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# NAG Toolbox: nag_lapack_zungqr (f08at)

## Purpose

nag_lapack_zungqr (f08at) generates all or part of the complex unitary matrix $Q$ from a $QR$ factorization computed by nag_lapack_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt).

## Syntax

[a, info] = f08at(a, tau, 'm', m, 'n', n, 'k', k)
[a, info] = nag_lapack_zungqr(a, tau, 'm', m, 'n', n, 'k', k)

## Description

nag_lapack_zungqr (f08at) is intended to be used after a call to nag_lapack_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt), which perform a $QR$ factorization of a complex matrix $A$. The unitary 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] = f08at(a, tau);
```
(note that the array a must have $m$ columns) or its leading $p$ columns by:
```[a, info] = f08at(a(1:p,:), tau);
```
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_zgeqrf (f08as) followed by nag_lapack_zungqr (f08at) 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 unitary matrix arising from this factorization can be computed by:
```[a, info] = f08at(a, tau);
```
or its leading $k$ columns by:
```[a, info] = f08at(a(:,1:k), tau);
```

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt).
2:     $\mathrm{tau}\left(:\right)$ – complex array
The dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$
Further details of the elementary reflectors, as returned by nag_lapack_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt).

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the first dimension of the array a.
$m$, the order of the unitary 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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
 $E2 = Oε ,$
where $\epsilon$ is the machine precision.

## Further Comments

The total number of real floating-point operations is approximately $16mnk-8\left(m+n\right){k}^{2}+\frac{16}{3}{k}^{3}$; when $n=k$, the number is approximately $\frac{8}{3}{n}^{2}\left(3m-n\right)$.
The real analogue of this function is nag_lapack_dorgqr (f08af).

## Example

This example forms the leading $4$ columns of the unitary matrix $Q$ from the $QR$ factorization of the matrix $A$, where
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i .$
The columns of $Q$ form an orthonormal basis for the space spanned by the columns of $A$.
```function f08at_example

fprintf('f08at example results\n\n');

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];
[m,n] = size(a);

% Compute the QR factorization of A
[qr, tau, info] = f08as(a);

% Form Q
[Q, info] = f08at(qr, tau);

mtitle = sprintf('The leading %2d columns of Q\n',n);
disp(mtitle);
disp(Q(:,1:n));

```
```f08at example results

The leading  4 columns of Q

-0.3110 + 0.2624i  -0.3175 + 0.4835i   0.4966 - 0.2997i  -0.0072 - 0.3718i
0.3175 - 0.6414i  -0.2062 + 0.1577i  -0.0793 - 0.3094i  -0.0282 - 0.1491i
-0.2008 + 0.1490i   0.4892 - 0.0900i   0.0357 - 0.0219i   0.5625 - 0.0710i
0.1199 - 0.1231i   0.2566 - 0.3055i   0.4489 - 0.2141i  -0.1651 + 0.1800i
-0.2689 - 0.1652i   0.1697 - 0.2491i  -0.0496 + 0.1158i  -0.4885 - 0.4540i
-0.3499 + 0.0907i  -0.0491 - 0.3133i  -0.1256 - 0.5300i   0.1039 + 0.0450i

```

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