demo_runge.m

Interpolation routines in matlab

%
%  interpolation of Runge's function (1+x^2)^(-1)
%  on the interval [-5, 5]
%

clear;


% Interval on which Runge's function is to be approximated
A = -5;
B =  5;


% evaluate Runge's function at z
m = 1000;
h = (B-A)/(m-1);
z = (A:h:B);
r = 1 ./(1+z.^2);



n = input(' number of interpolation points  n = ')

%   compute equidistant interpolation points
%   and function values to be interpolated

    h = (B-A)/(n-1);
    x = (A:h:B);     
    f = 1 ./ (1+x.^2);

%   compute Newton's interpolating polynomial

    a = NewtIntPol( x, f, 0 );

%   evaluate Newton's interpolating polynomial at the 
%   points z and plot it together with Runge's function.

    p = NewtIntPolEval( x, a, z );

%   plot the interpolating polynomial

    subplot(2,1,1)
    plot(z, p, 'r'); hold on   % plot polynomial
    plot(z, r, 'g'); hold off  % plot Runge's function
    title(['Interpolation of Runge''s Function at Equidistant Points (n=',...
            int2str(n),')'] )



%   compute Chebyshev interpolation points
%   and function values to be interpolated

    for i = 1:n
       x(i) = 0.5 * ( (A+B) + (B-A) * cos( (2*i-1)*pi/ (2*n) ) );     
       f(i) = 1 / (1+x(i)^2);
    end

%   compute Newton's interpolating polynomial

    a = NewtIntPol( x, f, 0 );

%   evaluate Newton's interpolating polynomial at the 
%   points z and plot it together with Runge's function.

    p = NewtIntPolEval( x, a, z );

%   plot the interpolating polynomial

    subplot(2,1,2)
    plot(z, p, 'r'); hold on   % plot polynomial
    plot(z, r, 'g'); hold off  % plot Runge's function
    title(['Interpolation of Runge''s Function at Chebyshev Points (n=',...
            int2str(n),')'] )