%% A wiggly function and its best approximations
% Ricardo Pachon and Nick Trefethen, November 2010

%%
% (Chebfun example approx/WigglyApprox.m)
% [Tags: #best, #REMEZ, #CHEBPOLY]

%%
% Ken Lord, whose doctoral supervisor was the Chebyshev technology wizard
% Charles Clenshaw, has explored functions of the form
%
% f(x) = T_m(x) + T_m+1(x) + ... + T_n(x),
%
% where T_k is the kth Chebyshev polynomial, as challenging functions for
% minimax approximation by polynomials of lower order.  We can construct
% such functions in a single Chebfun command:
fmn = @(m,n) sum(chebpoly(m:n),2);

%%
% For example, here we plot f(30,40) and its best approximation of degree 29:
LW = 'linewidth'; FS = 'fontsize'; fs = 14;
tic, m = 30; n = 40;
f = fmn(m,n);
subplot(2,2,1), plot(f,LW,1)
grid on, title('f(30,40)',FS,fs)
subplot(2,2,2), plot(f{.8,1},LW,1.6)
grid on, title('closeup',FS,fs)
p = remez(f,m-1); err = f-p;
subplot(2,2,3), plot(err,'r',LW,1)
grid on, title('f - p',FS,fs)
subplot(2,2,4), plot(err{.8,1},'r',LW,1.6)
grid on, title('closeup',FS,fs), toc
 
%%
% Here are f(200,220) and its best approximation of degree 199:
tic, m = 200; n = 220;
f = fmn(m,n);
subplot(2,2,1), plot(f,LW,.8)
grid on, title('f(200,220)',FS,fs)
subplot(2,2,2), plot(f{.995,1},LW,1.6)
grid on, title('closeup',FS,fs), xlim([.995 1])
p = remez(f,m-1); err = f-p;
subplot(2,2,3), plot(err,'r',LW,.8)
grid on, title('f - p',FS,fs)
subplot(2,2,4), plot(err{.995,1},'r',LW,1.6)
grid on, title('closeup',FS,fs), xlim([.995 1]), toc