%euler:  quick and dirty Matlab script to show Euler's polygonal method 
%        on simple logistic model:
% 
%           dx/dt = x ( 1 - x ) ,  0 < t < T ,  x(0) = x0
% 
% usage: edit below and run

% ~/matlab/misc/euler.m
% 
% C.G. (11-98)
% C.G. (02-06) minor touch ups

x0 = .01 ;          % initial data for x(0)

T = 10 ;            % time interval for integration

ntimes = 8 ;        % number of curves to generate and draw

n = 1 ;             % initial value for number of steps (doubled each time)

clf ,  hold on      % initialize plotting window

for time = 1 : ntimes

  n = 2 * n ;                      % double number of time steps

  dt = T / n ;                     % time-step size

  x = zeros( 1, n+1 ) ;            % pre-allocate x(1:n+1)

  x(1) = x0 ;                      % initial condition

  for k = 1 : n
    x(k+1) = x(k) + dt * x(k) * ( 1 - x(k) ) ;
  end

  t = linspace( 0, T, n+1 ) ;      % equally spaced 't' values for plotting

  plot( t, x )                     % plot polygonal approximation

end

% pretty up plot

set( gca, 'FontSize', 12 )

box on

xlabel( '\it t' )
ylabel( '\it x' )

title( 'Logistic Model: Euler''s Method ({\it n} = 2, 4, ... , 256 )', ...
       'FontSize', 14 )

axis tight