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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 cAx2  subject to  Bx = d
x
minimize x c-Ax2  subject to  Bx=d
where AA is an mm by nn matrix, BB is a pp by nn matrix, cc is an mm element vector and dd is a pp element vector. It is assumed that pnm + ppnm+p, rank(B) = prank(B)=p and rank(E) = nrank(E)=n, where E =
(A)
B
E= A B . These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized RQRQ factorization of the matrices BB and AA.

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)max(1,m)
The second dimension of the array must be at least max (1,n)max(1,n)
The mm by nn matrix AA.
2:     b(ldb, : :) – double array
The first dimension of the array b must be at least max (1,p)max(1,p)
The second dimension of the array must be at least max (1,n)max(1,n)
The pp by nn matrix BB.
3:     c(m) – double array
m, the dimension of the array, must satisfy the constraint m0m0.
The right-hand side vector cc 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 + p0pnm+p.
The right-hand side vector dd 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.)
mm, the number of rows of the matrix AA.
Constraint: m0m0.
2:     n – int64int32nag_int scalar
Default: The second dimension of the arrays a, b.
nn, the number of columns of the matrices AA and BB.
Constraint: n0n0.
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.)
pp, the number of rows of the matrix BB.
Constraint: 0pnm + p0pnm+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)max(1,m)
The second dimension of the array will be max (1,n)max(1,n)
2:     b(ldb, : :) – double array
The first dimension of the array b will be max (1,p)max(1,p)
The second dimension of the array will be max (1,n)max(1,n)
3:     c(m) – double array
The residual sum of squares for the solution vector xx is given by the sum of squares of elements c(np + 1),c(np + 2),,c(m)cn-p+1,cn-p+2,,cm; the remaining elements are overwritten.
4:     d(p) – double array
5:     x(n) – double array
The solution vector xx of the LSE problem.
6:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii 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 = 1INFO=1
The upper triangular factor RR associated with BB in the generalized RQRQ factorization of the pair (B,A)(B,A) is singular, so that rank(B) < prank(B)<p; the least squares solution could not be computed.
  INFO = 2INFO=2
The (NP)(N-P) by (NP)(N-P) part of the upper trapezoidal factor TT associated with AA in the generalized RQRQ factorization of the pair (B,A)(B,A) is singular, so that the rank of the matrix (EE) comprising the rows of AA and BB is less than nn; 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 = pmn=p, the total number of floating point operations is approximately (2/3)n2(6m + n)23n2(6m+n); if pnpn, the number reduces to approximately (2/3)n2(3mn)23n2(3m-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 xx and the linear constraints BxBx.

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



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Chapter Introduction
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