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NAG Toolbox: nag_fit_withdraw_1dmmax (e02ac)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_fit_1dmmax (e02ac) calculates a minimax polynomial fit to a set of data points.

Syntax

[a, ref] = e02ac(x, y, m1, 'n', n)
[a, ref] = nag_fit_withdraw_1dmmax(x, y, m1, 'n', n)

Description

Given a set of data points xi,yi, for i=1,2,,n, nag_fit_1dmmax (e02ac) uses the exchange algorithm to compute an mth-order polynomial
Px=a1+a2x+a3x2++am+1xm  
such that maxi2Pxi-yi is a minimum.
The function also returns a number whose absolute value is the final reference deviation (see Error Indicators and Warnings). The function is an adaptation of Boothroyd (1967).

References

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

Parameters

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:     m1 int64int32nag_int scalar
m+1, where m is the order of the polynomial to be found.
Constraint: m1<minn,100.

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.)
The number n of data points.

Output Parameters

1:     am1 – double array
The coefficients ai of the final polynomial, for i=1,2,,m+1.
2:     ref – double scalar
The final reference deviation (see Error Indicators and Warnings).

Error Indicators and Warnings

Accuracy

This is wholly dependent on the given data points.

Further Comments

The time taken increases with m.

Example

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 e02ac_example


fprintf('e02ac example results\n\n');

x = [0:0.05:1]';
y = exp(x);

m1 = int64(6);
[a, ref] = e02ac(x, y, m1);

disp('Polynomial coefficients');
disp(a);
fprintf('\nReference deviation = %10.2e\n\n',ref);

z = x(1:2:end);
pz = polyval(a(end:-1:1),z);
expz = exp(z);
resz = pz - expz;
fprintf('        x     Fit       exp(x) Residual(*10^-6)\n');
disp([z pz expz resz*10^6]);



e02ac example results

Polynomial coefficients
    1.0000
    1.0001
    0.4991
    0.1704
    0.0348
    0.0139


Reference deviation =   1.09e-06

        x     Fit       exp(x) Residual(*10^-6)
         0    1.0000    1.0000   -1.0915
    0.1000    1.1052    1.1052    0.9740
    0.2000    1.2214    1.2214   -0.7439
    0.3000    1.3499    1.3499   -0.9175
    0.4000    1.4918    1.4918    0.2988
    0.5000    1.6487    1.6487    1.0915
    0.6000    1.8221    1.8221    0.4586
    0.7000    2.0138    2.0138   -0.8163
    0.8000    2.2255    2.2255   -0.8419
    0.9000    2.4596    2.4596    0.8755
    1.0000    2.7183    2.7183   -1.0915


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