NAG Toolbox: nag_lapack_dggglm (f08zb)

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:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array

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

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

The $m$ by $p$ matrix $B$.

3:     $\mathrm{d}\left(\mathbf{m}\right)$ – double array

The left-hand side vector $d$ of the GLM equation.

Optional Input Parameters

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

Default: the dimension of the array d and the first dimension of the array b. (An error is raised if these dimensions are not equal.)

$m$, the number of rows of the matrices $A$ and $B$.

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: $0\le \mathbf{n}\le \mathbf{m}$.

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

Default: the second dimension of the array b.

$p$, the number of columns of the matrix $B$.

Constraint: $\mathbf{p}\ge \mathbf{m}-\mathbf{n}$.

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 $\max \left(1,\mathbf{n}\right)$.

2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array

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

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

3:     $\mathrm{d}\left(\mathbf{m}\right)$ – double array

4:     $\mathrm{x}\left(\mathbf{n}\right)$ – double array

The solution vector $x$ of the GLM problem.

5:     $\mathrm{y}\left(\mathbf{p}\right)$ – double array

The solution vector $y$ of the GLM problem.

6:     $\mathrm{info}$ – 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: p, 4: a, 5: lda, 6: b, 7: ldb, 8: d, 9: x, 10: y, 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.

   $\mathbf{info}=1$

The upper triangular factor $R$ associated with $A$ in the generalized $RQ$ factorization of the pair $\left(A,B\right)$ is singular, so that $\mathrm{rank}\left(A\right)<m$; the least squares solution could not be computed.

   $\mathbf{info}=2$

The bottom $\left(N-M\right)$ by $\left(N-M\right)$ part of the upper trapezoidal factor $T$ associated with $B$ in the generalized $QR$ factorization of the pair $\left(A,B\right)$ is singular, so that $\mathrm{rank}\left(\begin{array}{cc}A& B\end{array}\right)<N$; the least squares solutions could not be computed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08zb_example

function f08zb_example


fprintf('f08zb example results\n\n');

% Minimize ||y|| given Ax + By = d
a = [-0.57, -1.28, -0.39;
     -1.93,  1.08, -0.31;
      2.30,  0.24, -0.40;
     -0.02,  1.03, -1.43];
b = [ 0.5, 0, 0, 0;
      0,   1, 0, 0;
      0,   0, 2, 0;
      0,   0, 0, 5];
d = [ 1.32;
     -4.00;
      5.52;
      3.24];

[~, ~, ~, x, y, info] = f08zb( ...
                               a, b, d);

disp('Weighted least-squares solution');
fprintf('%12.4f',x);
fprintf('\n');
fprintf('Residual vector\n  ');
fprintf('%12.2e',y);
sqres = norm(y,2);
fprintf('\n\nSquare root of the residual sum of squares\n  %12.2e\n', ...
        sqres);

f08zb example results

Weighted least-squares solution
      1.9889     -1.0058     -2.9911
Residual vector
     -6.37e-04   -2.45e-03   -4.72e-03    7.70e-03

Square root of the residual sum of squares
      9.38e-03