% CLASSOCT1.TXT  Transcript of Matlab/Octave commands in  
%   Bueler's Math 310 class on 1 Oct, 2010.

% an example illustrating that for a function, some starting points
%   give fixed point iterations that converge, while some diverge

g = @(x) x.^3 - 1.5*x + 1
format long g
p=0.3, for n=1:20, p=g(p), end
p=0.3, for n=1:2000, p=g(p), end
p=1.33, for n=1:20, p=g(p), end
p=1.32, for n=1:20, p=g(p), end

% an example illustrating that it can be even worse than that; see:
%     http://en.wikipedia.org/wiki/Logistic_map
%     http://en.wikipedia.org/wiki/Cobweb_diagram

h = @(x) 4*x.*(1-x)
p=0.3, for n=1:200, p=h(p), end
p=0.3, for n=1:200000, p=h(p), end
p=0.753, for n=1:200000, p=h(p), end

% picture y=h(x); print this out and try doing a spider-web (cob-web)
%    diagram on it
x = 0:0.001:1; plot(x,h(x))
grid on, hold on, plot(x,x), axis equal, hold off