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 $\max (1, m)$ .
The second dimension of the array a must be at least $\max (1, n)$ .

The $m$ by $n$ matrix $A$ .
2: $b (ldb :)$ – double array: The first dimension of the array b must be at least $\max (1, p)$ .
The second dimension of the array b must be at least $\max (1, n)$ .

The $p$ by $n$ matrix $B$ .
3: $c (m)$ – double array: The right-hand side vector $c$ for the least squares part of the LSE problem.
4: $d (p)$ – double array: The right-hand side vector $d$ 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.)
$m$ , the number of rows of the matrix $A$ .

Constraint: $m \geq 0$ .
2: $n$ – int64int32nag_int scalar: Default: the second dimension of the arrays a, b.
$n$ , the number of columns of the matrices $A$ and $B$ .

Constraint: $n \geq 0$ .
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.)
$p$ , the number of rows of the matrix $B$ .

Constraint: $0 \leq p \leq n \leq m + p$ .

Output Parameters

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

Error Indicators and Warnings

$info = - i$: If $info = - i$ , 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.

$info = 1$: The upper triangular factor $R$ associated with $B$ in the generalized $R Q$ factorization of the pair $(B, A)$ is singular, so that $rank (B) < p$ ; the least squares solution could not be computed.

$info = 2$: The $(N - P)$ by $(N - P)$ part of the upper trapezoidal factor $T$ associated with $A$ in the generalized $R Q$ factorization of the pair $(B, A)$ 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).

Further Comments

When

m \geq n = p

, the total number of floating-point operations is approximately

\frac{2}{3} n^{2} (6 m + n)

; if

p ≪ n

, the number reduces to approximately

\frac{2}{3} n^{2} (3 m - n)

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

and the linear constraints

B x

Example

This example solves the least squares problem

\underset{x}{minimize} {‖c - A x‖}_{2} subject to B x = d

where

c = (\begin{array}{r} - 1.50 \\ - 2.14 \\ 1.23 \\ - 0.54 \\ - 1.68 \\ 0.82 \end{array}),

A = (\begin{array}{r} - 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 \end{array}),

B = (\begin{matrix} 1.0 & 0 & - 1.0 & 0 \\ 0 & 1.0 & 0 & - 1.0 \end{matrix})

and

d = (\begin{matrix} 0 \\ 0 \end{matrix}) .

The constraints

B x = d

correspond to

x_{1} = x_{3}

and

x_{2} = x_{4}

Open in the MATLAB editor: f08za_example

function f08za_example


fprintf('f08za example results\n\n');

% Minimize ||c - Ax|| given Bx=d
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];
c = [-1.50; -2.14;  1.23; -0.54;  -1.68;  0.82];
b = [ 1,     0,    -1,     0;
      0,     1,     0,    -1];
d = [ 0;
      0];

%
[~, ~, resid, ~, x, info] = f08za( ...
                                   a, b, c, d);

sqres = norm(resid(3:6),2);
disp('Constrained least-squares solution');
disp(x);
disp('Square root of the residual sum of squares');
disp(sqres);

f08za example results

Constrained least-squares solution
    0.4890
    0.9975
    0.4890
    0.9975

Square root of the residual sum of squares
    0.0251

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox