nag_dgels (f08aac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dgels (f08aac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgels (f08aac) solves linear least squares problems of the form
minx b-Ax2   or   minx b-ATx2 ,
where A is an m by n real matrix of full rank, using a QR or LQ factorization of A.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dgels (Nag_OrderType order, Nag_TransType trans, Integer m, Integer n, Integer nrhs, double a[], Integer pda, double b[], Integer pdb, NagError *fail)

3  Description

The following options are provided:
  1. If trans=Nag_NoTrans and mn: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
    minx b-Ax2 .
  2. If trans=Nag_NoTrans and m<n: find the minimum norm solution of an underdetermined system Ax=b.
  3. If trans=Nag_Trans and mn: find the minimum norm solution of an undetermined system ATx=b.
  4. If trans=Nag_Trans and m<n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
    minx b-ATx2 .
Several right-hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the m by r right-hand side matrix B and the n by r solution matrix X.

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 http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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:     transNag_TransTypeInput
On entry: if trans=Nag_NoTrans, the linear system involves A.
If trans=Nag_Trans, the linear system involves AT.
Constraint: trans=Nag_NoTrans or Nag_Trans.
3:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
4:     nIntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
5:     nrhsIntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrices B and X.
Constraint: nrhs0.
6:     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: if mn, a is overwritten by details of its QR factorization as returned by nag_dgeqrf (f08aec).
If m<n, a is overwritten by details of its LQ factorization as returned by nag_dgelqf (f08ahc).
7:     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.
8:     b[dim]doubleInput/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×nrhs when order=Nag_ColMajor;
  • max1,max1,m,n×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 matrix B of right-hand side vectors, stored in rows or columns; b is m by r if trans=Nag_NoTrans, or n by r if trans=Nag_Trans.
On exit: b is overwritten by the solution vectors, x, stored in rows or columns:
  • if trans=Nag_NoTrans and mn, or trans=Nag_Trans and m<n, elements 1 to minm,n in each column of b contain the least squares solution vectors; the residual sum of squares for the solution is given by the sum of squares of the modulus of elements minm,n+1  to maxm,n in that column;
  • otherwise, elements 1 to maxm,n in each column of b contain the minimum norm solution vectors.
9:     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,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
10:   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_FULL_RANK
Diagonal element value of the triangular factor of A is zero, so that A does not have full rank; the least squares solution could not be computed.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
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: pdamax1,m.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
NE_INT_3
On entry, pdb=value, m=value and n=value.
Constraint: pdbmax1,m,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.

7  Accuracy

See Section 4.5 of Anderson et al. (1999) for details of error bounds.

8  Further Comments

The total number of floating point operations required to factorize A is approximately 23 n2 3m-n  if mn and 23 m2 3n-m  otherwise. Following the factorization the solution for a single vector x requires O minm2,n2  operations.
The complex analogue of this function is nag_zgels (f08anc).

9  Example

This example solves the linear least squares problem
minx b-Ax2 ,
where
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   and   b= -2.67 -0.55 3.34 -0.77 0.48 4.10 .
The square root of the residual sum of squares is also output.

9.1  Program Text

Program Text (f08aace.c)

9.2  Program Data

Program Data (f08aace.d)

9.3  Program Results

Program Results (f08aace.r)


nag_dgels (f08aac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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