NAG Toolbox: nag_lapack_dgeqrf (f08ae)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

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

Further details of the orthogonal 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] = f08af(a, tau, 'k', min(m,n));

[a, info] = f08af(a, tau);

[c, info] = f08ag('Left', 'Transpose', a, tau, c, 'k', min(m,n));

Example

Open in the MATLAB editor: f08ae_example

function f08ae_example


fprintf('f08ae example results\n\n');

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
[a, tau, info] = f08ae(a);

% Compute C = (C1) = (Q^T)*B, storing the result in B (C2)
[b, info] = f08ag(...
                  'Left', 'Transpose', a, tau, b);

% Compute least-squares solutions by backsubstitution in R*X = C1
[b, info] = f07te(...
                  'Upper', 'No Transpose', 'Non-Unit', a, b, 'n', int64(4));
if (info >0)
  fprintf('The upper triangular factor, R, of A is singular,\n');
  fprintf('the least squares solution could not be computed.\n');
else
  % Print least-squares solutions
  [ifail] = x04ca('General', ' ', b(1:4,:), 'Least-squares solution(s)');
  % Compute and print estimates of the square roots of the residual
  % sums of squares
  rnorm = zeros(2,1);
  for j=1:2
    rnorm(j) = norm(b(5:6,j));
  end
  fprintf('\nSquare root(s) of the residual sum(s) of squares\n');
  fprintf('\t%11.2e    %11.2e\n', rnorm(1), rnorm(2));
end

f08ae example results

 Least-squares solution(s)
             1          2
 1      1.5339    -1.5753
 2      1.8707     0.5559
 3     -1.5241     1.3119
 4      0.0392     2.9585

Square root(s) of the residual sum(s) of squares
	   2.22e-02       1.38e-02