NAG Toolbox: nag_lapack_dgeqrt (f08ab)

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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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 orthogonal 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 orthogonal matrix $Q$ and the remaining elements store the corresponding elements of the $m$ by $n$ upper trapezoidal matrix $R$.

2:     $\mathrm{t}\left(\mathit{ldt},:\right)$ – double 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 orthogonal 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] = f08ac('Left', 'Transpose', nb, a, t, c, 'k', min(m,n));

Example

Open in the MATLAB editor: f08ab_example

function f08ab_example


fprintf('f08ab 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 = [-0.57, -1.28, -0.39,  0.25;
     -1.93,  1.08, -0.31, -2.14;
      2.30,  0.24,  0.40, -0.35;
     -1.93,  0.64, -0.66,  0.08;
      0.15,  0.30,  0.15, -2.13;
     -0.02,  1.03, -1.43,  0.50];
b = [-2.67,  0.41;
     -0.55, -3.10;
      3.34, -4.01;
     -0.77,  2.76;
      0.48, -6.17;
      4.10,  0.21];

% Compute the QR Factorisation of A
[QR, T, info] = f08ab(n,a);

% Compute C = (C1) = (Q^T)*B
[c1, info] = f08ac(...
                  'Left', 'Transpose', QR, T, b);

% Compute least-squares solutions by backsubstitution in R*X = C1
[x, info] = f07te(...
                  '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('Square roots of the residual sums of squares\n');
fprintf('%12.2e', rnorm);
fprintf('\n');

f08ab example results

Least-squares solutions
    1.5339   -1.5753
    1.8707    0.5559
   -1.5241    1.3119
    0.0392    2.9585

Square roots of the residual sums of squares
    2.22e-02    1.38e-02