**The Modern Taylor Method Solver of ODEs:
The **

*Keywords*: ODE Solver, Taylor Solver, Taylor Method,
Taylor Series, Power Series Method, Differential Transform Method,
Adjoint equations, Numeric Integration,
High Accuracy, Ultimate Accuracy, Finite Step, Dynamic Graphics, 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 (IVPs) for holomorphic Ordinary Differential Equations (ODEs) whose Right Hand Sides are general elementary functions. What distinguishes it from all other numerical methods for ODEs is that …

· The Taylor method computes the increments
of the
solution with principally unlimited order of approximation so that the
integration step must not approach zero whichever 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.

·
The Taylor method does
not use any finite
difference
formulas such as *f'(t) ≈ ( f(t+h) – f(t) ) / h* prone
to (catastrophic) subtraction error with float point numbers of fixed
length.

· The Taylor Center utilizes the 80 bit float
point type
*extended* with 64 bit mantissa generic for the processor *X87 **FPU* (while
other numeric programs usually use 64 bit type *double* with 52
bit
mantissa).

To download the Demo for Windows
click here, then download and unzip
the file ("*Save*",
don't "*Open*" it in your browser). Unzip and keep it in an
empty folder, *TCenter.exe* being the only executable to run.
Preserve these
files and sub-folders structure (in order that the program work
properly). See
a short guide navigating you through the
DEMO. You
can also download the full User
Manual (in MS Word doc format), or the articles (in pdf)
published in the Proceedings CSC
2005 , DMS 2007
, and
in CODEE
Journal,
September 2012.

From the algorithmic point of view, this software parses the right hand sides of the ODEs and auxiliary equations, and compiles them into a sequence of pseudo instructions of Automatic Differentiation. Then the programmatic emulator of those instructions runs them evaluating the derivatives and integrating the Initial Value Problem.

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 holomorphic ODEs in the standard format (whose Right Hand Sides are general elementary functions), meaning a system of explicit first 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 2nd degree multivariate polynomials;

- Perform numerical integration of Initial Value Problems with an arbitrary high accuracy for the standard 64 bit mantissa in PCs, 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 64-bit mantissa (contrary to the reduced 8 byte type*double*with 52 bit mantissa in Microsoft C++ as the highest precision); - Perform integration either "blindly"
(observing only the numerical changes), or graphically visualized.

- Graph color curves
(trajectories) for any pair of variables of the solution – up to 200 on
one screen – either as plane projections, or as 3D stereo images (for
triplets of variables) to view through anaglyphic (red/blue) glasses.
The 3D cursor with audio feedback enables "tactile" exploration of the
curves literally 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;

- Graph the enhanced field of directions (a
Phase portrait), actually the field of curvy segments, whose length is
proportional to the radius of convergence.

- Dynamically plot a triangle
based on the 3 bullets of the first 3 trajectories visualizing the
moments of syzygy (i.e. when the bodies in the 3-body problem are
collinear);

- Dynamically plot the 3 moving
axes of coordinates fixed within a moving solid body (or the respective
tetrahedron) based on the 4 bullets in order to visualize the motion of
a solid body in 3D;

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

- Terminate the integration either after a
given number of steps, or until an independent variable reaches a given
terminal value exactly, or until the control function reaches a given
terminal value approximately,
or when a
*dependent*variable reaches a given terminal value exactly in ODEs in a different state (as explained in the next item);

**Automatically generate the ODEs and switch integration between**(adjoint equations) defining the same trajectory, but with respect to different independent variables. For example, it is possible to switch the integration in respect to*several states*of ODEs*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 (guess) value is a nonzero and the terminal value is set to zero, the root (the zero) of the solution may be obtained directly without iterations;

**Automatically generate and simultaneously integrate an**for an 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*array*of Initial Value Problems*indefinite parameter*helps to estimate the solution of certain boundary value problems;**Automatically****perform massive sequential integration of an**for an array of initial vectors prepared in advance. The results of the massive computation are saved into a file.*array*of Initial Value Problems

- Integrate piecewise-analytical ODEs;

- 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;

- Explore a collection of 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, along with trying different initial configurations of special interest.

