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_dorgrq (f08cj)

## Purpose

nag_lapack_dorgrq (f08cj) generates all or part of the real $n$ by $n$ orthogonal matrix $Q$ from an $RQ$ factorization computed by nag_lapack_dgerqf (f08ch).

## Syntax

[a, info] = f08cj(a, tau, 'm', m, 'n', n, 'k', k)
[a, info] = nag_lapack_dorgrq(a, tau, 'm', m, 'n', n, 'k', k)

## Description

nag_lapack_dorgrq (f08cj) is intended to be used following a call to nag_lapack_dgerqf (f08ch), which performs an $RQ$ factorization of a real matrix $A$ and represents the orthogonal matrix $Q$ as a product of $k$ elementary reflectors of order $n$.
This function may be used to generate $Q$ explicitly as a square matrix, or to form only its trailing rows.
Usually $Q$ is determined from the $RQ$ factorization of a $p$ by $n$ matrix $A$ with $p\le n$. The whole of $Q$ may be computed by:
```[a, info] = f08cj(a, tau);
```
(note that the matrix $A$ must have at least $n$ rows), or its trailing $p$ rows as:
```[a, info] = f08cj(a(1:p,:), tau, 'k', p);
```
The rows of $Q$ returned by the last call form an orthonormal basis for the space spanned by the rows of $A$; thus nag_lapack_dgerqf (f08ch) followed by nag_lapack_dorgrq (f08cj) can be used to orthogonalize the rows of $A$.
The information returned by nag_lapack_dgerqf (f08ch) also yields the $RQ$ factorization of the trailing $k$ rows of $A$, where $k. The orthogonal matrix arising from this factorization can be computed by:
```[a, info] = f08cj(a, tau, 'k', k);
```
or its leading $k$ columns by:
```[a, info] = f08cj(a(1:k,:), tau, 'k', k);
```

## References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
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_dgerqf (f08ch).
2:     $\mathrm{tau}\left(:\right)$ – double array
The dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$
${\mathbf{tau}}\left(i\right)$ must contain the scalar factor of the elementary reflector ${H}_{i}$, as returned by nag_lapack_dgerqf (f08ch).

### 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 $Q$.
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: ${\mathbf{n}}\ge {\mathbf{m}}$.
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: ${\mathbf{m}}\ge {\mathbf{k}}\ge 0$.

### 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
 $E2 = O⁡ε$
and $\epsilon$ 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 $m=k$ this becomes $\frac{2}{3}{m}^{2}\left(3n-m\right)$.
The complex analogue of this function is nag_lapack_zungrq (f08cw).

## Example

This example generates the first four rows of the matrix $Q$ of the $RQ$ factorization of $A$ as returned by nag_lapack_dgerqf (f08ch), where
 $A = -0.57 -1.93 2.30 -1.93 0.15 -0.02 -1.28 1.08 0.24 0.64 0.30 1.03 -0.39 -0.31 0.40 -0.66 0.15 -1.43 0.25 -2.14 -0.35 0.08 -2.13 0.50 .$
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
```function f08cj_example

fprintf('f08cj example results\n\n');

a = [-0.57  -1.93   2.30  -1.93   0.15  -0.02;
-1.28   1.08   0.24   0.64   0.30   1.03;
-0.39  -0.31   0.40  -0.66   0.15  -1.43;
0.25  -2.14  -0.35   0.08  -2.13   0.50];

% Compute the RQ Factorisation of A
[rq, tau, info] = f08ch(a);

% Form Q
[Q, info] = f08cj(rq, tau);

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

```
```f08cj example results

Orthogonal factor Q
-0.0833    0.2972   -0.6404    0.4461   -0.2938   -0.4575
0.9100   -0.1080   -0.2351   -0.1620    0.2022   -0.1946
-0.2202   -0.2706    0.2220   -0.3866    0.0015   -0.8243
-0.0809    0.6922    0.1132   -0.0259    0.6890   -0.1617

```