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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dormqr (f08ag)

## Purpose

nag_lapack_dormqr (f08ag) multiplies an arbitrary real matrix $C$ by the real orthogonal matrix $Q$ from a $QR$ factorization computed by nag_lapack_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf).

## Syntax

[c, info] = f08ag(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)
[c, info] = nag_lapack_dormqr(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)

## Description

nag_lapack_dormqr (f08ag) is intended to be used after a call to nag_lapack_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf) which perform a $QR$ 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 ${\mathbf{c}}$ (which may be any real 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_dgeqrf (f08ae).

## 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, $\mathit{lda}$, 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_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf).
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_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf).
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 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:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – 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_zunmqr (f08au).

## Example

See Example in nag_lapack_dgeqrf (f08ae).
```function f08ag_example

fprintf('f08ag example results\n\n');

a = [-0.57, -1.28, -0.39,  0.25;
-1.93,  1.08, -0.31, -2.14;
2.30,  0.24,  0.40, -0.35;
-1.93,  0.64, -0.66,  0.08;
0.15,  0.30,  0.15, -2.13;
-0.02,  1.03, -1.43,  0.50];

% Compute the QR Factorisation of A
[a, tau, info] = f08ae(a);

% Apply Q^T to C from Left, i.e. B = Q^T*C
side = 'Left';
trans = 'Transpose';
c = [-2.67,  0.41;
-0.55, -3.10;
3.34, -4.01;
-0.77,  2.76;
0.48, -6.17;
4.10,  0.21];
[b, info] = f08ag(side, trans, a, tau, c);

disp('B = Q^T*C:');
disp(b);

```
```f08ag example results

B = Q^T*C:
3.2456   -2.6896
-4.5885    3.0574
-2.1500   -2.3696
-0.0931   -7.0279
-0.0085    0.0009
0.0204    0.0138

```

Chapter Contents
Chapter Introduction
NAG Toolbox

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