function [ ] = prob_2_16
% 
% Written by:
% -- 
% John L. Weatherwax                2005-05-07
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

nTs = 100;

ts = linspace( 0, 2*pi, nTs );
etp = exp( +sqrt(-1)*ts );
etm = exp( -sqrt(-1)*ts );

% Eulers method:
z = 1 - etp;
plotZ( z, 'Euler method' );

% Backward Euler:
z = 1 - 1./etp;
plotZ( z, 'Backward Euler' );

% Trapezoidal rule:
z = 2*( etp - 1 ) ./ ( etp + 1 );
plotZ( z, 'Trapezoidal Rule' );

% BDF2:
z = ( 1.5*etp.^2 - 2*etp + 0.5 ) ./ ( etp.^2 );
plotZ( z, 'BDF2' );

% AM3:
z = ( etp.^2 - 1 ) ./ ( (5/12)*etp.^2 + (8/12)*etp - (1/12) );
plotZ( z, 'AM3' );

% BDF3:
z = ( etp - (18/11) - (9/11)*etm - (2/11)*etm.^2 ) ./ ( (6/11)*etp );
plotZ( z, 'BDF3' );

return;



function [] = plotZ( z, titStg )

x = real( z ); y = imag( z );
figure;
plot( x, y, '-o' );
title( titStg );
grid on;

return;