nag_dggglm (f08zbc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG 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, 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×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: pdamax1,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 generalized 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 generalized 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  Parallelism and Performance

nag_dggglm (f08zbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dggglm (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 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 232m3-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)


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

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