% 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 lecture07

DRIVER_remez1
DRIVER_remez2
DRIVER_remez3

return

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

function DRIVER_remez1

x     = chebfun('x');
f     = min(abs(x+0.5),2*abs(x-0.5));
n     = 6;
pstar = minimax(f,n);
err   = max(abs(f-pstar));

xx = linspace(-1,1,1000);
figure(1)
subplot(1,2,1)
hold off
plot(xx,f(xx),'b',...
     xx,pstar(xx),'r',...
     'LineWidth',2)
legend('f','p^{*}','Location','NorthWest')
title(sprintf('Best approximant for n=%d',n))
subplot(1,2,2)
hold off
plot(xx,f(xx)-pstar(xx),'r',...
     xx, err*ones(size(xx)),'g',...
     xx,-err*ones(size(xx)),'g',...
     'LineWidth',2)
title(sprintf('Error "f - p^{*}r" for n=%d',n))
keyboard

return

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

function DRIVER_remez2

x     = chebfun('x');
f     = sin(exp(3*x));
n     = 28;
pstar = minimax(f,n);
err   = max(abs(f-pstar));

xx = linspace(-1,1,1000);
figure(1)
subplot(1,2,1)
hold off
plot(xx,f(xx),'b',...
     xx,pstar(xx),'r',...
     'LineWidth',2)
legend('f','p^{*}','Location','NorthWest')
title(sprintf('Best approximant for n=%d',n))
subplot(1,2,2)
hold off
plot(xx,f(xx)-pstar(xx),'r',...
     xx, err*ones(size(xx)),'g',...
     xx,-err*ones(size(xx)),'g',...
     'LineWidth',2)
title(sprintf('Error "f - p^{*}" for n=%d',n))
keyboard

return

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

function DRIVER_remez3

x = chebfun('x');
f = cumsum(sign(x+0.2));
%f = abs(x);
%f = cumsum(sign(x-0.5)-sign(x+0.5));

figure(1)
subplot(1,1,1)
hold off
plot(f)

keyboard

for n = 2.^(0:10)
  p = minimax(f,n);
  subplot(2,1,1)
  plot(f,'k',p,'r')
  title(sprintf('Functions: n = %d',n))
  legend('f','p')
  subplot(2,1,2)
  plot(f-p)
  legend('f-p')
  title(sprintf('Errors: n = %d',n))
  pause
end

return

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