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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zungql (f08ct)

## Purpose

nag_lapack_zungql (f08ct) generates all or part of the complex $m$ by $m$ unitary matrix $Q$ from a $QL$ factorization computed by nag_lapack_zgeqlf (f08cs).

## Syntax

[a, info] = f08ct(a, tau, 'm', m, 'n', n, 'k', k)
[a, info] = nag_lapack_zungql(a, tau, 'm', m, 'n', n, 'k', k)

## Description

nag_lapack_zungql (f08ct) is intended to be used after a call to nag_lapack_zgeqlf (f08cs), which performs a $QL$ 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 trailing columns.
Usually $Q$ is determined from the $QL$ factorization of an $m$ by $p$ matrix $A$ with $m\ge p$. The whole of $Q$ may be computed by:
```[a, info] = f08ct(a, tau, 'k', p);
```
(note that the array a must have at least $m$ columns) or its trailing $p$ columns by:
```[a, info] = f08ct(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_zgeqlf (f08cs) followed by nag_lapack_zungql (f08ct) can be used to orthogonalize the columns of $A$.
The information returned by nag_lapack_zgeqlf (f08cs) also yields the $QL$ factorization of the trailing $k$ columns of $A$, where $k. The unitary matrix arising from this factorization can be computed by:
```[a, info] = f08ct(a, tau);
```
or its trailing $k$ columns by:
```[a, info] = f08ct(a(:,1:k), tau);
```

## 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_zgeqlf (f08cs).
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_zgeqlf (f08cs).

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

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_dorgql (f08cf).

## Example

This example generates the first four columns of the matrix $Q$ of the $QL$ factorization of $A$ as returned by nag_lapack_zgeqlf (f08cs), where
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i .$
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 f08ct_example

fprintf('f08ct example results\n\n');

% Form Q from QL factorization of A
m = 6;
n = 4;
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];

% Compute the QL factorization of A
[ql, tau, info] = f08cs(a);

% Form Q
[Q, info] = f08ct(ql, tau);

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

```
```f08ct example results

Unitary Q from QL of A
1                 2                 3                 4
1  ( 0.2810, 0.5020) (-0.2051,-0.1092) ( 0.3083,-0.6874) ( 0.0181,-0.1483)
2  ( 0.2707,-0.3296) ( 0.5711, 0.0432) ( 0.2251,-0.1313) ( 0.2930,-0.2025)
3  (-0.2864,-0.0094) (-0.5416, 0.0454) (-0.2062, 0.0691) ( 0.4015,-0.2170)
4  ( 0.2262,-0.3854) (-0.3387, 0.2228) ( 0.3259, 0.1178) (-0.0796, 0.0723)
5  ( 0.0341,-0.0760) ( 0.0098,-0.0712) ( 0.0753, 0.1412) (-0.5317,-0.5751)
6  (-0.3936,-0.2083) (-0.1296, 0.3691) ( 0.0264,-0.4134) (-0.0940,-0.0940)
```