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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgglse (f08za)

## Purpose

nag_lapack_dgglse (f08za) solves a real linear equality-constrained least squares problem.

## Syntax

[a, b, c, d, x, info] = f08za(a, b, c, d, 'm', m, 'n', n, 'p', p)
[a, b, c, d, x, info] = nag_lapack_dgglse(a, b, c, d, 'm', m, 'n', n, 'p', p)

## Description

nag_lapack_dgglse (f08za) solves the real linear equality-constrained least squares (LSE) problem
 minimize ‖c − Ax‖2  subject to  Bx = d x
$minimize x ‖c-Ax‖2 subject to Bx=d$
where A$A$ is an m$m$ by n$n$ matrix, B$B$ is a p$p$ by n$n$ matrix, c$c$ is an m$m$ element vector and d$d$ is a p$p$ element vector. It is assumed that pnm + p$p\le n\le m+p$, rank(B) = p$\mathrm{rank}\left(B\right)=p$ and rank(E) = n$\mathrm{rank}\left(E\right)=n$, where E =
 ( A ) B
$E=\left(\begin{array}{c}A\\ B\end{array}\right)$. These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized RQ$RQ$ factorization of the matrices B$B$ and A$A$.

## 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:     a(lda, : $:$) – double array
The first dimension of the array a must be at least max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The m$m$ by n$n$ matrix A$A$.
2:     b(ldb, : $:$) – double array
The first dimension of the array b must be at least max (1,p)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The p$p$ by n$n$ matrix B$B$.
3:     c(m) – double array
m, the dimension of the array, must satisfy the constraint m0${\mathbf{m}}\ge 0$.
The right-hand side vector c$c$ for the least squares part of the LSE problem.
4:     d(p) – double array
p, the dimension of the array, must satisfy the constraint 0pnm + p$0\le {\mathbf{p}}\le {\mathbf{n}}\le {\mathbf{m}}+{\mathbf{p}}$.
The right-hand side vector d$d$ for the equality constraints.

### Optional Input Parameters

1:     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$m$, the number of rows of the matrix A$A$.
Constraint: m0${\mathbf{m}}\ge 0$.
2:     n – int64int32nag_int scalar
Default: The second dimension of the arrays a, b.
n$n$, the number of columns of the matrices A$A$ and B$B$.
Constraint: n0${\mathbf{n}}\ge 0$.
3:     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$p$, the number of rows of the matrix B$B$.
Constraint: 0pnm + p$0\le {\mathbf{p}}\le {\mathbf{n}}\le {\mathbf{m}}+{\mathbf{p}}$.

### Input Parameters Omitted from the MATLAB Interface

lda ldb work lwork

### Output Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a will be max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
2:     b(ldb, : $:$) – double array
The first dimension of the array b will be max (1,p)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
3:     c(m) – double array
The residual sum of squares for the solution vector x$x$ is given by the sum of squares of elements c(np + 1),c(np + 2),,c(m)${\mathbf{c}}\left({\mathbf{n}}-{\mathbf{p}}+1\right),{\mathbf{c}}\left({\mathbf{n}}-{\mathbf{p}}+2\right),\dots ,{\mathbf{c}}\left({\mathbf{m}}\right)$; the remaining elements are overwritten.
4:     d(p) – double array
5:     x(n) – double array
The solution vector x$x$ of the LSE problem.
6:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ 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.
INFO = 1${\mathbf{INFO}}=1$
The upper triangular factor R$R$ associated with B$B$ in the generalized RQ$RQ$ factorization of the pair (B,A)$\left(B,A\right)$ is singular, so that rank(B) < p$\mathrm{rank}\left(B\right); the least squares solution could not be computed.
INFO = 2${\mathbf{INFO}}=2$
The (NP)$\left(N-P\right)$ by (NP)$\left(N-P\right)$ part of the upper trapezoidal factor T$T$ associated with A$A$ in the generalized RQ$RQ$ factorization of the pair (B,A)$\left(B,A\right)$ is singular, so that the rank of the matrix (E$E$) comprising the rows of A$A$ and B$B$ is less than n$n$; 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 mn = p$m\ge n=p$, the total number of floating point operations is approximately (2/3)n2(6m + n)$\frac{2}{3}{n}^{2}\left(6m+n\right)$; if pn$p\ll n$, the number reduces to approximately (2/3)n2(3mn)$\frac{2}{3}{n}^{2}\left(3m-n\right)$.
nag_opt_lsq_lincon_solve (e04nc) may also be used to solve LSE problems. It differs from nag_lapack_dgglse (f08za) in that it uses an iterative (rather than direct) method, and that it allows general upper and lower bounds to be specified for the variables x$x$ and the linear constraints Bx$Bx$.

## Example

```function nag_lapack_dgglse_example
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 = [1, 0, -1, 0;
0, 1, 0, -1];
c = [-1.5;
-2.14;
1.23;
-0.54;
-1.68;
0.82];
d = [0;
0];
[aOut, bOut, cOut, dOut, x, info] = nag_lapack_dgglse(a, b, c, d)
```
```

aOut =

3.3264   -0.2354    1.5296   -0.8479
0.3955    2.0364    0.8978   -1.3588
-0.4767    0.1207   -1.3372   -0.4674
0.4573   -0.2835   -0.2594   -2.6447
-0.0530    0.5095    0.3009   -0.0678
0.2560   -0.4662    0.3760    0.4784

bOut =

-0.4142         0    1.4142         0
0   -0.4142         0    1.4142

cOut =

0.6916
1.4107
-0.0052
-0.0153
-0.0126
-0.0144

dOut =

0
0

x =

0.4890
0.9975
0.4890
0.9975

info =

0

```
```function f08za_example
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 = [1, 0, -1, 0;
0, 1, 0, -1];
c = [-1.5;
-2.14;
1.23;
-0.54;
-1.68;
0.82];
d = [0;
0];
[aOut, bOut, cOut, dOut, x, info] = f08za(a, b, c, d)
```
```

aOut =

3.3264   -0.2354    1.5296   -0.8479
0.3955    2.0364    0.8978   -1.3588
-0.4767    0.1207   -1.3372   -0.4674
0.4573   -0.2835   -0.2594   -2.6447
-0.0530    0.5095    0.3009   -0.0678
0.2560   -0.4662    0.3760    0.4784

bOut =

-0.4142         0    1.4142         0
0   -0.4142         0    1.4142

cOut =

0.6916
1.4107
-0.0052
-0.0153
-0.0126
-0.0144

dOut =

0
0

x =

0.4890
0.9975
0.4890
0.9975

info =

0

```