% Lab. 5 - Spectral, pseudo-spectral methods
clear all
% 1. Spectral differentiation
%
%
I=sqrt(-1); np=1;
for n=16:16:256
  h=2*pi/n;
  x=(0:n-1)*h;
  %-----------------------------------------------
  % Choose function you want to differentiate here
  %-----------------------------------------------
  f=sin(cos(x))+cos(sin(x));
  fp=-sin(x).*cos(cos(x))-cos(x).*sin(sin(x));
  %f=sin(x);
  %fp=cos(x);
  plot(x,f,'r',x,fp,'b');
  fhat=fft(f);
  for j=1:n
     if j<=n/2+1
        k=j-1;
     else
        k=j-1-n;
     end
    fphat(j)=k*I*fhat(j);
    if (j>1) & (j<n)
      fpFD(j)=(f(j+1)-f(j-1))/(2*h);
    end
  end
  fpSpectral=ifft(fphat);
  figure(1);
  plot(x,f,'r',x,fp,'b',x,real(fpSpectral),'bo',x,imag(fpSpectral),'bx');
  pause;
  err(np)=log10(norm(abs(fpSpectral-fp)));
  errFD(np)=log10(norm(abs(fpFD(2:n-1)-fp(2:n-1))));
  herr(np)=log10(h);
  np=np+1;
end
figure(2);
plot(herr,err,'ro',herr,errFD,'bx');
title('Convergence plot: o = spectral, x = finite difference');