Reliable integration of differential equations

This document shows how to use reliable integration for the classical pendulum problem, within the Mathemagix interpreter.

Pendulum vector field 

Mmx]  
use "continewz"
 
Mmx]  
bit_precision := 128; significant_digits := 6;
 
Mmx]  
z == analytic (ball 0.0, ball 1.0);
 
Mmx]  
vector_field (v) == [ v[1], -v[2], v[1]*v[3], -v[1]*v[2] ];
 
Mmx]  
initial_conditions == [ ball 0.0, ball 1.0, ball 0.0, ball 1.0 ];
 
Mmx]  
pendulum == integrate_analytic (vector_field, initial_conditions);
 
Mmx]  
pendulum[0]
 

Mmx]  
significant_digits := 0;
 
Mmx]  
pendulum[0][99]
 

Mmx]  
pendulum[0] (0.0)
 

Mmx]  
pendulum[0] (0.1)
 

Mmx]  
radius pendulum[0] (0.2)
 

Mmx]  
significant_digits := 15;
 
Mmx]  
points == [ (0.25 * t, pendulum[0] (0.25 * t)) || t in 0 to 20 ]
 

Mmx]  
include "graphix/simple_plot.mmx";
$draw_diagram ([ [Re center points[i,0],
                  Re center points[i,1]] | i in 0..rows points ])

© 2012 Joris van der Hoeven and Grégoire Lecerf
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU General Public License. If you don't have this file, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.