NAG CL Interface
f08zbc (dggglm)

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1 Purpose

f08zbc solves a real general Gauss–Markov linear (least squares) model problem.

2 Specification

#include <nag.h>
void  f08zbc (Nag_OrderType order, Integer m, Integer n, Integer p, double a[], Integer pda, double b[], Integer pdb, double d[], double x[], double y[], NagError *fail)
The function may be called by the names: f08zbc, nag_lapackeig_dggglm or nag_dggglm.

3 Description

f08zbc solves the real general Gauss–Markov linear model (GLM) problem
minimize x y2  subject to  d=Ax+By  
where A is an m×n matrix, B is an m×p matrix and d is an m element vector. It is assumed that nmn+p, rank(A)=n and rank(E)=m, where E= ( A B ) . Under these assumptions, the problem has a unique solution x and a minimal 2-norm solution y, which is obtained using a generalized QR factorization of the matrices A and B.
In particular, if the matrix B is square and nonsingular, then the GLM problem is equivalent to the weighted linear least squares problem
minimize x B-1(d-Ax)2 .  

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

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 matrices A and B.
Constraint: m0.
3: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: 0nm.
4: p Integer Input
On entry: p, the number of columns of the matrix B.
Constraint: pm-n.
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×p) when order=Nag_ColMajor;
  • max(1,m×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 m×p 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,m);
  • if order=Nag_RowMajor, pdbmax(1,p).
9: d[m] double Input/Output
On entry: the left-hand side vector d of the GLM equation.
On exit: d is overwritten.
10: x[n] double Output
On exit: the solution vector x of the GLM problem.
11: y[p] double Output
On exit: the solution vector y of the GLM 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, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, m=value and n=value.
Constraint: 0nm.
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 m=value.
Constraint: pdbmax(1,m).
On entry, pdb=value and p=value.
Constraint: pdbmax(1,p).
NE_INT_3
On entry, p=value, m=value and n=value.
Constraint: pm-n.
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 bottom (m-n)×(m-n) part of the upper trapezoidal factor T associated with B in the generalized QR factorization of the pair (A,B) is singular, so that rank(AB)<n; the least squares solutions could not be computed.
The upper triangular factor R associated with A in the generalized RQ factorization of the pair (A,B) is singular, so that rank(A)<n; the least squares solution could not be computed.

7 Accuracy

For an error analysis, see Anderson et al. (1992). 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.
f08zbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08zbc 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 p=mn, the total number of floating-point operations is approximately 23(2m3-n3)+4nm2; when p=m=n, the total number of floating-point operations is approximately 143m3.

10 Example

This example solves the weighted least squares problem
minimize x B-1(d-Ax)2 ,  
where
B = ( 0.5 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 2.0 0.0 0.0 0.0 0.0 5.0 ) ,   d= ( 1.32 -4.00 5.52 3.24 )   and   A= ( -0.57 -1.28 -0.39 -1.93 1.08 -0.31 2.30 0.24 -0.40 -0.02 1.03 -1.43 ) .  

10.1 Program Text

Program Text (f08zbce.c)

10.2 Program Data

Program Data (f08zbce.d)

10.3 Program Results

Program Results (f08zbce.r)