NAG Toolbox: nag_lapack_dgeqlf (f08ce)

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 lower triangle of the subarray $\mathbf{a}\left(m-n+1:m,1:n\right)$ contains the $n$ by $n$ lower triangular matrix $L$.

If $m\le n$, the elements on and below the $\left(n-m\right)$th superdiagonal contain the $m$ by $n$ lower trapezoidal matrix $L$. The remaining elements, with the array tau, represent the orthogonal matrix $Q$ as a product of elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).

2:     $\mathrm{tau}\left(:\right)$ – double array

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

The scalar factors of the elementary reflectors (see Further Comments).

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] = f08cf(a(:,1:m), tau);

[a, info] = f08cf(a, tau);

[c, info] = f08cg('Left','Transpose', a(:,1:min(m,n)), tau, c);

Example

Open in the MATLAB editor: f08ce_example

function f08ce_example


fprintf('f08ce example results\n\n');

% Find least squares solution of Ax=B (m>n) via QL factorization.
m = 6;
n = 4;
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,  0.3,   0.15, -2.13;
     -0.02,  1.03, -1.43,  0.5];
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 QL factorization of A
[ql, tau, info] = f08ce(a);

% LX = Q^T B = C; compute C = (Q^T)*B
[c, info] = f08cg( ...
		   'Left', 'Transpose', ql, tau, b);

% Least-squares solution X = L^-1 C (lower n part)
il = m-n+1;
[x, info] = f07te( ...
		   'Lower', 'NoTrans', 'Non-Unit', ql(il:m,:), c(il:m,:));

% Print least-squares solutions
[ifail] = x04ca( ...
		 'General', ' ', x, 'Least-squares solution(s)');
% Compute estimates of the square roots of the residual sums of squares
rnorm = zeros(2,1);
for j=1:2
  rnorm(j) = norm(c(1:2,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));

f08ce 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