NAG Toolbox: nag_lapack_zgglse (f08zn)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array

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

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

The $p$ by $n$ matrix $B$.

3:     $\mathrm{c}\left(\mathbf{m}\right)$ – complex array

The right-hand side vector $c$ for the least squares part of the LSE problem.

4:     $\mathrm{d}\left(\mathbf{p}\right)$ – complex array

The right-hand side vector $d$ for the equality constraints.

Optional Input Parameters

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

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

$m$, 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 arrays a, b.

$n$, the number of columns of the matrices $A$ and $B$.

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

3:     $\mathrm{p}$ – 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.)

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

Constraint: $0\le \mathbf{p}\le \mathbf{n}\le \mathbf{m}+\mathbf{p}$.

Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array

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

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

3:     $\mathrm{c}\left(\mathbf{m}\right)$ – complex array

The residual sum of squares for the solution vector $x$ is given by the sum of squares of elements $\mathbf{c}\left(\mathbf{n}-\mathbf{p}+1\right),\mathbf{c}\left(\mathbf{n}-\mathbf{p}+2\right),\dots ,\mathbf{c}\left(\mathbf{m}\right)$; the remaining elements are overwritten.

4:     $\mathrm{d}\left(\mathbf{p}\right)$ – complex array

5:     $\mathrm{x}\left(\mathbf{n}\right)$ – complex array

The solution vector $x$ of the LSE 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: c, 9: d, 10: x, 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 $B$ in the generalized $RQ$ factorization of the pair $\left(B,A\right)$ is singular, so that $\mathrm{rank}\left(B\right)<p$; the least squares solution could not be computed.

   $\mathbf{info}=2$

The $\left(N-P\right)$ by $\left(N-P\right)$ part of the upper trapezoidal factor $T$ associated with $A$ in the generalized $RQ$ factorization of the pair $\left(B,A\right)$ is singular, so that the rank of the matrix ($E$) comprising the rows of $A$ and $B$ is less than $n$; the least squares solutions could not be computed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08zn_example

function f08zn_example


fprintf('f08zn example results\n\n');

% Solve the equality-constrained least-squares problem
% minimize ||c - A*x|| (in the 2-norm) subject to B*x = D

a = [ 0.96 - 0.81i, -0.03 + 0.96i, -0.91 + 2.06i, -0.05 + 0.41i;
     -0.98 + 1.98i, -1.20 + 0.19i, -0.66 + 0.42i, -0.81 + 0.56i;
      0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i, -1.11 + 0.60i;
      0.37 + 0.38i,  0.19 - 0.54i, -0.98 - 0.36i,  0.22 - 0.20i;
      0.83 + 0.51i,  0.20 + 0.01i, -0.17 - 0.46i,  1.47 + 1.59i;
      1.08 - 0.28i,  0.20 - 0.12i, -0.07 + 1.23i,  0.26 + 0.26i];
b = complex([1,  0, -1,  0;
             0,  1,  0, -1]);
c = [-2.54 + 0.09i;
      1.65 - 2.26i;
     -2.11 - 3.96i;
      1.82 + 3.30i;
     -6.41 + 3.77i;
      2.07 + 0.66i];
d = [complex(0);
      0 + 0i];

%Solve
[~, ~, c, ~, x, info] = f08zn( ...
                               a, b, c, d);

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

rnorm = norm(c(3:6));
fprintf('Square root of the residual sum of squares\n%11.2e\n',rnorm);

f08zn example results


Constrained least-squares solution
   1.0874 - 1.9621i
  -0.7409 + 3.7297i
   1.0874 - 1.9621i
  -0.7409 + 3.7297i

Square root of the residual sum of squares
   1.59e-01