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# NAG Toolbox: nag_lapack_zunmrq (f08cx)

## Purpose

nag_lapack_zunmrq (f08cx) multiplies a general complex $m$ by $n$ matrix $C$ by the complex unitary matrix $Q$ from an $RQ$ factorization computed by nag_lapack_zgerqf (f08cv).

## Syntax

[a, c, info] = f08cx(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)
[a, c, info] = nag_lapack_zunmrq(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)

## Description

nag_lapack_zunmrq (f08cx) 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 elementary reflectors.
This function may be used to form one of the matrix products
 $QC , QHC , CQ , CQH ,$
overwriting the result on $C$, which may be any complex rectangular $m$ by $n$ matrix.
A common application of this function is in solving underdetermined linear least squares problems, as described in the F08 Chapter Introduction, and illustrated in Example in nag_lapack_zgerqf (f08cv).

## 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:     $\mathrm{side}$ – 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 ${\mathbf{side}}=\text{'R'}$.
The $\mathit{i}$th row of a must contain the vector which defines the elementary reflector ${H}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$, as returned by nag_lapack_zgerqf (f08cv).
4:     $\mathrm{tau}\left(:\right)$ – complex array
The dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$
${\mathbf{tau}}\left(i\right)$ must contain the scalar factor of the elementary reflector ${H}_{i}$, as returned by nag_lapack_zgerqf (f08cv).
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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{side}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{side}}=\text{'R'}$.
Is modified by nag_lapack_zunmrq (f08cx) but restored on exit.
2:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – 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.
3:     $\mathrm{info}$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
 $E2 = O⁡ε C2$
where $\epsilon$ is the machine precision.

The total number of 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_dormrq (f08ck).

## Example

See Example in nag_lapack_zgerqf (f08cv).
```function f08cx_example

fprintf('f08cx example results\n\n');

% Find x that minimizes norm(c-Ax) subject to Bx = d .
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);
p = 2;

b = complex([ 1 0 -1  0;
0 1  0 -1]);
c = [-2.54 + 0.09i;
1.65 - 2.26i;
-2.11 - 3.96i;
1.82 + 3.30i;
-6.41 + 3.77i;
2.07 + 0.66i];
d = complex([0;0]);

% Compute the generalized RQ factorization of (B,A) as
% A = ZRQ, B = TQ
[TQ, taub, ZR, taua, info] = ...
f08zt(b, a);

% Set Qx = y. The problem reduces to:
% minimize (Ry - Z^Hc) subject to Ty = d

% Update c = Z^H*c -> minimize (Ry-c)
[cup, info] = f08au( ...
'Left','Conjugate Transpose',ZR,taua,c);

% Solve Ty = d for last p elements
T12 = complex(triu(TQ(1:p,n-p+1:n)));

[y2, info] = f07ts( ...
'Upper', 'No transpose', 'Non-unit', T12, d);

% (from Ry-c) R11*y1 + R12*y2 = c1 --> R11*y1 = c1 - R12*y2
% Update c1
c1 = cup(1:n-p) - ZR(1:n-p,n-p+1:n)*y2;

% Solve R11*y1 = c1 for y1
R11 = complex(triu(ZR(1:n-p,1:n-p)));
[y1, info] = f07ts( ...
'Upper', 'No transpose', 'Non-unit', R11, c1);

% Contruct y and backtransform for x = Q^Hy
y = [y1;y2];
[~, x, info] = f08cx( ...
'Left', 'Conjugate Transpose', TQ, taub, y);
fprintf('Constrained least squares solution\n');
disp(x);

fprintf('Residuals computed directly\n');
res = a*x - c;
disp(res);
fprintf('Residual norm\n');
disp(norm(res));

```
```f08cx example results

Constrained least squares solution
1.0874 - 1.9621i
-0.7409 + 3.7297i
1.0874 - 1.9621i
-0.7409 + 3.7297i

Residuals computed directly
-0.0035 - 0.1423i
-0.0325 + 0.0351i
-0.0052 - 0.0100i
0.0121 - 0.0039i
0.0209 + 0.0326i
0.0293 + 0.0033i

Residual norm
0.1587

```

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