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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dorglq (f08aj)

Purpose

nag_lapack_dorglq (f08aj) generates all or part of the real orthogonal matrix $Q$ from an $LQ$ factorization computed by nag_lapack_dgelqf (f08ah).

Syntax

[a, info] = f08aj(a, tau, 'm', m, 'n', n, 'k', k)
[a, info] = nag_lapack_dorglq(a, tau, 'm', m, 'n', n, 'k', k)

Description

nag_lapack_dorglq (f08aj) 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 generate $Q$ explicitly as a square matrix, or to form only its leading rows.
Usually $Q$ is determined from the $LQ$ factorization of a $p$ by $n$ matrix $A$ with $p\le n$. The whole of $Q$ may be computed by:
```[a, info] = f08aj(a, tau);
```
(note that the array a must have at least $n$ rows) or its leading $p$ rows by:
```[a, info] = f08aj(a(1:p,:), tau);
```
The rows of $Q$ returned by the last call form an orthonormal basis for the space spanned by the rows of $A$; thus nag_lapack_dgelqf (f08ah) followed by nag_lapack_dorglq (f08aj) can be used to orthogonalize the rows of $A$.
The information returned by the $LQ$ factorization functions also yields the $LQ$ factorization of the leading $k$ rows of $A$, where $k. The orthogonal matrix arising from this factorization can be computed by:
```[a, info] = f08aj(a, tau);
```
or its leading $k$ rows by:
```[a, info] = f08aj(a(1:k,:), tau);
```

References

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

Parameters

Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dgelqf (f08ah).
2:     $\mathrm{tau}\left(:\right)$ – 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).

Optional Input Parameters

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

Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $n$ matrix $Q$.
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: 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 $Q$ differs from an exactly orthogonal matrix by a matrix $E$ such that
 $E2 = Oε ,$
where $\epsilon$ is the machine precision.

The total number of floating-point operations is approximately $4mnk-2\left(m+n\right){k}^{2}+\frac{4}{3}{k}^{3}$; when $m=k$, the number is approximately $\frac{2}{3}{m}^{2}\left(3n-m\right)$.
The complex analogue of this function is nag_lapack_zunglq (f08aw).

Example

This example forms the leading $4$ rows of the orthogonal matrix $Q$ from the $LQ$ factorization of the matrix $A$, where
 $A = -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50 .$
The rows of $Q$ form an orthonormal basis for the space spanned by the rows of $A$.
```function f08aj_example

fprintf('f08aj 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];

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

% Generate Q
[q, info] = f08aj(lq, tau);

disp('Orthogonal factor Q');
disp(q);

```
```f08aj example results

Orthogonal factor Q
-0.7104    0.4299   -0.4824    0.0354    0.2700    0.0603
-0.2412   -0.5323   -0.4845   -0.1595   -0.6311   -0.0027
0.1287   -0.2619   -0.2108   -0.7447    0.5227   -0.2063
-0.3403   -0.0921    0.4546   -0.3869   -0.0465    0.7191

```