NAG Toolbox: nag_lapack_zgeqrf (f08as)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array

The first dimension of the array a must be at least $\max \left(1,\mathbf{m}\right)$.

The second dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

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 $\max \left(1,\mathbf{m}\right)$.

The second dimension of the array a will be $\max \left(1,\mathbf{n}\right)$.

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 $m<n$, 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 $\max \left(1,\min \left(\mathbf{m},\mathbf{n}\right)\right)$

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

Further Comments

[a, info] = f08at(a, tau);

[a, info] = f08at(a(:,1:n), tau);

[c, info] = f08au('Left', 'Conjugate Transpose', a, tau, c);

Example

Open in the MATLAB editor: f08as_example

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