NAG CL Interface
f08zac (dgglse)

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1 Purpose

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

2 Specification

#include <nag.h>
void  f08zac (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)
The function may be called by the names: f08zac, nag_lapackeig_dgglse or nag_dgglse.

3 Description

f08zac solves the real linear equality-constrained least squares (LSE) problem
minimize x c-Ax2  subject to  Bx=d  
where A is an m×n matrix, B is a p×n matrix, c is an m element vector and d is a p element vector. It is assumed that pnm+p, rank(B)=p and rank(E)=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: order Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3: n Integer Input
On entry: n, the number of columns of the matrices A and B.
Constraint: n0.
4: p Integer Input
On entry: p, the number of rows of the matrix B.
Constraint: 0pnm+p.
5: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least
  • max(1,pda×n) when order=Nag_ColMajor;
  • max(1,m×pda) when order=Nag_RowMajor.
The (i,j)th 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×n matrix A.
On exit: a is overwritten.
6: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if order=Nag_ColMajor, pdamax(1,m);
  • if order=Nag_RowMajor, pdamax(1,n).
7: b[dim] double Input/Output
Note: the dimension, dim, of the array b must be at least
  • max(1,pdb×n) when order=Nag_ColMajor;
  • max(1,p×pdb) when order=Nag_RowMajor.
The (i,j)th 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×n matrix B.
On exit: b is overwritten.
8: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax(1,p);
  • if order=Nag_RowMajor, pdbmax(1,n).
9: c[m] double Input/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] double Input/Output
On entry: the right-hand side vector d for the equality constraints.
On exit: d is overwritten.
11: x[n] double Output
On exit: the solution vector x of the LSE problem.
12: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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: pdamax(1,m).
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
On entry, pdb=value and p=value.
Constraint: pdbmax(1,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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
The (N-P)×(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 rank(B)<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

f08zac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also 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 23n2(6m+n); if pn, the number reduces to approximately 23n2(3m-n).
e04ncc may also be used to solve LSE problems. It differs from 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)