A Rigorous ODE solver and Smale's 14th Problem

This paper consists of two parts, (1) the paper itself, and (2) the codes.

(1) The paper:

JFoCM.pdf This is the published PDF file,
rodes.ps This is a normal PostScrip file,
rodes.ps.gz and this is a gzipped PostScript file.

(2) The codes:

All my programs use the PROFIL/BIAS package written by Olaf Knuppel. The package and its manual are available via anonymous ftp from ftp://ti3sun.ti3.tu-harburg.de, and can also be downloaded from the web at http://www.ti3.tu-harburg.de. The package provides modules for interval arithmetic with directed rounding.


Unpack the tar file by typing 'tar -xvzf programs.tar' in the directory where you want it to unfold. The README file will then explain how to build the programs.

A sanity check:

I understand that when it comes to large programs such as rodes, it is difficult to verify the correctness of the code. The following smaller program verifies (in a non-rigorous manner) that the results are not inconsistent with standard numerical methods. Simply put, every rectangle is filled with a large number of points that are iterated, one at a time, until they return to the Poincare' plane. The integration is done by the standard 4th order Runge-Kutta method. At the same time, the partial derivatives of the Poincare map are computed. By comparing these non-rigorous data with the output from the rodes program, we can check for consistency, both for the images of the initial rectangles, and for the tangent vectors and their expansion estimates.

Here is the outcome of the run I used to prove the existence of the Lorenz attractor. To understand what the numbers represent you must read the article above.


And here is the small verifying program and its header file.


The comments in the code explain how to compile the program.