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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgglse (f08zn)

## Purpose

nag_lapack_zgglse (f08zn) solves a complex linear equality-constrained least squares problem.

## Syntax

[a, b, c, d, x, info] = f08zn(a, b, c, d, 'm', m, 'n', n, 'p', p)
[a, b, c, d, x, info] = nag_lapack_zgglse(a, b, c, d, 'm', m, 'n', n, 'p', p)

## Description

nag_lapack_zgglse (f08zn) solves the complex linear equality-constrained least squares (LSE) problem
 $minimize x c-Ax2 subject to Bx=d$
where $A$ is an $m$ by $n$ matrix, $B$ is a $p$ by $n$ matrix, $c$ is an $m$ element vector and $d$ is a $p$ element vector. It is assumed that $p\le n\le m+p$, $\mathrm{rank}\left(B\right)=p$ and $\mathrm{rank}\left(\mathrm{E}\right)=n$, where $E=\left(\begin{array}{c}A\\ B\end{array}\right)$. These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized $RQ$ factorization of the matrices $B$ and $A$.

## References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
Anderson E, Bai Z and Dongarra J (1992) Generalized QR factorization and its applications Linear Algebra Appl. (Volume 162–164) 243–271
Eldèn L (1980) Perturbation theory for the least squares problem with linear equality constraints SIAM J. Numer. Anal. 17 338–350

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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); 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

For an error analysis, see Anderson et al. (1992) and Eldèn (1980). See also Section 4.6 of Anderson et al. (1999).

When $m\ge n=p$, the total number of real floating-point operations is approximately $\frac{8}{3}{n}^{2}\left(6m+n\right)$; if $p\ll n$, the number reduces to approximately $\frac{8}{3}{n}^{2}\left(3m-n\right)$.

## Example

This example solves the least squares problem
 $minimize x c-Ax2 subject to Bx=d$
where
 $c = -2.54+0.09i 1.65-2.26i -2.11-3.96i 1.82+3.30i -6.41+3.77i 2.07+0.66i ,$
and
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i 0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i ,$
 $B = 1.0+0.0i 0.0i+0.0 -1.0+0.0i 0.0i+0.0 0.0i+0.0 1.0+0.0i 0.0i+0.0 -1.0+0.0i$
and
 $d = 0 0 .$
The constraints $Bx=d$ correspond to ${x}_{1}={x}_{3}$ and ${x}_{2}={x}_{4}$.
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
```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
```