% 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 lecture08

DRIVER1
DRIVER2
DRIVER3

return

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

function DRIVER1

x = chebfun('x');
hold off, plot(real(x),imag(x),'r')
semi = 2*exp(0.5i*pi*x);
S = join(x-2i, 1+semi, 2i-x, -1-semi);
hold on, plot(S,'k'), axis equal off
z = exp(1i*pi*x);
Gamma = (2.8+.2i)*(sinh(z)+.5*real(z));
plot(Gamma,'b')
text(4.2,2,'\Gamma','color','b','fontsize',12)
text(3.1,.7,'S','fontsize',12)
text(.9,-.3,'1','color','r')
text(-1.4,-.3,'-1','color','r')
return

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

function DRIVER2

np  = 8; 
xj  = chebpts(np); 
%xj  = linspace(-1,1,np);

figure(1)
hold off
tt  = linspace(-1.5,1.5);
plot(tt,node_poly(tt,xj),'r',...
     xj,0*xj,'.k',...
     'LineWidth',3,'MarkerSize',30)

figure(2)
hold off
tt  = linspace(-1.5,1.5,10000);
semilogy(tt,abs(node_poly(tt,xj)),'r',...
     'LineWidth',3,'MarkerSize',30)

figure(3)
[XX,YY] = meshgrid(linspace(-1.5,1.5,401),linspace(-1,1,301));
    ZZ = XX + 1i*YY;
[C,h] = contour(XX,YY,log10(abs(node_poly(ZZ,xj))),(-8):0);
clabel(C,h)
title('log_{10}(abs(l(z)))')
keyboard

return

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

function DRIVER3

np     = 30; 
xxcheb = chebpts(np); 
%xxuni  = linspace(-1,1,np);
xxuni  = -1 + 2*rand(1,np);

figure(1)
hold off
tt  = linspace(-1.5,1.5);
plot(tt,node_poly(tt,xxcheb),'r',...
     tt,node_poly(tt,xxuni),'k',...
     'LineWidth',3)
legend('cheb','uni')

figure(2)
hold off
tt  = linspace(-1.5,1.5,10000);
semilogy(tt,abs(node_poly(tt,xxcheb)),'r',...
         tt,abs(node_poly(tt,xxuni)),'k',...
         'LineWidth',3,'MarkerSize',30)
legend('cheb','uni')

figure(3)
[XX,YY] = meshgrid(linspace(-2,2,161),linspace(-1,1,81));
ZZ = XX + 1i*YY;
subplot(1,2,1)
[C,h] = contour(XX,YY,log10(abs(node_poly(ZZ,xxcheb))),[-5,-4,-3,-2,-1,0,1,2]);
clabel(C,h)
title('Chebyshev nodes')
subplot(1,2,2)
[C,h] = contour(XX,YY,log10(abs(node_poly(ZZ,xxuni))),[-5,-4,-3,-2,-1,0,1,2]);
clabel(C,h)
title('Uniform nodes')
keyboard

return

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

function ll = node_poly(xx,rr)

ll = ones(size(xx));

for i = 1:length(rr)
  ll = ll.*(xx - rr(i));
end

return