NAG Toolbox: nag_lapack_zgeqrt (f08ap)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{nb}$ – int64int32nag_int scalar

The explicitly chosen block size to be used in computing the $QR$ factorization. See Further Comments for details.

Constraints:

• $\mathbf{nb}\ge 1$;
• if $\min \left(\mathbf{m},\mathbf{n}\right)>0$, $\mathbf{nb}\le \min \left(\mathbf{m},\mathbf{n}\right)$.

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

The first dimension of the array t will be $\mathbf{nb}$.

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

Further details of the unitary matrix $Q$. The number of blocks is $b=⌈\frac{k}{\mathbf{nb}}⌉$, where $k=\min \left(m,n\right)$ and each block is of order nb except for the last block, which is of order $k-\left(b-1\right)\times \mathbf{nb}$. For each of the blocks, an upper triangular block reflector factor is computed: ${\mathit{T}}_1,{\mathit{T}}_2,\dots ,{\mathit{T}}_b$. These are stored in the $\mathbf{nb}$ by $n$ matrix $T$ as $\mathit{T}=\left[{\mathit{T}}_1\left|{\mathit{T}}_2\right|\dots |{\mathit{T}}_b\right]$.

3:     $\mathrm{info}$ – int64int32nag_int scalar

$\mathbf{info}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

Accuracy

Further Comments

[t, c, info] = f08aq('Left', 'Conjugate Transpose', nb, a, t, c);

Example

Open in the MATLAB editor: f08ap_example

function f08ap_example


fprintf('f08ap example results\n\n');

% Minimize ||Ax - b|| using recursive QR for m-by-n A and m-by-p B

m = int64(6);
n = int64(4);
p = int64(2);

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 Factorisation of A
[QR, T, info] = f08ap(n,a);

% Compute C = (C1) = (Q^H)*B
[c1, info] = f08aq(...
                  'Left', 'Conjugate Transpose', QR, T, b);

% Compute least-squares solutions by backsubstitution in R*X = C1
[x, info] = f07ts(...
                  'Upper', 'No Transpose', 'Non-Unit', QR, c1, 'n', n);

% Print least-squares solutions
disp('Least-squares solutions');
disp(x(1:n,:));

% Compute and print estimates of the square roots of the residual
% sums of squares
for j=1:p
  rnorm(j) = norm(x(n+1:m,j));
end
fprintf('\nSquare roots of the residual sums of squares\n');
fprintf('%12.2e', rnorm);
fprintf('\n');

f08ap example results

Least-squares solutions
  -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 roots of the residual sums of squares
    6.88e-02    1.87e-01