%% Extrema of a complicated function
% Nick Trefethen, September 2010

%%
% (Chebfun example opt/ExtremeExtrema.m)
% [Tags: #optimization, #localextrema, #piecewise, #rootfinding]
  
%%
% Here is the function cos(x)*sin(exp(x)) on the interval [0,6]:
tic, x = chebfun('x',[0 6]);
f = cos(x).*sin(exp(x));
length(f)
plot(f,'color',[0 .7 0],'numpts',10000)
FS = 'fontsize';
title('A complicated function',FS,14)

%%
% Here's its absolute value:
g = abs(f);
plot(g,'m','numpts',10000)
title('Absolute value',FS,14)

%%
% Here's the minimum of that function and x/8:
h = min(g,x/8);
plot(h,'numpts',10000)
title('Minimum with x/8',FS,14)

%%
% We can find the maximum over the interval [0,5] like this:
[maxval,maxpos] = max(h{0,5})
hold on, plot(maxpos,maxval,'.r','markersize',40)
title('Global maximum',FS,14)

%%
% Let's add all the local extrema to the plot:
hp = diff(h);
extrema = roots(hp);
plot(extrema,h(extrema),'.k','markersize',6)
title('Local extrema',FS,14)

%%
% These computations showcase the fact that Chebfun
% optimization is global -- whether in the
% sense of finding a global extremum, or in the sense
% of globally finding all the local extrema.

%%
% They also showcase the treatment of discontinuities.
% To find extrema, Chebfun looks for zeros of the derivative.
% In some cases those are points where the derivative is
% continuous and passes through zero. In others,
% like the leftmost black dot in the plot above, they are
% points where the derivative jumps from positive to negative
% or vice versa.

%%
% The time for this whole sequence of computations:
Total_time = toc