% October 2017
%
% Written by Per-Gunnar Martinsson, University of Oxford
%
% The methods follow Lloyd N. Trefethen's 
%
%    "Approximation Theory and Approximation Practice"
%
% textbook published by SIAM in 2013. Many code snippets are taken from 
% this text. They have been modified freey, however, and any errors are 
% due to PGM. 

function lecture04

DRIVER_n0
DRIVER_n1
DRIVER_n3
DRIVER_analytic
DRIVER_entire
DRIVER_Cinfty

return

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function DRIVER_n0

x  = chebfun('x'); 
f  = sign(x);
nn = (2.^(0:8))-1;

err = 0*nn;

for j = 1:length(nn)
  n      = nn(j); 
  fn     = chebfun(f,n+1); 
  err(j) = norm(f-fn,inf);
end

hold off
figure(1)
nmax = max(nn);
fn = chebfun(f,2*nmax);
kk = 0:nmax;
aa = chebcoeffs(fn);
loglog(kk,1./(kk.^1),'r',...
       kk,abs(aa(1:(nmax+1))),'b.',....
       'MarkerSize',35,...
       'LineWidth',3)
title('Magnitude of Chebyshev coefficient: f(x) = sign(x)')
xlabel('k')
ylabel('|a_k|')
legend('Exact k^{-1}','|a_{k}|')

hold off
figure(2)
loglog(nn,err,'b.',....
       'MarkerSize',35,...
       'LineWidth',3)
title('Errors of Chebyshev interpolants in sup norm: f(x) = sign(x)')
xlabel('n')
ylabel('||f - f_n||')
keyboard

return

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function DRIVER_n1

x  = chebfun('x'); 
f  = abs(x);
nn = (2.^(0:8))-1;

err = 0*nn;

for j = 1:length(nn)
  n      = nn(j); 
  fn     = chebfun(f,n+1); 
  err(j) = norm(f-fn,inf);
end

hold off
figure(1)
nmax = max(nn);
fn = chebfun(f,2*nmax);
kk = 0:nmax;
aa = chebcoeffs(fn);
loglog(kk,1./(kk.^2),'r',...
       kk,abs(aa(1:(nmax+1))),'b.',....
       'MarkerSize',35,...
       'LineWidth',3)
title('Magnitude of Chebyshev coefficient: f(x) = abs(x)')
xlabel('k')
ylabel('|a_k|')
legend('Exact k^{-2}','|a_{k}|')

hold off
figure(2)
loglog(nn,1./nn,'r',...
       nn,err,'b.',....
       'MarkerSize',35,...
       'LineWidth',3)
title('Errors of Chebyshev interpolants in sup norm: f(x) = abs(x)')
xlabel('n')
ylabel('||f - f_n||')
legend('Exact n^{-1}','||f - f_{n}||')
keyboard

return

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function DRIVER_n3

x  = chebfun('x'); 
f  = abs(x).^3;
nn = (2.^(0:8))-1;

err = 0*nn;

for j = 1:length(nn)
  n      = nn(j); 
  fn     = chebfun(f,n+1); 
  err(j) = norm(f-fn,inf);
end

hold off
figure(1)
nmax = max(nn);
fn = chebfun(f,2*nmax);
kk = 0:nmax;
aa = chebcoeffs(fn);
loglog(kk,1./(kk.^4),'r',...
       kk,abs(aa(1:(nmax+1))),'b.',....
       'MarkerSize',35,...
       'LineWidth',3)
title('Magnitude of Chebyshev coefficient: f(x) = abs(x)^3')
xlabel('k')
ylabel('|a_k|')
legend('Exact k^{-4}','|a_{k}|')

hold off
figure(2)
loglog(nn,1./(nn.^3),'r',...
       nn,err,'b.',....
       'MarkerSize',35,...
       'LineWidth',3)
title('Errors of Chebyshev interpolants in sup norm: f(x) = abs(x)^3')
xlabel('n')
ylabel('||f - f_n||')
legend('Exact n^{-3}','||f - f_{n}||')
keyboard

return

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function DRIVER_analytic

x  = chebfun('x'); 
f  = 1/(1 + 25*x*x);
nn = (2.^(0:12))-1;

err = 0*nn;

for j = 1:length(nn)
  n      = nn(j); 
  fn     = chebfun(f,n+1); 
  err(j) = norm(f-fn,inf);
end

hold off
figure(1)
nmax = max(nn);
fn = chebfun(f,2*nmax);
kk = 0:nmax;
aa = chebcoeffs(fn);
loglog(kk,abs(aa(1:(nmax+1))),'b.',....
       'MarkerSize',30,...
       'LineWidth',3)
title('Magnitude of Chebyshev coefficient: f(x) = 1/(1 + 25*x*x)')
xlabel('k')
ylabel('|a_k|')

hold off
figure(2)
loglog(nn,err,'b.',....
       'MarkerSize',35,...
       'LineWidth',3)
title('Errors of Chebyshev interpolants in sup norm: f(x) = 1/(1 + 25*x*x)')
xlabel('n')
ylabel('||f - f_n||')
keyboard

return

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function DRIVER_entire

x  = chebfun('x'); 
f  = sin(27.3*x);
nn = (2.^(0:12))-1;

err = 0*nn;

for j = 1:length(nn)
  n      = nn(j); 
  fn     = chebfun(f,n+1); 
  err(j) = norm(f-fn,inf);
end

hold off
figure(1)
nmax = max(nn);
fn = chebfun(f,2*nmax);
kk = 0:nmax;
aa = chebcoeffs(fn);
loglog(kk,abs(aa(1:(nmax+1))),'b.',....
       'MarkerSize',30,...
       'LineWidth',3)
title('Magnitude of Chebyshev coefficient: f(x) = sin(27.3*x)')
xlabel('k')
ylabel('|a_k|')

hold off
figure(2)
loglog(nn,err,'b.',....
       'MarkerSize',35,...
       'LineWidth',3)
title('Errors of Chebyshev interpolants in sup norm: f(x) = sin(27.3*x)')
xlabel('n')
ylabel('||f - f_n||')
keyboard

return

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function DRIVER_Cinfty

x  = chebfun('x'); 
f  = exp(-1./((1-x)*(1+x)));
nn = (2.^(0:12))-1;

err = 0*nn;

for j = 1:length(nn)
  n      = nn(j); 
  fn     = chebfun(f,n+1); 
  err(j) = norm(f-fn,inf);
end

figure(1)
hold off
nmax = max(nn);
fn = chebfun(f,2*nmax);
kk = 0:nmax;
aa = chebcoeffs(fn);
loglog(kk,abs(aa(1:(nmax+1))),'b.',....
       'MarkerSize',30,...
       'LineWidth',3)
title('Magnitude of Chebyshev coefficient: f(x) = sin(27.3*x)')
xlabel('k')
ylabel('|a_k|')

figure(2)
hold off
loglog(nn,err,'b.',....
       'MarkerSize',35,...
       'LineWidth',3)
title('Errors of Chebyshev interpolants in sup norm: f(x) = sin(27.3*x)')
xlabel('n')
ylabel('||f - f_n||')

figure(3)
hold off
tt = linspace(-1,1,1000);
fff = fn(tt);
plot(tt,fff)
keyboard

return

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%