Keywords: ODE, Taylor Method, Taylor Solver, Taylor Series, Numeric Integration, High Accuracy Integration, Ultimate Accuracy, Finite Step, Dynamic Graphic, 3D Stereo, 3D Cursor, Trajectories, Interactive environment, Windows, Linux.

The Modern Taylor Method is a descender of its classical counterpart. It is an efficient method for numerical integration of the Initial Value Problems for Ordinary Differential Equations (ODEs). What distinguishes it from all other numerical methods for ODEs is that only the Taylor Method can compute the increments of the solution with principally unlimited order of approximation so that the integration step does not approach zero whatever high accuracy is specified. That is possible because the method performs the automatic differentiation - exact computing of the derivatives up to any desired order N, allowing to obtain the Taylor series of any length for the solution components.

To download the Demo for Windows (95, NT and later), click here, then download and unzip the file ("Save", don't "Open" it in your browser). See for a short guide navigating you through the DEMO. You can also download the full User Manual (in MS Word doc format), or the recent article published in the Proceedings of the 2005 International Conference on Scientific Computing CSC'05.

To make inquires, please contact Alexander Gofen at galex@ski.org .

With the current version of the product you can:

• Specify and study the Initial Value Problems for virtually any system of ODEs in the standard format with numeric and symbolic constants and parameters;

• Perform numerical integration of Initial Value Problems with the highest possible accuracy while the step of integration remains finite and does not approach zero (instead, the order of approximation is high);

• Apply the order of approximation as high as you like (30, or 40 or whatever), and get the solution in the form of the set of analytical elements - Taylor expansions covering the required domain;

• Study Taylor expansions and the radius of convergence for the solution at all points of interest up to any high order (with the only limitation that the terms in the  series do not exceed the maximum value of about 10⁴⁹³² implied by the 10-byte implementation of the real type extended);

• Perform integration either "blindly", or graphically visualized a given number of steps, or until an independent variable reaches a terminal value, or until a dependent variable reaches a terminal value (as explained in the next item);

• Switch integration between several versions of ODEs defining the same trajectory, but with respect to different independent variables. For example, it is possible to switch the integration from that with respect to t to that by x, or by y in order to reach the terminal value (or zeros) of a dependent variable;

• Integrate piecewise-analytical ODEs;

• Specify different methods of controlling the accuracy and the step size;

• Specify accuracy for individual components either as an absolute or relative error tolerance, or both;

• Graph color curves (trajectories) for any pair of variables of the solution - up to 7 on one screen - either as plane projections, or as 3D stereo images (for triplets of variables) to be viewed through anaglyphic (Red/Blue) glasses. The 3D cursor (controlled by a conventional mouse) with audio feedback enables "tactile" exploration of the curves literally "hanging in thin air";

• Play dynamically the near-real time motion along the computed trajectories either as 2D or 3D stereo animation of moving bullets;

• Graph the field of directions, actually the field of curvy segments, whose length is proportional to the radius of convergence.

• Explore several meaningful examples supplied with the package such as the problem of Three and Four Bodies. Symbolic constants and expressions make possible parameterization of the equations and initial values, and trying different initial configurations of special interest.

 
   The future versions will include everything above plus the following:

• It will be supplied not only as the Taylor Center GUI executable, but also as the separate Delphi Units (to include them directly in Delphi projects) and also as DLLs to use in other environments;

• It will implement the Merge procedure and a library of ODEs – definitions of a large variety of commonly used elementary functions. (Presently, the functions which are not in the allowed list, may be used also – providing that the user declares the ODEs defining them and properly links them with the source ODEs (more about that in Help for Merge). Also, it will include a larger variety of the "calculator" functions for specifying even more complex relationships between constants, parameters and the initial values.

• It will work with complex numbers, so that integration along any pass in complex plane is possible in order to study the solutions, to locate and explore their singularities;

• The application will be ported to windowed Linux;

• The set of the internal differentiation instructions will be translated into the machine code - to reach the highest possible speed for massive computations. (Meanwhile it is an emulator written in Delphi which runs these instructions). Also, it may be translated into instructions in Pascal, C or Fortran to be further compiled and linked with other applications;

• Integrating by a parameter or integrating boundary value problems .

      Here is how the front panel of the Taylor Center looks like:

And here are the trajectories of a slightly disturbed (Lagrange) case of the Three Body Problem with the central symmetry (Demo/Three bodies/Disturbed/2D). All possible pairs of the three bodies couple randomly in turn. You would prefer to watch it as a real time motion in the Demo rather than as a still image.