack_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf).

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the first dimension of the array a.
$m$, the order of the orthogonal matrix $Q$.
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 $Q$.
Constraint: ${\mathbf{m}}\ge {\mathbf{n}}\ge 0$.
3:     $\mathrm{k}$int64int32nag_int scalar
Default: the dimension of the array tau.
$k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraint: ${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.

### 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $n$ matrix $Q$.
2:     $\mathrm{info}$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: k, 4: a, 5: lda, 6: tau, 7: work, 8: lwork, 9: 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 matrix $Q$ differs from an exactly orthogonal matrix by a matrix $E$ such that
 $E2 = Oε ,$
where $\epsilon$ is the machine precision.

The total number of floating-point operations is approximately $4mnk-2\left(m+n\right){k}^{2}+\frac{4}{3}{k}^{3}$; when $n=k$, the number is approximately $\frac{2}{3}{n}^{2}\left(3m-n\right)$.
The complex analogue of this function is nag_lapack_zungqr (f08at).

## Example

This example forms the leading $4$ columns of the orthogonal matrix $Q$ from the $QR$ factorization of the matrix $A$, where
 $A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 .$
The columns of $Q$ form an orthonormal basis for the space spanned by the columns of $A$.
```function f08af_example

fprintf('f08af 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];

% Compute the QR Factorisation of A
[a, tau, info] = f08ae(a);

% Generate q
[q, info] = f08af(a, tau);

disp('Orthogonal factor Q');
disp(q);

```
```f08af example results

Orthogonal factor Q
-0.1576    0.6744   -0.4571    0.4489
-0.5335   -0.3861    0.2583    0.3898
0.6358   -0.2928    0.0165    0.1930
-0.5335   -0.1692   -0.0834   -0.2350
0.0415   -0.1593    0.1475    0.7436
-0.0055   -0.5064   -0.8339    0.0335

```

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