nag_dgglse (f08zac) (PDF version)
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NAG 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, pdamax1,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: pdamax1,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 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.
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  Parallelism and Performance

nag_dgglse (f08zac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dgglse (f08zac) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the Users' Note for your implementation for any additional implementation-specific information.

9  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.

10  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 .

10.1  Program Text

Program Text (f08zace.c)

10.2  Program Data

Program Data (f08zace.d)

10.3  Program Results

Program Results (f08zace.r)


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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014