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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dormqr (f08ag)

## Purpose

nag_lapack_dormqr (f08ag) multiplies an arbitrary real matrix C$C$ by the real orthogonal matrix Q$Q$ from a QR$QR$ 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$QR$ factorization of a real matrix A$A$. The orthogonal matrix Q$Q$ 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 , $QC , QTC , CQ ​ or ​ CQT ,$
overwriting the result on c${\mathbf{c}}$ (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 Section [Example] in (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:     side – string (length ≥ 1)
Indicates how Q$Q$ or QT${Q}^{\mathrm{T}}$ is to be applied to C$C$.
side = 'L'${\mathbf{side}}=\text{'L'}$
Q$Q$ or QT${Q}^{\mathrm{T}}$ is applied to C$C$ from the left.
side = 'R'${\mathbf{side}}=\text{'R'}$
Q$Q$ or QT${Q}^{\mathrm{T}}$ is applied to C$C$ from the right.
Constraint: side = 'L'${\mathbf{side}}=\text{'L'}$ or 'R'$\text{'R'}$.
2:     trans – string (length ≥ 1)
Indicates whether Q$Q$ or QT${Q}^{\mathrm{T}}$ is to be applied to C$C$.
trans = 'N'${\mathbf{trans}}=\text{'N'}$
Q$Q$ is applied to C$C$.
trans = 'T'${\mathbf{trans}}=\text{'T'}$
QT${Q}^{\mathrm{T}}$ is applied to C$C$.
Constraint: trans = 'N'${\mathbf{trans}}=\text{'N'}$ or 'T'$\text{'T'}$.
3:     a(lda, : $:$) – double array
The first dimension, lda, of the array a must satisfy
• if side = 'L'${\mathbf{side}}=\text{'L'}$, lda max (1,m) $\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if side = 'R'${\mathbf{side}}=\text{'R'}$, lda max (1,n) $\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array must be at least max (1,k)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$
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:     tau( : $:$) – double array
Note: the dimension of the array tau must be at least max (1,k)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
Further details of the elementary reflectors, as returned by nag_lapack_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf).
5:     c(ldc, : $:$) – double array
The first dimension of the array c must be at least max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The m$m$ by n$n$ matrix C$C$.

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array c.
m$m$, the number of rows of the matrix C$C$.
Constraint: m0${\mathbf{m}}\ge 0$.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array c.
n$n$, the number of columns of the matrix C$C$.
Constraint: n0${\mathbf{n}}\ge 0$.
3:     k – int64int32nag_int scalar
Default: The second dimension of the arrays a, tau.
k$k$, the number of elementary reflectors whose product defines the matrix Q$Q$.
Constraints:
• if side = 'L'${\mathbf{side}}=\text{'L'}$, m k 0 ${\mathbf{m}}\ge {\mathbf{k}}\ge 0$;
• if side = 'R'${\mathbf{side}}=\text{'R'}$, n k 0 ${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

lda ldc work lwork

### Output Parameters

1:     c(ldc, : $:$) – double array
The first dimension of the array c will be max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldcmax (1,m)$\mathit{ldc}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
c stores QC$QC$ or QTC${Q}^{\mathrm{T}}C$ or CQ$CQ$ or CQT$C{Q}^{\mathrm{T}}$ as specified by side and trans.
2:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ 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$E$ such that
 ‖E‖2 = O(ε) ‖C‖2 , $‖E‖2 = O(ε) ‖C‖2 ,$
where ε$\epsilon$ is the machine precision.

The total number of floating point operations is approximately 2nk (2mk) $2nk\left(2m-k\right)$ if side = 'L'${\mathbf{side}}=\text{'L'}$ and 2mk (2nk) $2mk\left(2n-k\right)$ if side = 'R'${\mathbf{side}}=\text{'R'}$.
The complex analogue of this function is nag_lapack_zunmqr (f08au).

## Example

```function nag_lapack_dormqr_example
side = 'Left';
trans = 'Transpose';
a = [3.61767881382524, -0.5565999923223895, 0.847366545721238, 0.7460032078266114;
0.4608758421558694, -2.028077032202356, 0.5513872350020937, 1.16996276895585;
-0.5492302782168392, -0.04571098289280237, 1.374460641222295, -1.410473781059997;
0.4608758421558694, 0.2828431690617352, 0.004430814804361739, -2.375527319588618;
-0.03581936597066342, 0.0796426824688576, -0.07728561757441148, -0.5213744847432364;
0.004775915462755124, 0.3002942085609617, 0.801665355572228, 0.2558113872182322];
tau = [1.157559592582321;
1.696915139470381;
1.213106371299621;
1.495583371241627];
c = [-2.67, 0.41;
-0.55, -3.1;
3.34, -4.01;
-0.77, 2.76;
0.48, -6.17;
4.1, 0.21];
[cOut, info] = nag_lapack_dormqr(side, trans, a, tau, c)
```
```

cOut =

3.2456   -2.6896
-4.5885    3.0574
-2.1500   -2.3696
-0.0931   -7.0279
-0.0085    0.0009
0.0204    0.0138

info =

0

```
```function f08ag_example
side = 'Left';
trans = 'Transpose';
a = [3.61767881382524, -0.5565999923223895, 0.847366545721238, 0.7460032078266114;
0.4608758421558694, -2.028077032202356, 0.5513872350020937, 1.16996276895585;
-0.5492302782168392, -0.04571098289280237, 1.374460641222295, -1.410473781059997;
0.4608758421558694, 0.2828431690617352, 0.004430814804361739, -2.375527319588618;
-0.03581936597066342, 0.0796426824688576, -0.07728561757441148, -0.5213744847432364;
0.004775915462755124, 0.3002942085609617, 0.801665355572228, 0.2558113872182322];
tau = [1.157559592582321;
1.696915139470381;
1.213106371299621;
1.495583371241627];
c = [-2.67, 0.41;
-0.55, -3.1;
3.34, -4.01;
-0.77, 2.76;
0.48, -6.17;
4.1, 0.21];
[cOut, info] = f08ag(side, trans, a, tau, c)
```
```

cOut =

3.2456   -2.6896
-4.5885    3.0574
-2.1500   -2.3696
-0.0931   -7.0279
-0.0085    0.0009
0.0204    0.0138

info =

0

```