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# NAG Toolbox: nag_lapack_zgemqrt (f08aq)

## Purpose

nag_lapack_zgemqrt (f08aq) multiplies an arbitrary complex matrix $C$ by the complex unitary matrix $Q$ from a $QR$ factorization computed by nag_lapack_zgeqrt (f08ap).

## Syntax

[c, info] = f08aq(side, trans, v, t, c, 'm', m, 'n', n, 'k', k, 'nb', nb)
[c, info] = nag_lapack_zgemqrt(side, trans, v, t, c, 'm', m, 'n', n, 'k', k, 'nb', nb)

## Description

nag_lapack_zgemqrt (f08aq) is intended to be used after a call to nag_lapack_zgeqrt (f08ap), which performs 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 form one of the matrix products
 $QC , QHC , CQ ​ or ​ CQH ,$
overwriting the result on $C$ (which may be any complex rectangular 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_zgeqrt (f08ap).

## References

Golub G H and Van Loan C F (2012) Matrix Computations (4th 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:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – complex array
The first dimension, $\mathit{ldv}$, of the array v must satisfy
• if ${\mathbf{side}}=\text{'L'}$, $\mathit{ldv}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{side}}=\text{'R'}$, $\mathit{ldv}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array v 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_zgeqrt (f08ap) in the first $k$ columns of its array argument a.
4:     $\mathrm{t}\left(\mathit{ldt},:\right)$ – complex array
The first dimension of the array t must be at least ${\mathbf{nb}}$.
The second dimension of the array t must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
Further details of the unitary matrix $Q$ as returned by nag_lapack_zgeqrt (f08ap). The number of blocks is $b=⌈\frac{k}{{\mathbf{nb}}}⌉$, where $k=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ and each block is of order nb except for the last block, which is of order $k-\left(b-1\right)×{\mathbf{nb}}$. For the $b$ blocks the upper triangular block reflector factors ${\mathbit{T}}_{1},{\mathbit{T}}_{2},\dots ,{\mathbit{T}}_{b}$ are stored in the ${\mathbf{nb}}$ by $n$ matrix $T$ as $\mathbit{T}=\left[{\mathbit{T}}_{1}|{\mathbit{T}}_{2}|\dots |{\mathbit{T}}_{b}\right]$.
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 v, t.
$k$, the number of elementary reflectors whose product defines the matrix $Q$. Usually ${\mathbf{k}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({m}_{A},{n}_{A}\right)$ where ${m}_{A}$, ${n}_{A}$ are the dimensions of the matrix $A$ supplied in a previous call to nag_lapack_zgeqrt (f08ap).
Constraints:
• if ${\mathbf{side}}=\text{'L'}$, ${\mathbf{m}}\ge {\mathbf{k}}\ge 0$;
• if ${\mathbf{side}}=\text{'R'}$, ${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.
4:     $\mathrm{nb}$int64int32nag_int scalar
Default: the first dimension of the array t.
The block size used in the $QR$ factorization performed in a previous call to nag_lapack_zgeqrt (f08ap); this value must remain unchanged from that call.
Constraints:
• ${\mathbf{nb}}\ge 1$;
• if ${\mathbf{k}}>0$, ${\mathbf{nb}}\le {\mathbf{k}}$.

### 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

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 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_dgemqrt (f08ac).

## Example

See Example in nag_lapack_zgeqrt (f08ap).
```function f08aq_example

fprintf('f08aq example results\n\n');

% Minimize ||Ax - b|| using recursive QR for m-by-n A and m-by-p B

m = int64(6);
n = int64(4);
p = int64(2);

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 QR Factorisation of A
[QR, T, info] = f08ap(n,a);

% Compute C = (C1) = (Q^H)*B
[c1, info] = f08aq(...
'Left', 'Conjugate Transpose', QR, T, b);

% Compute least-squares solutions by backsubstitution in R*X = C1
[x, info] = f07ts(...
'Upper', 'No Transpose', 'Non-Unit', QR, c1, 'n', n);

% Print least-squares solutions
disp('Least-squares solutions');
disp(x(1:n,:));

% Compute and print estimates of the square roots of the residual
% sums of squares
for j=1:p
rnorm(j) = norm(x(n+1:m,j));
end
fprintf('\nSquare roots of the residual sums of squares\n');
fprintf('%12.2e', rnorm);
fprintf('\n');

```
```f08aq example results

Least-squares solutions
-0.5044 - 1.2179i   0.7629 + 1.4529i
-2.4281 + 2.8574i   5.1570 - 3.6089i
1.4872 - 2.1955i  -2.6518 + 2.1203i
0.4537 + 2.6904i  -2.7606 + 0.3318i

Square roots of the residual sums of squares
6.88e-02    1.87e-01
```

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