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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zungrq (f08cw)

## Purpose

nag_lapack_zungrq (f08cw) generates all or part of the complex $n$ by $n$ unitary matrix $Q$ from an $RQ$ factorization computed by nag_lapack_zgerqf (f08cv).

## Syntax

[a, info] = f08cw(a, tau, 'm', m, 'n', n, 'k', k)
[a, info] = nag_lapack_zungrq(a, tau, 'm', m, 'n', n, 'k', k)

## Description

nag_lapack_zungrq (f08cw) is intended to be used following a call to nag_lapack_zgerqf (f08cv), which performs an $RQ$ factorization of a complex matrix $A$ and represents the unitary 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] = f08cw(a, tau);
```
(note that the matrix $A$ must have at least $n$ rows), or its trailing $p$ rows as:
```[a, info] = f08cw(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_zgerqf (f08cv) followed by nag_lapack_zungrq (f08cw) can be used to orthogonalize the rows of $A$.
The information returned by nag_lapack_zgerqf (f08cv) also yields the $RQ$ factorization of the trailing $k$ rows of $A$, where $k. The unitary matrix arising from this factorization can be computed by:
```[a, info] = f08cw(a, tau, 'k', k);
```
or its leading $k$ columns by:
```[a, info] = f08cw(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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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_zgerqf (f08cv).
2:     $\mathrm{tau}\left(:\right)$ – complex 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_zgerqf (f08cv).

### 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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 unitary 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 $16mnk-8\left(m+n\right){k}^{2}+\frac{16}{3}{k}^{3}$; when $m=k$ this becomes $\frac{8}{3}{m}^{2}\left(3n-m\right)$.
The real analogue of this function is nag_lapack_dorgrq (f08cj).

## Example

This example generates the first four rows of the matrix $Q$ of the $RQ$ factorization of $A$ as returned by nag_lapack_zgerqf (f08cv), where
 $A = 0.96-0.81i -0.98+1.98i 0.62-0.46i -0.37+0.38i 0.83+0.51i 1.08-0.28i -0.03+0.96i -1.20+0.19i 1.01+0.02i 0.19-0.54i 0.20+0.01i 0.20-0.12i -0.91+2.06i -0.66+0.42i 0.63-0.17i -0.98-0.36i -0.17-0.46i -0.07+1.23i -0.05+0.41i -0.81+0.56i -1.11+0.60i 0.22-0.20i 1.47+1.59i 0.26+0.26i .$
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 f08cw_example

fprintf('f08cw example results\n\n');

% Form Q from RQ factorization of A
a = [ 0.96 - 0.81i, -0.03 + 0.96i,  -0.91 + 2.06i, -0.05 + 0.41i;
-0.98 + 1.98i, -1.20 + 0.19i,  -0.66 + 0.42i, -0.81 + 0.56i;
0.62 - 0.46i,  1.01 + 0.02i,   0.63 - 0.17i, -1.11 + 0.60i;
-0.37 + 0.38i,  0.19 - 0.54i,  -0.98 - 0.36i,  0.22 - 0.20i;
0.83 + 0.51i,  0.20 + 0.01i,  -0.17 - 0.46i,  1.47 + 1.59i;
1.08 - 0.28i,  0.20 - 0.12i,  -0.07 + 1.23i,  0.26 + 0.26i];
a = transpose(a);
[m,n] = size(a);

% Compute the RQ factorization of A
[rq, tau, info] = f08cv(a);

% Form Q
[Q, info] = f08cw(rq, tau);

%Print Q
ncols  = int64(80);
indent = int64(0);
[ifail] = x04db( ...
'General', ' ', Q, 'Bracketed', 'F7.4', ...
'Unitary Q from RQ of A', 'Integer', 'Integer', ...
ncols, indent);

```
```f08cw example results

Unitary Q from RQ of A
1                 2                 3                 4
1  ( 0.2810, 0.5020) ( 0.2707,-0.3296) (-0.2864,-0.0094) ( 0.2262,-0.3854)
2  (-0.2051,-0.1092) ( 0.5711, 0.0432) (-0.5416, 0.0454) (-0.3387, 0.2228)
3  ( 0.3083,-0.6874) ( 0.2251,-0.1313) (-0.2062, 0.0691) ( 0.3259, 0.1178)
4  ( 0.0181,-0.1483) ( 0.2930,-0.2025) ( 0.4015,-0.2170) (-0.0796, 0.0723)

5                 6
1  ( 0.0341,-0.0760) (-0.3936,-0.2083)
2  ( 0.0098,-0.0712) (-0.1296, 0.3691)
3  ( 0.0753, 0.1412) ( 0.0264,-0.4134)
4  (-0.5317,-0.5751) (-0.0940,-0.0940)
```