nag_linf_fit (e02gcc) (PDF version)
e02 Chapter Contents
e02 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_linf_fit (e02gcc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_linf_fit (e02gcc) calculates an l solution to an over-determined system of linear equations.

2  Specification

#include <nag.h>
#include <nage02.h>
void  nag_linf_fit (Nag_OrderType order, Integer m, Integer n, double a[], double b[], double tol, double *relerr, double x[], double *resmax, Integer *rank, Integer *iter, NagError *fail)

3  Description

Given a matrix A with m rows and n columns mn and a vector b with m elements, the function calculates an l solution to the over-determined system of equations
Ax=b.
That is to say, it calculates a vector x, with n elements, which minimizes the l norm of the residuals (the absolutely largest residual)
rx = max 1im ri
where the residuals ri are given by
ri = bi - j=1n aij xj ,   i=1,2,,m .
Here aij is the element in row i and column j of A, bi is the ith element of b and xj the jth element of x. The matrix A need not be of full rank. The solution is not unique in this case, and may not be unique even if A is of full rank.
Alternatively, in applications where a complete minimization of the l norm is not necessary, you may obtain an approximate solution, usually in shorter time, by giving an appropriate value to the argument relerr.
Typically in applications to data fitting, data consisting of m points with coordinates ti,yi is to be approximated in the l norm by a linear combination of known functions ϕjt,
α1ϕ1t+α2ϕ2t++αnϕnt.
This is equivalent to finding an l solution to the over-determined system of equations
j=1n ϕj ti αj = yi ,   i=1,2,,m .
Thus if, for each value of i and j the element aij of the matrix A above is set equal to the value of ϕjti and bi is set equal to yi, the solution vector x will contain the required values of the αj. Note that the independent variable t above can, instead, be a vector of several independent variables (this includes the case where each ϕi is a function of a different variable, or set of variables).
The algorithm is a modification of the simplex method of linear programming applied to the dual formation of the l problem (see Barrodale and Phillips (1974) and Barrodale and Phillips (1975)). The modifications are designed to improve the efficiency and stability of the simplex method for this particular application.

4  References

Barrodale I and Phillips C (1974) An improved algorithm for discrete Chebyshev linear approximation Proc. 4th Manitoba Conf. Numerical Mathematics 177–190 University of Manitoba, Canada
Barrodale I and Phillips C (1975) Solution of an overdetermined system of linear equations in the Chebyshev norm [F4] (Algorithm 495) ACM Trans. Math. Software 1(3) 264–270

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: the number of equations, m (the number of rows of the matrix A).
Constraint: mn.
3:     nIntegerInput
On entry: the number of unknowns, n (the number of columns of the matrix A).
Constraint: n1.
4:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least n+3×m+1.
Where Aj,i appears in this document, it refers to the array element
  • a[i-1×n+3+j-1] when order=Nag_ColMajor;
  • a[j-1×m+1+i-1] when order=Nag_RowMajor.
On entry: Aj,i must contain aij, the element in the ith row and jth column of the matrix A, for i=1,2,,m and j=1,2,,n, (that is, the transpose of the matrix). The remaining elements need not be set. Preferably, the columns of the matrix A (rows of the argument a) should be scaled before entry: see Section 7.
On exit: contains the last simplex tableau.
5:     b[m]doubleInput/Output
On entry: b[i-1] must contain bi, the ith element of the vector b, for i=1,2,,m.
On exit: the ith residual ri corresponding to the solution vector x, for i=1,2,,m. Note however that these residuals may contain few significant figures, especially when resmax is within one or two orders of magnitude of tol. Indeed if resmaxtol, the elements b[i-1] may all be set to zero. It is therefore often advisable to compute the residuals directly.
6:     toldoubleInput
On entry: a threshold below which numbers are regarded as zero. The recommended threshold value is 10.0×ε, where ε is the machine precision. If tol0.0 on entry, the recommended value is used within the function. If premature termination occurs, a larger value for tol may result in a valid solution.
Suggested value: 0.0.
7:     relerrdouble *Input/Output
On entry: must be set to a bound on the relative error acceptable in the maximum residual at the solution.
If relerr0.0, then the l solution is computed, and relerr is set to 0.0 on exit.
If relerr>0.0, then the function obtains instead an approximate solution for which the largest residual is less than 1.0+relerr times that of the l solution; on exit, relerr contains a smaller value such that the above bound still applies. (The usual result of this option, say with relerr=0.1, is a saving in the number of simplex iterations).
On exit: is altered as described above.
8:     x[n]doubleOutput
On exit: if an optimal but not necessarily unique solution is found, x[j-1] contains the jth element of the solution vector x, for j=1,2,,n. Whether this is an l solution or an approximation to one, depends on the value of relerr on entry.
9:     resmaxdouble *Output
On exit: if an optimal but not necessarily unique solution is found, resmax contains the absolute value of the largest residual(s) for the solution vector x. (See b.)
10:   rankInteger *Output
On exit: if an optimal but not necessarily unique solution is found, rank contains the computed rank of the matrix A.
11:   iterInteger *Output
On exit: if an optimal but not necessarily unique solution is found, iter contains the number of iterations taken by the simplex method.
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, n=value.
Constraint: n1.
NE_INT_2
On entry, m=value and n=value.
Constraint: mn.
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_NON_UNIQUE
An optimal solution has been obtained, but may not be unique.
NE_TERMINATION_FAILURE
Premature termination due to rounding errors. Try using larger value of tol: tol=value.

7  Accuracy

Experience suggests that the computational accuracy of the solution x is comparable with the accuracy that could be obtained by applying Gaussian elimination with partial pivoting to the n+1 equations which have residuals of largest absolute value. The accuracy therefore varies with the conditioning of the problem, but has been found generally very satisfactory in practice.

8  Further Comments

The effects of m and n on the time and on the number of iterations in the simplex method vary from problem to problem, but typically the number of iterations is a small multiple of n and the total time is approximately proportional to mn2.
It is recommended that, before the function is entered, the columns of the matrix A are scaled so that the largest element in each column is of the order of unity. This should improve the conditioning of the matrix, and also enable the argument tol to perform its correct function. The solution x obtained will then, of course, relate to the scaled form of the matrix. Thus if the scaling is such that, for each j=1,2,,n, the elements of the jth column are multiplied by the constant kj, the element xj of the solution vector x must be multiplied by kj if it is desired to recover the solution corresponding to the original matrix A.

9  Example

This example approximates a set of data by a curve of the form
y=Ket+Le-t+M
where K, L and M are unknown. Given values yi at 5 points ti we may form the over-determined set of equations for K, L and M 
etiK+e-tiL+M=yi,  i=1,2,,5.
nag_linf_fit (e02gcc) is used to solve these in the l sense.

9.1  Program Text

Program Text (e02gcce.c)

9.2  Program Data

Program Data (e02gcce.d)

9.3  Program Results

Program Results (e02gcce.r)


nag_linf_fit (e02gcc) (PDF version)
e02 Chapter Contents
e02 Chapter Introduction
NAG C Library Manual

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