nag_dggglm (f08zbc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dggglm (f08zbc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dggglm (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)

3  Description

nag_dggglm (f08zbc) solves the real general Gauss–Markov linear model (GLM) problem
minimize x y2  subject to  d=Ax+By
where A is an m by n matrix, B is an m by p matrix and d is an m element vector. It is assumed that nmn+p, rankA=n and rankE=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:     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 matrices A and B.
Constraint: m0.
3:     nIntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: 0nm.
4:     pIntegerInput
On entry: p, the number of columns of the matrix B.
Constraint: pm-n.
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×p when order=Nag_ColMajor;
  • max1,m×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 m by p 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,m;
  • if order=Nag_RowMajor, pdbmax1,p.
9:     d[m]doubleInput/Output
On entry: the left-hand side vector d of the GLM equation.
On exit: d is overwritten.
10:   x[n]doubleOutput
On exit: the solution vector x of the GLM problem.
11:   y[p]doubleOutput
On exit: the solution vector y of the GLM 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, 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: pda max1,m .
On entry, pda=value and n=value.
Constraint: pdamax1,n.
On entry, pdb=value and m=value.
Constraint: pdbmax1,m.
On entry, pdb=value and p=value.
Constraint: pdbmax1,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.
NE_SINGULAR
The bottom N-M by N-M part of the upper trapezoidal factor T associated with B in the generalised QR factorization of the pair A,B is singular, so that rankAB<n; the least squares solutions could not be computed.
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 rankBA<n; the least squares solutions 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  Further Comments

When p=mn, the total number of floating point operations is approximately 232m3-n3+4nm2; when p=m=n, the total number of floating point operations is approximately 143m3.

9  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 .

9.1  Program Text

Program Text (f08zbce.c)

9.2  Program Data

Program Data (f08zbce.d)

9.3  Program Results

Program Results (f08zbce.r)


nag_dggglm (f08zbc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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