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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgerqf (f08cv)

## Purpose

nag_lapack_zgerqf (f08cv) computes an RQ factorization of a complex $m$ by $n$ matrix $A$.

## Syntax

[a, tau, info] = f08cv(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_zgerqf(a, 'm', m, 'n', n)

## Description

nag_lapack_zgerqf (f08cv) 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 $n$ unitary 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 $n$ unitary 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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 unitary 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:     $\mathrm{tau}\left(:\right)$ – complex 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 unitary matrix $Q$ nag_lapack_zgerqf (f08cv) may be followed by a call to nag_lapack_zungrq (f08cw):
```[a, info] = f08cw(a, tau, 'k', min(m,n));
```
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_zgerqf (f08cv). 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] = f08cw(a, tau);
```
To apply $Q$ to an arbitrary real rectangular matrix $C$, nag_lapack_zgerqf (f08cv) may be followed by a call to nag_lapack_zunmrq (f08cx). For example:
```[a, c, info] = f08cx('Left','C', a, tau, c);
```
forms $C={Q}^{\mathrm{H}}C$, where $C$ is $n$ by $p$.
The real analogue of this function is nag_lapack_dgerqf (f08ch).

## Example

This example finds the minimum norm solution to the underdetermined equations
 $Ax=b$
where
 $A = 0.28-0.36i 0.50-0.86i -0.77-0.48i 1.58+0.66i -0.50-1.10i -1.21+0.76i -0.32-0.24i -0.27-1.15i 0.36-0.51i -0.07+1.33i -0.75+0.47i -0.08+1.01i$
and
 $b = -1.35+0.19i 9.41-3.56i -7.57+6.93i .$
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 f08cv_example

fprintf('f08cv example results\n\n');

% Minimum norm solution of AX = B, m<n
m = 3;
n = 4;
a = [ 0.28 - 0.36i,  0.50 - 0.86i, -0.77 - 0.48i,  1.58 + 0.66i;
-0.50 - 1.10i, -1.21 + 0.76i, -0.32 - 0.24i, -0.27 - 1.15i;
0.36 - 0.51i, -0.07 + 1.33i, -0.75 + 0.47i, -0.08 + 1.01i];
b = [ -1.35 + 0.19i;
9.41 - 3.56i;
-7.57 + 6.93i];

% Compute the RQ factorization of A
[rq, tau, info] = f08cv(a);

% RQX = B ==> C = QX = R^-1 B
c = zeros(n, 1);
il = n - m + 1;
[c(il:n,:), info] = f07ts( ...
'Upper', 'No transpose','Non-Unit', rq(:,il:n), b);

% QX = C ==> X = Q^H C
[rq, x, info] = f08cx( ...
'Left', 'Conjugate Transpose', rq, tau, c);

fprintf('Minimum-norm solution\n');
disp(x);

```
```f08cv example results

Minimum-norm solution
-2.8501 + 6.4683i
1.6264 - 0.7799i
6.9290 + 4.6481i
1.4048 + 3.2400i

```