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NAG Toolbox

NAG Toolbox: nag_lapack_dgglse (f08za)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

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 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 pnm+p, rankB=p and rankE=n, where E= A B . 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:     alda: – double array
The first dimension of the array a must be at least max1,m.
The second dimension of the array a must be at least max1,n.
The m by n matrix A.
2:     bldb: – double array
The first dimension of the array b must be at least max1,p.
The second dimension of the array b must be at least max1,n.
The p by n matrix B.
3:     cm – double array
The right-hand side vector c for the least squares part of the LSE problem.
4:     dp – 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: m0.
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: n0.
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: 0pnm+p.

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,m.
The second dimension of the array a will be max1,n.
2:     bldb: – double array
The first dimension of the array b will be max1,p.
The second dimension of the array b will be max1,n.
3:     cm – double array
The residual sum of squares for the solution vector x is given by the sum of squares of elements cn-p+1,cn-p+2,,cm; the remaining elements are overwritten.
4:     dp – double array
5:     xn – 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 RQ factorization of the pair B,A is singular, so that rankB<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 RQ 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 mn=p, the total number of floating-point operations is approximately 23n26m+n; if pn, the number reduces to approximately 23n23m-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 Bx.

Example

This example solves the least squares problem
minimize x c-Ax2   subject to   Bx=d  
where
c = -1.50 -2.14 1.23 -0.54 -1.68 0.82 ,  
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 ,  
B = 1.0 0 -1.0 0 0 1.0 0 -1.0  
and
d = 0 0 .  
The constraints Bx=d  correspond to x1 = x3  and x2 = x4 .
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


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