nag_dgglse (f08zac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dgglse (f08zac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgglse (f08zac) solves a real linear equality-constrained least squares problem.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dgglse (Nag_OrderType order, Integer m, Integer n, Integer p, double a[], Integer pda, double b[], Integer pdb, double c[], double d[], double x[], NagError *fail)

3  Description

nag_dgglse (f08zac) 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.

4  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

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3:     nIntegerInput
On entry: n, the number of columns of the matrices A and B.
Constraint: n0.
4:     pIntegerInput
On entry: p, the number of rows of the matrix B.
Constraint: 0pnm+p.
5:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least
  • max1,pda×n when order=Nag_ColMajor;
  • max1,m×pda when order=Nag_RowMajor.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the m by n matrix A.
On exit: a is overwritten.
6:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if order=Nag_ColMajor, pda max1,m ;
  • if order=Nag_RowMajor, pdamax1,n.
7:     b[dim]doubleInput/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×n when order=Nag_ColMajor;
  • max1,p×pdb when order=Nag_RowMajor.
The i,jth element of the matrix B is stored in
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the p by n matrix B.
On exit: b is overwritten.
8:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,p;
  • if order=Nag_RowMajor, pdbmax1,n.
9:     c[m]doubleInput/Output
On entry: the right-hand side vector c for the least squares part of the LSE problem.
On exit: the residual sum of squares for the solution vector x is given by the sum of squares of elements c[n-p],c[n-p+1],,c[m-1]; the remaining elements are overwritten.
10:   d[p]doubleInput/Output
On entry: the right-hand side vector d for the equality constraints.
On exit: d is overwritten.
11:   x[n]doubleOutput
On exit: the solution vector x of the LSE problem.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pda=value and m=value.
Constraint: pda max1,m .
On entry, pda=value and n=value.
Constraint: pdamax1,n.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and p=value.
Constraint: pdbmax1,p.
NE_INT_3
On entry, p=value, m=value and n=value.
Constraint: 0pnm+p.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_SINGULAR
The N-P by N-P part of the upper trapezoidal factor T associated with A in the generalised 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.
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.

7  Accuracy

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

8  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_lin_lsq (e04ncc) may also be used to solve LSE problems. It differs from nag_dgglse (f08zac) 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.

9  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 .

9.1  Program Text

Program Text (f08zace.c)

9.2  Program Data

Program Data (f08zace.d)

9.3  Program Results

Program Results (f08zace.r)


nag_dgglse (f08zac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012