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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgeqrf (f08as)

## Purpose

nag_lapack_zgeqrf (f08as) computes the QR$QR$ factorization of a complex m$m$ by n$n$ 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$QR$ factorization of an arbitrary rectangular complex m$m$ by n$n$ matrix. No pivoting is performed.
If mn$m\ge n$, the factorization is given by:
A = Q
 ( R ) 0
,
$A = Q R 0 ,$
where R$R$ is an n$n$ by n$n$ upper triangular matrix (with real diagonal elements) and Q$Q$ is an m$m$ by m$m$ unitary matrix. It is sometimes more convenient to write the factorization as
A =
 ( Q1 Q2 )
 ( R ) 0
,
$A = Q1 Q2 R 0 ,$
which reduces to
 A = Q1R , $A = Q1R ,$
where Q1${Q}_{1}$ consists of the first n$n$ columns of Q$Q$, and Q2${Q}_{2}$ the remaining mn$m-n$ columns.
If m < n$m, R$R$ is trapezoidal, and the factorization can be written
A = Q
 ( R1 R2 )
,
$A = Q R1 R2 ,$
where R1${R}_{1}$ is upper triangular and R2${R}_{2}$ is rectangular.
The matrix Q$Q$ is not formed explicitly but is represented as a product of min (m,n)$\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with Q$Q$ in this representation (see Section [Further Comments]).
Note also that for any k < n$k, the information returned in the first k$k$ columns of the array a represents a QR$QR$ factorization of the first k$k$ columns of the original matrix A$A$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     a(lda, : $:$) – complex 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The m$m$ by n$n$ matrix A$A$.

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
m$m$, the number of rows of the matrix A$A$.
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 A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda work lwork

### Output Parameters

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

The total number of real floating point operations is approximately (8/3) n2 (3mn) $\frac{8}{3}{n}^{2}\left(3m-n\right)$ if mn$m\ge n$ or (8/3) m2 (3nm) $\frac{8}{3}{m}^{2}\left(3n-m\right)$ if m < n$m.
To form the unitary matrix Q$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 mn$m\ge n$, it is often only the first n$n$ columns of Q$Q$ that are required, and they may be formed by the call:
```[a, info] = f08at(a(:,1:n), tau);
```
To apply Q$Q$ to an arbitrary complex rectangular matrix C$C$, 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 = QHC$C={Q}^{\mathrm{H}}C$, where C$C$ is m$m$ by p$p$.
To compute a QR$QR$ factorization with column pivoting, use nag_lapack_zgeqpf (f08bs).
The real analogue of this function is nag_lapack_dgeqrf (f08ae).

## Example

```function nag_lapack_zgeqrf_example
a = [ 0.96 - 0.81i,  -0.03 + 0.96i,  -0.91 + 2.06i, ...
-0.05 + 0.41i;
-0.98 + 1.98i,  -1.2 + 0.19i,  -0.66 + 0.42i, ...
-0.81 + 0.56i;
0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i, ...
-1.11 + 0.6i;
-0.37 + 0.38i,  0.19 - 0.54i,  -0.98 - 0.36i, ...
0.22 - 0.2i;
0.83 + 0.51i,  0.2 + 0.01i,  -0.17 - 0.46i, ...
1.47 + 1.59i;
1.08 - 0.28i,  0.2 - 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
[a, tau, info] = nag_lapack_zgeqrf(a);

% Compute c = [c1;C2] = (q**h)*b, storing the result in b
[b, info] = nag_lapack_zunmqr('Left', 'Conjugate transpose', a, tau, b);

% Compute least-squares solutions by backsubstitution in r*x = c1
[bOut, info] = nag_lapack_ztrtrs('Upper', 'No transpose', 'Non-Unit', a(1:4,:), b(1:4,:));

if (info > 0)
fprintf('\nThe upper triangular factor, R, of A is singular,\n');
fprintf('the least squares solution could not be computed');
else
% Print least-squares solutions
fprintf('\nLeast-squares solution(s)\n');
disp(bOut);

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

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

```
```function f08as_example
a = [ 0.96 - 0.81i,  -0.03 + 0.96i,  -0.91 + 2.06i, ...
-0.05 + 0.41i;
-0.98 + 1.98i,  -1.2 + 0.19i,  -0.66 + 0.42i, ...
-0.81 + 0.56i;
0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i, ...
-1.11 + 0.6i;
-0.37 + 0.38i,  0.19 - 0.54i,  -0.98 - 0.36i, ...
0.22 - 0.2i;
0.83 + 0.51i,  0.2 + 0.01i,  -0.17 - 0.46i, ...
1.47 + 1.59i;
1.08 - 0.28i,  0.2 - 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
[a, tau, info] = f08as(a);

% Compute c = [c1;C2] = (q**h)*b, storing the result in b
[b, info] = f08au('Left', 'Conjugate transpose', a, tau, b);

% Compute least-squares solutions by backsubstitution in r*x = c1
[bOut, info] = f07ts('Upper', 'No transpose', 'Non-Unit', a(1:4,:), b(1:4,:));

if (info > 0)
fprintf('\nThe upper triangular factor, R, of A is singular,\n');
fprintf('the least squares solution could not be computed');
else
% Print least-squares solutions
fprintf('\nLeast-squares solution(s)\n');
disp(bOut);

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

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

```