hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_1d_minimax_polynomial (e02al)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_1d_minimax_polynomial (e02al) calculates a minimax polynomial fit to a set of data points.


[a, ref, ifail] = e02al(x, y, m, 'n', n)
[a, ref, ifail] = nag_1d_minimax_polynomial(x, y, m, 'n', n)


Given a set of data points xi,yi, for i=1,2,,n, nag_1d_minimax_polynomial (e02al) uses the exchange algorithm to compute an mth-degree polynomial
Px = a0 + a1x + a2 x2 + + am xm  
such that maxiPxi-yi is a minimum.
The function also returns a number whose absolute value is the final reference deviation (see Arguments). The function is an adaptation of Boothroyd (1967).


Boothroyd J B (1967) Algorithm 318 Comm. ACM 10 801
Stieffel E (1959) Numerical methods of Tchebycheff approximation On Numerical Approximation (ed R E Langer) 217–232 University of Wisconsin Press


Compulsory Input Parameters

1:     xn – double array
The values of the x coordinates, xi, for i=1,2,,n.
Constraint: x1<x2<<xn.
2:     yn – double array
The values of the y coordinates, yi, for i=1,2,,n.
3:     m int64int32nag_int scalar
m, where m is the degree of the polynomial to be found.
Constraint: 0m<min100,n-1.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the arrays x, y. (An error is raised if these dimensions are not equal.)
n, the number of data points.
Constraint: n1.

Output Parameters

1:     am+1 – double array
The coefficients ai of the minimax polynomial, for i=0,1,,m.
2:     ref – double scalar
The final reference deviation, i.e., the maximum deviation of the computed polynomial evaluated at xi from the reference values yi, for i=1,2,,n. ref may return a negative value which indicates that the algorithm started to cycle due to round-off errors.
3:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
Constraint: n1.
Constraint: m<100.
Constraint: m<n-1.
Constraint: m0.
Constraint: xi+1>xi.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


This is dependent on the given data points and on the degree of the polynomial. The data points should represent a fairly smooth function which does not contain regions with markedly different behaviours. For large numbers of data points (n>100, say), rounding error will affect the computation regardless of the quality of the data; in this case, relatively small degree polynomials (mn) may be used when this is consistent with the required approximation. A limit of 99 is placed on the degree of polynomial since it is known from experiment that a complete loss of accuracy often results from using such high degree polynomials in this form of the algorithm.

Further Comments

The time taken increases with m.


This example calculates a minimax fit with a polynomial of degree 5 to the exponential function evaluated at 21 points over the interval 0,1. It then prints values of the function and the fitted polynomial.
function e02al_example

fprintf('e02al example results\n\n');

% Minimax polynomial of degree 5 that fits exp(x) on grid in [0,1].
x = [0:0.05:1];
y = exp(x);
m = int64(5);
[a, ref, ifail] = e02al(x, y, m);

fprintf('   Polynomial coefficients\n');
fprintf('        %7.4f\n',a(1:m+1));
fprintf('\n   Reference deviation = %8.2e\n\n', ref);
fprintf('   x     Fit      exp(x)   Residual\n');

xx = [0:0.1:1];
p  = a(m+1:-1:1);
s  = polyval(p,xx);
yy = exp(xx);

for i=1:11
  fprintf('%6.2f%9.4f%9.4f%11.2e\n',xx(i), s(i), yy(i), s(i) - yy(i));

e02al example results

   Polynomial coefficients

   Reference deviation = 1.09e-06

   x     Fit      exp(x)   Residual
  0.00   1.0000   1.0000  -1.09e-06
  0.10   1.1052   1.1052   9.74e-07
  0.20   1.2214   1.2214  -7.44e-07
  0.30   1.3499   1.3499  -9.18e-07
  0.40   1.4918   1.4918   2.99e-07
  0.50   1.6487   1.6487   1.09e-06
  0.60   1.8221   1.8221   4.59e-07
  0.70   2.0138   2.0138  -8.16e-07
  0.80   2.2255   2.2255  -8.42e-07
  0.90   2.4596   2.4596   8.75e-07
  1.00   2.7183   2.7183  -1.09e-06

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015