NAG Toolbox: nag_lapack_dormql (f08cg)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{side}$ – string (length ≥ 1)

Indicates how $Q$ or $Q^{\mathrm{T}}$ is to be applied to $C$.

$\mathbf{side}=\text{'L'}$

$Q$ or $Q^{\mathrm{T}}$ is applied to $C$ from the left.

$\mathbf{side}=\text{'R'}$

$Q$ or $Q^{\mathrm{T}}$ is applied to $C$ from the right.

Constraint: $\mathbf{side}=\text{'L'}$ or $\text{'R'}$.

2:     $\mathrm{trans}$ – string (length ≥ 1)

Indicates whether $Q$ or $Q^{\mathrm{T}}$ is to be applied to $C$.

$\mathbf{trans}=\text{'N'}$

$Q$ is applied to $C$.

$\mathbf{trans}=\text{'T'}$

$Q^{\mathrm{T}}$ is applied to $C$.

Constraint: $\mathbf{trans}=\text{'N'}$ or $\text{'T'}$.

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

The first dimension, $\mathit{lda}$, of the array a must satisfy

• if $\mathbf{side}=\text{'L'}$, $\mathit{lda}\ge \max \left(1,\mathbf{m}\right)$;
• if $\mathbf{side}=\text{'R'}$, $\mathit{lda}\ge \max \left(1,\mathbf{n}\right)$.

The second dimension of the array a must be at least $\max \left(1,\mathbf{k}\right)$.

Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dgeqlf (f08ce).

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

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

Further details of the elementary reflectors, as returned by nag_lapack_dgeqlf (f08ce).

5:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double array

The first dimension of the array c must be at least $\max \left(1,\mathbf{m}\right)$.

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

The $m$ by $n$ matrix $C$.

Optional Input Parameters

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

Default: the first dimension of the array c.

$m$, the number of rows of the matrix $C$.

Constraint: $\mathbf{m}\ge 0$.

2:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the second dimension of the array c.

$n$, the number of columns of the matrix $C$.

Constraint: $\mathbf{n}\ge 0$.

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

Default: the second dimension of the arrays a, tau.

$k$, the number of elementary reflectors whose product defines the matrix $Q$.

Constraints:

• if $\mathbf{side}=\text{'L'}$, $\mathbf{m}\ge \mathbf{k}\ge 0$;
• if $\mathbf{side}=\text{'R'}$, $\mathbf{n}\ge \mathbf{k}\ge 0$.

Output Parameters

1:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double array

The first dimension of the array c will be $\max \left(1,\mathbf{m}\right)$.

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

c stores $QC$ or $Q^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$ as specified by side and trans.

2:     $\mathrm{info}$ – 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: side, 2: trans, 3: m, 4: n, 5: k, 6: a, 7: lda, 8: tau, 9: c, 10: ldc, 11: work, 12: lwork, 13: 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

Example

Open in the MATLAB editor: f08cg_example

function f08cg_example


fprintf('f08cg example results\n\n');

a = [-0.57, -1.28, -0.39,  0.25;
     -1.93,  1.08, -0.31, -2.14;
      2.3,   0.24,  0.4,  -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
[a, tau, info] = f08ce(a);

% Compute C = (C1) = (Q^T)*b
%             (C2)
[c, info] = f08cg( ...
		   'Left', 'Transpose', a, tau, b);

% Compute least-squares solutions by backsubstitution in L*X = C2
[b, info] = f07te( ...
		   'Lower', 'No Transpose', 'Non-Unit', a(3:6,:), c(3:6,:));

if (info > 0)
  fprintf('The lower triangular factor, L, of A is singular,\n');
  fprintf('the least squares solution could not be computed.\n');
else
  % Print least-squares solutions
  [ifail] = x04ca( ...
		   'General', ' ', b, '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(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));
end

f08cg 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