Automatic implementation of the numerical Taylor series method: A Mathematica and Sage approach

In the last few years, the requirements in the numerical solution of ordinary differential equations in physics and in dynamical systems have pointed to new kind of methods capable to maintain geometric properties of the equations, or looking for high-precision, or solving variational equations. One method that can solve most of these problems is the Taylor series method. TIDES is a free software based on the Taylor series method that uses an optimized variable-stepsize variable-order formulation. The kernel of this software consists of a C library that permits to compute up to any precision level (by using multiple precision libraries for high precision when needed) the solution of an ordinary differential system from a C driver program containing the equations of the ODE. In this paper we present the symbolic methods, implemented in a computer algebra system (CAS), used to write, automatically, the code based on the automatic differentiation processes that integrates a particular differential system by means of the Taylor method. The precompiler has been written in Mathematica and Sage (which includes it by default since version 6.4). The software has been done to be extremely easy to use. The Mathematica version also permits to compute in a direct way not only the solution of the differential system, but also the partial derivatives, up to any order, of the solution with respect to the initial conditions or any parameter of the system.