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# NAG Toolbox: nag_lapack_zunmql (f08cu)

## Purpose

nag_lapack_zunmql (f08cu) multiplies a general complex $m$ by $n$ matrix $C$ by the complex unitary matrix $Q$ from a $QL$ factorization computed by nag_lapack_zgeqlf (f08cs).

## Syntax

[c, info] = f08cu(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)
[c, info] = nag_lapack_zunmql(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)

## Description

nag_lapack_zunmql (f08cu) is intended to be used following a call to nag_lapack_zgeqlf (f08cs), which performs a $QL$ 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 linear least squares problems, as described in the F08 Chapter Introduction, and illustrated in Example in nag_lapack_zgeqlf (f08cs).

## 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

## 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, $\mathit{lda}$, of the array a must satisfy
• if ${\mathbf{side}}=\text{'L'}$, $\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{side}}=\text{'R'}$, $\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_zgeqlf (f08cs).
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_zgeqlf (f08cs).
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 second dimension of the arrays a, 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
 $E2 = O⁡ε C2$
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_dormql (f08cg).

## Example

See Example in nag_lapack_zgeqlf (f08cs).
```function f08cu_example

fprintf('f08cu example results\n\n');

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];

b = [-2.09 + 1.93i,  3.26 - 2.70i;
3.34 - 3.53i, -6.22 + 1.16i;
-4.94 - 2.04i,  7.94 - 3.13i;
0.17 + 4.23i,  1.04 - 4.26i;
-5.19 + 3.63i, -2.31 - 2.12i;
0.98 + 2.53i, -1.39 - 4.05i];

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

% LX = Q^H B = C; compute C = (Q^H)*B
[c, info] = f08cu( ...
'Left', 'ConjTrans', ql, tau, b);

% Least-squares solution X = L^-1 C (lower n part)
il = m-n+1;
[x, info] = f07ts( ...
'Lower', 'Notrans','Non-Unit', ql(il:m,:), c(il:m,:));

% Print least-squares solutions
ncols  = int64(80);
indent = int64(0);
[ifail] = x04db( ...
'General', ' ', x, 'Bracketed', 'F7.4', ...
'Least-squares solution(s)', 'Integer', 'Integer', ...
ncols, indent);

% Compute estimates of the square roots of the residual sums of squares.
rnorm = zeros(2,1);
for j=1:2
rnorm(j) = norm(c(1:m-n,j));
end
fprintf('\nSquare root(s) of the residual sum(s) of squares\n');
fprintf('\t%11.2e    %11.2e\n', rnorm(1), rnorm(2));

```
```f08cu example results

Least-squares solution(s)
1                 2
1  (-0.5044,-1.2179) ( 0.7629, 1.4529)
2  (-2.4281, 2.8574) ( 5.1570,-3.6089)
3  ( 1.4872,-2.1955) (-2.6518, 2.1203)
4  ( 0.4537, 2.6904) (-2.7606, 0.3318)

Square root(s) of the residual sum(s) of squares
6.88e-02       1.87e-01
```

Chapter Contents
Chapter Introduction
NAG Toolbox

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