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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dorgqr (f08af)

## Purpose

nag_lapack_dorgqr (f08af) generates all or part of 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

[a, info] = f08af(a, tau, 'm', m, 'n', n, 'k', k)
[a, info] = nag_lapack_dorgqr(a, tau, 'm', m, 'n', n, 'k', k)

## Description

nag_lapack_dorgqr (f08af) 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 generate Q$Q$ explicitly as a square matrix, or to form only its leading columns.
Usually Q$Q$ is determined from the QR$QR$ factorization of an m$m$ by p$p$ matrix A$A$ with mp$m\ge p$. The whole of Q$Q$ may be computed by:
```[a, info] = f08af(a, tau, 'k', p);
```
(note that the array a must have m$m$ columns) or its leading p$p$ columns by:
```[a, info] = f08af(a(:,1:p), tau, 'k', p);
```
The columns of Q$Q$ returned by the last call form an orthonormal basis for the space spanned by the columns of A$A$; thus nag_lapack_dgeqrf (f08ae) followed by nag_lapack_dorgqr (f08af) can be used to orthogonalize the columns of A$A$.
The information returned by the QR$QR$ factorization functions also yields the QR$QR$ factorization of the leading k$k$ columns of A$A$, where k < p$k. The orthogonal matrix arising from this factorization can be computed by:
```[a, info] = f08af(a, tau, 'k', k);
```
or its leading k$k$ columns by:
```[a, info] = f08af(a(:,1:p), tau, 'k', k);
```

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
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).
2:     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).

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
m$m$, the order of the orthogonal matrix Q$Q$.
Constraint: m0${\mathbf{m}}\ge 0$.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array a.
n$n$, the number of columns of the matrix Q$Q$.
Constraint: mn0${\mathbf{m}}\ge {\mathbf{n}}\ge 0$.
3:     k – int64int32nag_int scalar
Default: The dimension of the array tau.
k$k$, the number of elementary reflectors whose product defines the matrix Q$Q$.
Constraint: nk0${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.

lda work lwork

### Output Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,m)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The m$m$ by n$n$ matrix Q$Q$.
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: m, 2: n, 3: k, 4: a, 5: lda, 6: tau, 7: work, 8: lwork, 9: 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 matrix Q$Q$ differs from an exactly orthogonal matrix by a matrix E$E$ such that
 ‖E‖2 = O(ε) , $‖E‖2 = O(ε) ,$
where ε$\epsilon$ is the machine precision.

The total number of floating point operations is approximately 4mnk2 (m + n) k2 + (4/3) k3 $4mnk-2\left(m+n\right){k}^{2}+\frac{4}{3}{k}^{3}$; when n = k$n=k$, the number is approximately (2/3) n2 (3mn) $\frac{2}{3}{n}^{2}\left(3m-n\right)$.
The complex analogue of this function is nag_lapack_zungqr (f08at).

## Example

```function nag_lapack_dorgqr_example
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];
[aOut, info] = nag_lapack_dorgqr(a, tau)
```
```

aOut =

-0.1576    0.6744   -0.4571    0.4489
-0.5335   -0.3861    0.2583    0.3898
0.6358   -0.2928    0.0165    0.1930
-0.5335   -0.1692   -0.0834   -0.2350
0.0415   -0.1593    0.1475    0.7436
-0.0055   -0.5064   -0.8339    0.0335

info =

0

```
```function f08af_example
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];
[aOut, info] = f08af(a, tau)
```
```

aOut =

-0.1576    0.6744   -0.4571    0.4489
-0.5335   -0.3861    0.2583    0.3898
0.6358   -0.2928    0.0165    0.1930
-0.5335   -0.1692   -0.0834   -0.2350
0.0415   -0.1593    0.1475    0.7436
-0.0055   -0.5064   -0.8339    0.0335

info =

0

```