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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dormlq (f08ak)

Purpose

nag_lapack_dormlq (f08ak) multiplies an arbitrary real matrix $C$ by the real orthogonal matrix $Q$ from an $LQ$ factorization computed by nag_lapack_dgelqf (f08ah).

Syntax

[c, info] = f08ak(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)
[c, info] = nag_lapack_dormlq(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)

Description

nag_lapack_dormlq (f08ak) is intended to be used after a call to nag_lapack_dgelqf (f08ah), which performs an $LQ$ factorization of a real matrix $A$. The orthogonal matrix $Q$ is represented as a product of elementary reflectors.
This function may be used to form one of the matrix products
 $QC , QTC , CQ ​ or ​ CQT ,$
overwriting the result on $C$ (which may be any real 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 ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{side}}=\text{'L'}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$ from the left.
${\mathbf{side}}=\text{'R'}$
$Q$ or ${Q}^{\mathrm{T}}$ 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 ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{trans}}=\text{'N'}$
$Q$ is applied to $C$.
${\mathbf{trans}}=\text{'T'}$
${Q}^{\mathrm{T}}$ is applied to $C$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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_dgelqf (f08ah).
4:     $\mathrm{tau}\left(:\right)$ – double 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_dgelqf (f08ah).
5:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double 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:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double 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 ${Q}^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$ 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 floating-point operations is approximately $2nk\left(2m-k\right)$ if ${\mathbf{side}}=\text{'L'}$ and $2mk\left(2n-k\right)$ if ${\mathbf{side}}=\text{'R'}$.
The complex analogue of this function is nag_lapack_zunmlq (f08ax).

Example

See Example in nag_lapack_dgelqf (f08ah).
```function f08ak_example

fprintf('f08ak example results\n\n');

a = [ -5.42   3.28  -3.68   0.27   2.06   0.46;
-1.65  -3.40  -3.20  -1.03  -4.06  -0.01;
-0.37   2.35   1.90   4.31  -1.76   1.13;
-3.15  -0.11   1.99  -2.70   0.26   4.50];
m = int64(4);
n = 6;
nrhs = 2;

% Compute the LQ Factorisation of A
[lq, tau, info] = f08ah(a);

% Solve L*Y = B
b = [ -2.87  -5.23;
1.63   0.29;
-3.52   4.76;
0.45  -8.41];
[y, info] = f07te(...
'Lower', 'Notrans', 'Non-unit', lq(1:m,1:m), b);

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

% Compute minimum-norm solution X = (Q**T)*Y
side = 'Left';
trans = 'Transpose';
[x, info] = f08ak( ...
side, trans, lq, tau, y);

disp('Least squares solution X');
disp(x);

```
```f08ak example results

Least squares solution X
0.2371    0.7383
-0.4575    0.0158
-0.0085   -0.0161
-0.5192    1.0768
0.0239   -0.6436
-0.0543   -0.6613

```