NAG Toolbox: nag_lapack_dgeqpf (f08be)

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 $A$.

2:     $\mathrm{jpvt}\left(:\right)$ – int64int32nag_int array

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

If $\mathbf{jpvt}\left(i\right)\ne 0$, then the $i$ th column of $A$ is moved to the beginning of $AP$ before the decomposition is computed and is fixed in place during the computation. Otherwise, the $i$ th column of $A$ is a free column (i.e., one which may be interchanged during the computation with any other free column).

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:     $\mathrm{jpvt}\left(:\right)$ – int64int32nag_int array

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

Details of the permutation matrix $P$. More precisely, if $\mathbf{jpvt}\left(i\right)=k$, then the $k$th column of $A$ is moved to become the $i$ th column of $AP$; in other words, the columns of $AP$ are the columns of $A$ in the order $\mathbf{jpvt}\left(1\right),\mathbf{jpvt}\left(2\right),\dots ,\mathbf{jpvt}\left(n\right)$.

3:     $\mathrm{tau}\left(\min \left(\mathbf{m},\mathbf{n}\right)\right)$ – double array

Further details of the orthogonal matrix $Q$.

4:     $\mathrm{info}$ – 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: jpvt, 6: tau, 7: work, 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(:,1:m), tau);

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

[c, info] = f08ag('Left','Transpose', a(:,1:min(m,n)), tau, c);

Example

Open in the MATLAB editor: f08be_example

function f08be_example


fprintf('f08be example results\n\n');

a = [-0.09,  0.14, -0.46,  0.68,  1.29;
     -1.56,  0.2,   0.29,  1.09,  0.51;
     -1.48, -0.43,  0.89, -0.71, -0.96;
     -1.09,  0.84,  0.77,  2.11, -1.27;
      0.08,  0.55, -1.13,  0.14,  1.74;
     -1.59, -0.72,  1.06,  1.24,  0.34];
b = [-0.01, -0.04;
      0.04, -0.03;
      0.05,  0.01;
     -0.03, -0.02;
      0.02,  0.05;
     -0.06,  0.07];
jpvt = [int64(0);0;0;0;0];

% Compute the QR factorization of a
[a, jpvt, tau, info] = f08be( ...
			      a, jpvt);

% Choose tol to reflect the relative accuracy of the input data
tol = 0.01;

% Determine which columns of R to use
k = find(abs(diag(a)) <= tol*abs(a(1,1)));
if numel(k) == 0
  k = numel(diag(a));
else
  k = k(1)-1;
end

% Compute c = (q^t)*b,
[c, info] = f08ag( ...
		   'Left', 'Transpose', a, tau, b);

% Compute least-squares solution by backsubstitution in r*b = c
c(1:k, :) = inv(triu(a(1:k,1:k)))*c(1:k,:);
c(k+1:5, :) = 0;

% Unscramble the least-squares solution stored in c
x = zeros(5, 2);
for i=1:5
  x(jpvt(i), :) = c(i, :);
end

fprintf('\nLeast-squares solution\n');
disp(x);

f08be example results


Least-squares solution
   -0.0370   -0.0044
    0.0647   -0.0335
         0         0
   -0.0515    0.0018
    0.0066    0.0102