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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgeqrf (f08as)

## Purpose

nag_lapack_zgeqrf (f08as) computes the $QR$ factorization of a complex $m$ by $n$ matrix.

## Syntax

[a, tau, info] = f08as(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_zgeqrf(a, 'm', m, 'n', n)

## Description

nag_lapack_zgeqrf (f08as) forms the $QR$ factorization of an arbitrary rectangular complex $m$ by $n$ matrix. No pivoting is performed.
If $m\ge n$, the factorization is given by:
 $A = Q R 0 ,$
where $R$ is an $n$ by $n$ upper triangular matrix (with real diagonal elements) and $Q$ is an $m$ by $m$ unitary matrix. It is sometimes more convenient to write the factorization as
 $A = Q1 Q2 R 0 ,$
which reduces to
 $A = Q1R ,$
where ${Q}_{1}$ consists of the first $n$ columns of $Q$, and ${Q}_{2}$ the remaining $m-n$ columns.
If $m, $R$ is trapezoidal, and the factorization can be written
 $A = Q R1 R2 ,$
where ${R}_{1}$ is upper triangular and ${R}_{2}$ is rectangular.
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).
Note also that for any $k, the information returned in the first $k$ columns of the array a represents a $QR$ factorization of the first $k$ columns of the original matrix $A$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 $m\ge n$, the elements below the diagonal store details of the unitary matrix $Q$ and the upper triangle stores the corresponding elements of the $n$ by $n$ upper triangular matrix $R$.
If $m, the strictly lower triangular part stores details of the unitary matrix $Q$ and the remaining elements store the corresponding elements of the $m$ by $n$ upper trapezoidal matrix $R$.
The diagonal elements of $R$ are real.
2:     $\mathrm{tau}\left(:\right)$ – complex array
The dimension of the array tau will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$
Further details of the unitary matrix $Q$.
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 $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision.

The total number of real floating-point operations is approximately $\frac{8}{3}{n}^{2}\left(3m-n\right)$ if $m\ge n$ or $\frac{8}{3}{m}^{2}\left(3n-m\right)$ if $m.
To form the unitary matrix $Q$ nag_lapack_zgeqrf (f08as) may be followed by a call to nag_lapack_zungqr (f08at):
```[a, info] = f08at(a, tau);
```
but note that the second dimension of the array a must be at least m, which may be larger than was required by nag_lapack_zgeqrf (f08as).
When $m\ge n$, it is often only the first $n$ columns of $Q$ that are required, and they may be formed by the call:
```[a, info] = f08at(a(:,1:n), tau);
```
To apply $Q$ to an arbitrary complex rectangular matrix $C$, nag_lapack_zgeqrf (f08as) may be followed by a call to nag_lapack_zunmqr (f08au). For example,
```[c, info] = f08au('Left', 'Conjugate Transpose', a, tau, c);
```
forms $C={Q}^{\mathrm{H}}C$, where $C$ is $m$ by $p$.
To compute a $QR$ factorization with column pivoting, use nag_lapack_zgeqpf (f08bs).
The real analogue of this function is nag_lapack_dgeqrf (f08ae).

## Example

This example solves the linear least squares problems
 $minimize⁡ Axi - bi 2 , i=1,2$
where ${b}_{1}$ and ${b}_{2}$ are the columns of the matrix $B$,
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i$
and
 $B = -1.54+0.76i 3.17-2.09i 0.12-1.92i -6.53+4.18i -9.08-4.31i 7.28+0.73i 7.49+3.65i 0.91-3.97i -5.63-2.12i -5.46-1.64i 2.37+8.03i -2.84-5.86i .$
```function f08as_example

fprintf('f08as example results\n\n');

a = [ 0.96 - 0.81i, -0.03 + 0.96i, -0.91 + 2.06i, -0.05 + 0.41i;
-0.98 + 1.98i, -1.20 + 0.19i, -0.66 + 0.42i, -0.81 + 0.56i;
0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i, -1.11 + 0.60i;
-0.37 + 0.38i,  0.19 - 0.54i, -0.98 - 0.36i,  0.22 - 0.20i;
0.83 + 0.51i,  0.20 + 0.01i, -0.17 - 0.46i,  1.47 + 1.59i;
1.08 - 0.28i,  0.20 - 0.12i, -0.07 + 1.23i,  0.26 + 0.26i];
b = [-2.09 + 1.93i,  3.26 - 2.70i;
3.34 - 3.53i, -6.22 + 1.16i;
-4.94 - 2.04i,  7.94 - 3.13i;
0.17 + 4.23i,  1.04 - 4.26i;
-5.19 + 3.63i, -2.31 - 2.12i;
0.98 + 2.53i, -1.39 - 4.05i];
% Compute the QR factorization of A
[qr, tau, info] = f08as(a);

% Compute C = [c1;C2] = (Q^H)*B
[c, info] = f08au(...
'Left', 'Conjugate transpose', qr, tau, b);

% Compute least-squares solutions by backsubstitution in R*x = C1
[x, info] = f07ts(...
'Upper', 'No transpose', 'Non-Unit', qr(1:4,:), c(1:4,:));

fprintf('Least-squares solution(s)\n');
disp(x);

fprintf('Square root(s) of the residual sum(s) of squares\n');
for i=1:2
fprintf('%8.3f ',norm(c(5:6,i)));
end
fprintf('\n');

```
```f08as example results

Least-squares solution(s)
-0.5044 - 1.2179i   0.7629 + 1.4529i
-2.4281 + 2.8574i   5.1570 - 3.6089i
1.4872 - 2.1955i  -2.6518 + 2.1203i
0.4537 + 2.6904i  -2.7606 + 0.3318i

Square root(s) of the residual sum(s) of squares
0.069    0.187
```