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: $x (n)$ – double array: The values of the $x$ coordinates, $x_{i}$ , for $i = 1, 2, \dots, n$ .

Constraint: $x_{1} < x_{2} < \dots < x_{n}$ .
2: $y (n)$ – double array: The values of the $y$ coordinates, $y_{i}$ , for $i = 1, 2, \dots, n$ .
3: $m1$ – int64int32nag_int scalar: $m + 1$ , where $m$ is the order of the polynomial to be found.

Constraint: $m1 < \min (n, 100)$ .

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: $a (m1)$ – double array: The coefficients $a_{i}$ of the final polynomial, for $i = 1, 2, \dots, m + 1$ .
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.

Open in the MATLAB editor: e02ac_example

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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox