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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zunmlq (f08ax)

## Purpose

nag_lapack_zunmlq (f08ax) multiplies an arbitrary complex matrix $C$ by the complex unitary matrix $Q$ from an $LQ$ factorization computed by nag_lapack_zgelqf (f08av).

## Syntax

[c, info] = f08ax(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)
[c, info] = nag_lapack_zunmlq(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)

## Description

nag_lapack_zunmlq (f08ax) is intended to be used after a call to nag_lapack_zgelqf (f08av), which performs an $LQ$ factorization of a complex matrix $A$. The unitary matrix $Q$ is represented as a product of elementary reflectors.
This function may be used to form one of the matrix products
 $QC , QHC , CQ ​ or ​ CQH ,$
overwriting the result on $C$ (which may be any complex rectangular matrix).

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{side}$ – string (length ≥ 1)
Indicates how $Q$ or ${Q}^{\mathrm{H}}$ is to be applied to $C$.
${\mathbf{side}}=\text{'L'}$
$Q$ or ${Q}^{\mathrm{H}}$ is applied to $C$ from the left.
${\mathbf{side}}=\text{'R'}$
$Q$ or ${Q}^{\mathrm{H}}$ is applied to $C$ from the right.
Constraint: ${\mathbf{side}}=\text{'L'}$ or $\text{'R'}$.
2:     $\mathrm{trans}$ – string (length ≥ 1)
Indicates whether $Q$ or ${Q}^{\mathrm{H}}$ is to be applied to $C$.
${\mathbf{trans}}=\text{'N'}$
$Q$ is applied to $C$.
${\mathbf{trans}}=\text{'C'}$
${Q}^{\mathrm{H}}$ is applied to $C$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'C'}$.
3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{side}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{side}}=\text{'R'}$.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_zgelqf (f08av).
4:     $\mathrm{tau}\left(:\right)$ – 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_zgelqf (f08av).
5:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – complex array
The first dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $n$ matrix $C$.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the first dimension of the array c.
$m$, the number of rows of the matrix $C$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array c.
$n$, the number of columns of the matrix $C$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathrm{k}$int64int32nag_int scalar
Default: the first dimension of the array a and the dimension of the array tau.
$k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraints:
• if ${\mathbf{side}}=\text{'L'}$, ${\mathbf{m}}\ge {\mathbf{k}}\ge 0$;
• if ${\mathbf{side}}=\text{'R'}$, ${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.

### Output Parameters

1:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – complex array
The first dimension of the array c will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array c will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
c stores $QC$ or ${Q}^{\mathrm{H}}C$ or $CQ$ or $C{Q}^{\mathrm{H}}$ as specified by side and trans.
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: side, 2: trans, 3: m, 4: n, 5: k, 6: a, 7: lda, 8: tau, 9: c, 10: ldc, 11: work, 12: lwork, 13: 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 result differs from the exact result by a matrix $E$ such that
 $E2 = Oε C2 ,$
where $\epsilon$ is the machine precision.

The total number of real floating-point operations is approximately $8nk\left(2m-k\right)$ if ${\mathbf{side}}=\text{'L'}$ and $8mk\left(2n-k\right)$ if ${\mathbf{side}}=\text{'R'}$.
The real analogue of this function is nag_lapack_dormlq (f08ak).

## Example

See Example in nag_lapack_zgelqf (f08av).
```function f08ax_example

fprintf('f08ax example results\n\n');

m = 3;
n = 4;
a = [ 0.28 - 0.36i,  0.50 - 0.86i, -0.77 - 0.48i,  1.58 + 0.66i;
-0.50 - 1.10i, -1.21 + 0.76i, -0.32 - 0.24i, -0.27 - 1.15i;
0.36 - 0.51i, -0.07 + 1.33i, -0.75 + 0.47i, -0.08 + 1.01i];

nrhs = 2;
b = [-1.35 + 0.19i,  4.83 - 2.67i;
9.41 - 3.56i, -7.28 + 3.34i;
-7.57 + 6.93i,  0.62 + 4.53i];

% Compute the LQ factorization of A
[lq, tau, info] = f08av(a);

% Solve L*Y = B
[y, info] = f07ts(...
'Lower', 'No trans', 'Non-Unit', lq(1:m,1:m), b);

% Add zero rows to B
y(m+1:n,1:nrhs) = 0 + 0i;

% Compute minimum norm solution X = Q^H Y
side = 'Left';
trans = 'Conjugate transpose';
[x, info] = f08ax(side, trans, lq, tau, y);

disp('Mimimum norm soltion:');
disp(x);

```
```f08ax example results

Mimimum norm soltion:
-2.8501 + 6.4683i  -1.1682 - 1.8886i
1.6264 - 0.7799i   2.8377 + 0.7654i
6.9290 + 4.6481i  -1.7610 - 0.7041i
1.4048 + 3.2400i   1.0518 - 1.6365i

```