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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgerqf (f08ch)

## Purpose

nag_lapack_dgerqf (f08ch) computes an RQ factorization of a real $m$ by $n$ matrix $A$.

## Syntax

[a, tau, info] = f08ch(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_dgerqf(a, 'm', m, 'n', n)

## Description

nag_lapack_dgerqf (f08ch) forms the $RQ$ factorization of an arbitrary rectangular real $m$ by $n$ matrix. If $m\le n$, the factorization is given by
 $A = 0 R Q ,$
where $R$ is an $m$ by $m$ lower triangular matrix and $Q$ is an $n$ by $n$ orthogonal matrix. If $m>n$ the factorization is given by
 $A =RQ ,$
where $R$ is an $m$ by $n$ upper trapezoidal matrix and $Q$ is again an $n$ by $n$ orthogonal matrix. In the case where $m the factorization can be expressed as
 $A = 0 R Q1 Q2 =RQ2 ,$
where ${Q}_{1}$ consists of the first $\left(n-m\right)$ rows of $Q$ and ${Q}_{2}$ the remaining $m$ rows.
The matrix $Q$ is not formed explicitly, but is represented as a product of $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with $Q$ in this representation (see Further Comments).

## References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $n$ matrix $A$.

### 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 $A$.
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 $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If $m\le n$, the upper triangle of the subarray ${\mathbf{a}}\left(1:m,n-m+1:n\right)$ contains the $m$ by $m$ upper triangular matrix $R$.
If $m\ge n$, the elements on and above the $\left(m-n\right)$th subdiagonal contain the $m$ by $n$ upper trapezoidal matrix $R$; the remaining elements, with the array tau, represent the orthogonal matrix $Q$ as a product of $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).
2:     $\mathrm{tau}\left(:\right)$ – double array
The dimension of the array tau will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$
The scalar factors of the elementary reflectors.
3:     $\mathrm{info}$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: a, 4: lda, 5: tau, 6: work, 7: lwork, 8: 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 factorization is the exact factorization of a nearby matrix $A+E$, where
 $E2 = O⁡ε A2$
and $\epsilon$ is the machine precision.

The total number of floating-point operations is approximately $\frac{2}{3}{m}^{2}\left(3n-m\right)$ if $m\le n$, or $\frac{2}{3}{n}^{2}\left(3m-n\right)$ if $m>n$.
To form the orthogonal matrix $Q$ nag_lapack_dgerqf (f08ch) may be followed by a call to nag_lapack_dorgrq (f08cj):
```[a, info] = f08cj(a, tau);
```
but note that the first dimension of the array a must be at least n, which may be larger than was required by nag_lapack_dgerqf (f08ch). When $m\le n$, it is often only the first $m$ rows of $Q$ that are required and they may be formed by the call:
```[a, info] = f08cj(a(1:m,1:n), tau);
```
To apply $Q$ to an arbitrary real rectangular matrix $C$, nag_lapack_dgerqf (f08ch) may be followed by a call to nag_lapack_dormrq (f08ck). For example:
```[a, c, info] = f08ck('Left','Transpose', a, tau, c);
```
forms $C={Q}^{\mathrm{T}}C$, where $C$ is $n$ by $p$.
The complex analogue of this function is nag_lapack_zgerqf (f08cv).

## Example

This example finds the minimum norm solution to the underdetermined equations
 $Ax=b$
where
 $A = -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50 and b= -2.87 1.63 -3.52 0.45 .$
The solution is obtained by first obtaining an $RQ$ factorization of the matrix $A$.
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
```function f08ch_example

fprintf('f08ch 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];
b = [-2.87;  1.63; -3.52;  0.45];
% Compute the RQ factorization of a
[a, tau, info] = f08ch(a);

% Solve R*y2 = b
c = zeros(6,1);
[c(3:6), info] = f07te( ...
'Upper', 'No transpose', 'Non-Unit', a(:, 3:6), b);

if (info >0)
fprintf('The upper triangular factor, R, of A is singular,\n');
fprintf('the least squares solution could not be computed.\n');
else
% Compute the minimum-norm solution x = (Q^T)*y
[a, c, info] = f08ck( ...
'Left', 'Transpose', a, tau, c);

fprintf('Minimum-norm solution\n');
disp(transpose(c));
end

```
```f08ch example results

Minimum-norm solution
0.2371   -0.4575   -0.0085   -0.5192    0.0239   -0.0543

```