NAG Toolbox: nag_lapack_dormrq (f08ck)

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

The second dimension of the array a must be at least $\max \left(1,\mathbf{m}\right)$ if $\mathbf{side}=\text{'L'}$ and at least $\max \left(1,\mathbf{n}\right)$ if $\mathbf{side}=\text{'R'}$.

The $\mathit{i}$th row of a must contain the vector which defines the elementary reflector $H_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$, as returned by nag_lapack_dgerqf (f08ch).

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

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

$\mathbf{tau}\left(i\right)$ must contain the scalar factor of the elementary reflector $H_i$, as returned by nag_lapack_dgerqf (f08ch).

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 first dimension of the array a and the dimension of the array 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array

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

The second dimension of the array a will be $\max \left(1,\mathbf{m}\right)$ if $\mathbf{side}=\text{'L'}$ and at least $\max \left(1,\mathbf{n}\right)$ if $\mathbf{side}=\text{'R'}$.

Is modified by nag_lapack_dormrq (f08ck) but restored on exit.

2:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – 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.

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: 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: f08ck_example

function f08ck_example


fprintf('f08ck example results\n\n');

% Find x that minimizes norm(c-Ax) subject to Bx = d .

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 = [1, 0, -1, 0;
     0, 1, 0, -1];
c = [-1.50; -2.14;  1.23; -0.54; -1.68; 0.82];
d = [0; 0];

% Compute the generalized RQ factorization of (B,A) as
% A = ZRQ, B = TQ
[TQ, taub, ZR, taua, info] = ... 
  f08zf(b, a);

% Set Qx = y. The problem reduces to:
% minimize (Ry - Z^Tc) subject to Ty = d

% Update c = Z^T*c -> minimize (Ry-c)
[cup, info] = f08ag( ...
		     'Left','Transpose',ZR,taua,c);

% Solve Ty = d for last p elements
T12 = triu(TQ(1:p,n-p+1:n));

[y2, info] = f07te( ...
		    'Upper', 'No transpose', 'Non-unit', T12, d);

% (from Ry-c) R11*y1 + R12*y2 = c1 --> R11*y1 = c1 - R12*y2
% Update c1
c1 = cup(1:n-p) - ZR(1:n-p,n-p+1:n)*y2;

% Solve R11*y1 = c1 for y1
R11 = triu(ZR(1:n-p,1:n-p));
[y1, info] = f07te( ...
		    'Upper', 'No transpose', 'Non-unit', R11, c1);

% Contruct y and backtransform for x = Q^Ty
y = [y1;y2];
[~, x, info] = f08ck( ...
		      'Left', 'Transpose', TQ, taub, y);
fprintf('Constrained least squares solution\n');
disp(x);

fprintf('Residuals computed directly\n');
res = a*x - c;
disp(res);
fprintf('Residual norm\n');
disp(norm(res));

f08ck example results

Constrained least squares solution
    0.4890
    0.9975
    0.4890
    0.9975

Residuals computed directly
    0.0030
   -0.0129
   -0.0193
   -0.0084
    0.0012
   -0.0029

Residual norm
    0.0251