NAG FL Interface
f08zaf (dgglse)

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1 Purpose

f08zaf solves a real linear equality-constrained least squares problem.

2 Specification

Fortran Interface
Subroutine f08zaf ( m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)
Integer, Intent (In) :: m, n, p, lda, ldb, lwork
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), c(m), d(p)
Real (Kind=nag_wp), Intent (Out) :: x(n), work(max(1,lwork))
C Header Interface
#include <nag.h>
void  f08zaf_ (const Integer *m, const Integer *n, const Integer *p, double a[], const Integer *lda, double b[], const Integer *ldb, double c[], double d[], double x[], double work[], const Integer *lwork, Integer *info)
The routine may be called by the names f08zaf, nagf_lapackeig_dgglse or its LAPACK name dgglse.

3 Description

f08zaf 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: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
2: n Integer Input
On entry: n, the number of columns of the matrices A and B.
Constraint: n0.
3: p Integer Input
On entry: p, the number of rows of the matrix B.
Constraint: 0pnm+p.
4: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the m×n matrix A.
On exit: a is overwritten.
5: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08zaf is called.
Constraint: ldamax(1,m).
6: b(ldb,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least max(1,n).
On entry: the p×n matrix B.
On exit: b is overwritten.
7: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f08zaf is called.
Constraint: ldbmax(1,p).
8: c(m) Real (Kind=nag_wp) array 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+1),c(n-p+2),,c(m); the remaining elements are overwritten.
9: d(p) Real (Kind=nag_wp) array Input/Output
On entry: the right-hand side vector d for the equality constraints.
On exit: d is overwritten.
10: x(n) Real (Kind=nag_wp) array Output
On exit: the solution vector x of the LSE problem.
11: work(max(1,lwork)) Real (Kind=nag_wp) array Workspace
On exit: if info=0, work(1) contains the minimum value of lwork required for optimal performance.
12: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08zaf is called.
If lwork=−1, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, lworkp+min(m,n)+max(m,n)×nb, where nb is the optimal block size.
Constraint: lwork max(1,m+n+p) or lwork=−1.
13: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
info=1
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.
info=2
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.

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

Background information to multithreading can be found in the Multithreading documentation.
f08zaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08zaf 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 routine. 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).
e04ncf/​e04nca may also be used to solve LSE problems. It differs from f08zaf 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 (f08zafe.f90)

10.2 Program Data

Program Data (f08zafe.d)

10.3 Program Results

Program Results (f08zafe.r)