Discrte Cosine Series Tutorial 2

DCS2.m

n  = 15;              % number of collocation points is (n+1)
nmodes = 5;           % number of modes of the DCS

xi = [0:n]'/(n+1);       % collocation points for discrete cosine series
V  = zeros(n+1, nmodes); 

%
% V: stack the nmodes DCS basis functions columnwise
%
for j = 1:nmodes
  V(:,j) = cos(2*pi*(j-1)*xi);  % evaluation of phi_j(x_i) at all the n+1 nodes
end

				%
				% Sample Function
				%

fi = (1/2-xi).^2 + cos(2*pi*xi).^2;

				%
				% if nmodes = n+1: interpolation
				%    nmodes < n+1: approximation

nmodes = 5;
a = zeros(nmodes,1);

a=2/(n+1)*V'*fi;
a(1) = a(1)/2;    % a(0) needs a factor of 1/2


%
% Plot data, and the approximating function
%
x = linspace(0,1)';
V  = zeros(numel(x), nmodes); 
for j = 1:nmodes
  V(:,j) = cos(2*pi*(j-1)*x);
endfor

g = V*a;

plot(xi,fi,'o',x,g,'LineWidth',3)
xlabel('x')
ylabel('f(x)')
a