AD Tool: The Taylor Center
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The Taylor Center

ODE Solver for Initial Value Problems (IVPs) given in the form of a system of 1st order explicit ODEs. The integration is based on Automatic Differentiation of the right hand sides entered in a conventional mathematical notation.

The package is an All-In-One advanced GUI application with near real time animation of the solution in 2D or in 3D stereo with a conventional monitor and Red/Blue glasses.


  • Alexander Gofen

Mode: Forward
Supported Language: Delphi
Language independent

Alexander Gofen
Interactive Environment for the Taylor Integration (in 3D Stereo)
Conference proceeding, CSREA Press, 2005

Alexander Gofen
Visual Environment for the Taylor integration in 3D Stereo
Conference proceeding, 2007

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, meaning a system of explicit 1st order ODEs, derivatives in the left hand sides and arithmetic expressions in the right hand. The standard elementary functions, numeric and symbolic constants and parameters may be used;

* Enter arithmetic expression in the standard Pascal syntax either through the editor windows, or via the Polynomial Designer for cumbersome polynomial expressions;

* Perform numerical integration of Initial Value Problems with an arbitrary high accuracy along a path without singularities, while the step of integration remains finite and does not approach zero (presuming the order of approximation or the number of terms could increase to infinity with the length of mantissa unlimited);

* Apply an arbitrary high order of approximation (by default 30), 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. An upper limit for the terms in the series is as high as 10^4932 implied by the Intel generic 10-byte float point type extended with 63-bit mantissa (contrary to the reduced 8-byte 48 bit mantissa in Microsoft C++ for this type);

* Graph Taylor expansions as bar diagrams and vary the step h observing its effect on the bell shape bulge;

* Perform integration either "blindly" (observing only the numerical changes), or graphically visualized. The visualization comprises graphs of the solution, real time motion along the trajectories, and the field of directions, or rather a phase portrait comprised of particular families of trajectories;

* Terminate the integration either after a given number of steps, or when an independent variable reaches a given terminal value, or when a dependent variable reaches a given terminal value (as explained in the next item);

* Automatically generate the ODEs and switch integration between several states of ODEs defining the same trajectory, but with respect to different independent variables. For example, it is possible to switch the integration in respect to t to that by x, or by y in order to reach the terminal value (or zeros) of a former dependent variable (x, or y). In particular, if the initial value is a nonzero (guess) value and the terminal value is set to zero, the root (the zero) of the solution may be obtained directly without iterations;

* Automatically generate and integrate an array of Initial Value Problems for the array of initial vectors. The solutions of these IVPs are displayed in one plot resembling a phase portrait. An array of IVPs considered as an IVP with an indefinite parameter helps to estimate the solution of certain boundary value problems;

* Integrate piecewise-analytical ODEs;

* Integrate IVPs in complex variables along an arbitrary path in acomplex plane – by automatic (since version 29) transformation of the source ODEs into a system over the real and imaginary parts of the respective variables;

* Specify different methods to control 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 99 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 virtually "hanging in thin air";

* Graph non-planar curves as though tubes of a required thickness implementing the proper skew resolution at points of illusory intersections;

* 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 strokes, 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 allow parameterization of the equations and initial values, and trying different initial configurations of special interest.

* Automatically generate ODEs for the classical Newtonian n-body problem for n < 100, and then integrate and explore the motion. In the case of n=99 there are 595 ODEs, 19404 auxiliary equations, compiled into over 132000 variables and over 130000 AD processor's instructions: a "heavy duty" integration!

* Use a DLL (for any programming languages under Windows) which provides functions computing the vector-function of a solution of ODEs (first obtained with the Taylor Center, but later – without using the Taylor Center). Instead, this DLL implements the optimized (Horner algorithm) for computation of the desired vector function using the polynomial expansions obtained from the Taylor integration and saved in a binary file.

* Integrate a few special instances of singular ODEs having regular solutions at the points of the so called "regular singularities".

Supported Platforms:
  • Windows

Licensing: license

References on The Taylor Center in our publication database:  2

'05 '07

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