- Automatically generate ODEs for the
classical Newtonian
*n*-body problem for*n*up to 99, 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 accepting the real 80 byte type
*extended*). This DLL (without using the Taylor Center) provides functions for computing the vector-function of a solution of ODEs earlier obtained with the Taylor Center. The DLL implements the optimized (Horner algorithm) for computation of the desired vector function using the polynomial expansions obtained from the Taylor integration and saved as a binary file (to re-use the integration results obtained in the Taylor Center in other user applications);

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

In particular, the Demo includes fascinating examples of the so called choreography for the 3 body problem: 345 of them (courtesy of Dr. Carles Simò), plus 204 cases of periodic orbits of unusual shapes (courtesy of Ana Hudomal). Click here to learn more about the Choreographies of N-body problem and how to "feed" their ODEs and initial values into the Taylor Center, plot the curves and play the motion in the real-time mode: all in the same place. Similarly, click here for playing with the periodic orbits for the 3 body problem from the list of Ana Hudomal. These orbits are closed curves (as intuitively expected from periodic orbits). Here however you may see the newly discovered periodic orbits which are finite curved segments, whose extremes are resting points in the 3-body problem, so that the bodies periodically fulfill a free fall along these segmented orbits (the data was kindly provided by Xiaoming LI and Shijun LIAO). Another recent fascinating example of the four body non-planar trajectories inscribed in a cube discovered by Cris Moore & Michael Nauenberg (also here) is incorporated too. Here is the list of brief explanations for all several hundreds samples pre-loaded with the program.

As of now (and in foreseeable future), the Taylor Center will remain a 32-bit application run on
the *x87 **FPU*. This
processor was designed to address only 32 bit address space, i.e. no
more than
4 Gb memory, or 400 millions of variables and their expansions (as 10
byte
float point numbers).

In order to randomly
address a memory
space larger than 4 Gb, Intel (and other companies) enhanced the *x87
**FPU*
processor by adding a set of instruction *SSE* capable to
randomly access
a 64 bit address space. However in doing so, Intel designed the SSE
instructions to operate only on the 8 byte data types abandoning the 10
byte
type *extended*. Though it is logically possible to expand the
addressing
ability of the x87 FPU by combining its 32-bit instructions with the
SSE
instructions to randomly address the 64 bit memory space, such a trick
prohibitively slows the overall operation speed of x87 FPU on the 10
byte type.

For the program like the Taylor Center it's crucial to operated with the highest precision available in a processor. Therefore, with such a design blunder by Intel, it makes sense to maintain the Taylor Center only as a 32 bit application operating at the x87 FPU with the 10 byte type at the highest speed.

The memory
consumption
depends on the number of variables *VarNum* (a function of the
number of
ODEs and their complexity) and on the specified *Order* of
approximation.
If the expansions are not stored, the program takes *2*VarNum*Order*10*
bytes of memory. If the expansions in *P* points are stored, it
additionally requires *P*VarNum*Order*10* bytes.

Here is a benchmark for the 10 body planar problem comprised of
2*(10+10)+1=41
ODEs and 10*9/2*(2+1)=135 auxiliary equations, which are parsed into
811 AD
instructions. For this problem 10000 steps of integration took 32 s (or
3.2 ms
per step) on 2.4 GHz Pentium with polynomial expressions spelled out,
and it
took 29 s (or 2.9 ms per step) with polynomial expressions encoded.

For a benchmark for the 99 body spatial problem comprised of
3*(99+99)+1=595
ODEs and 99*98/2*(3+1)=19404 auxiliary equations, 100 steps took 65 s
with
polynomial expressions spelled out, and took 42.5 s with polynomial
expressions
encoded.

Generally for each system of ODEs there exists such a small value of
the
accuracy requirement, that at this high accuracy the Taylor methods
beats any
fixed order method due to the unlimited order of approximation in the
Taylor
method.

The future version will include the following:

- It will be supplied not only as the Taylor Center GUI executable, but also as the separate Delphi component (to include them directly in Delphi projects);
- To integrate IVPs in complex variables
along an arbitrary path in a complex plane.

- 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.

- The application will be ported to windowed Linux/Kylex;

- 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, the set of the inernal AD instructions may be translated into instructions in Pascal, C or Fortran to be further compiled and linked with other applications;

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 (Demo/Three bodies/Disturbed/2D). However there is a dramatic difference between viewing a still image like in this page vs. dynamically evolving real time motion displayed in the running Taylor Center. In the Demo you will be fascinated to watch all possible pairs of the three bodies coupling randomly in turn with acceleration or deceleration.

Here is a Double spiral (*Demo/Spirals/Double
spiral*) – a solution of the system

*x' = x² - y²+
2xy - x -3.5y +
1, x(0)=0.1
y' = -x² + y² + 2xy - y +3.5x - 1,
y(0)=0 *

corresponding to one complex ODE *z' = (1 - i)z² + (-1 +
3.5i)z + 1 -
i, z = x + iy. *Similarly to the previous
example, it is
one thing to see this Double Spiral as* *a static image

It is quite different thing to watch the *real time* motion along
this
spiral observing that every lap of the spiral takes exactly the same
time to
run (with a huge acceleration in big laps), as it follows from the
complex
analysis.

And here is an array of 5 Initial Value Problems integrated
simultaneously for
the same system of ODEs with various initial values:

The unique feature of this software is that 3D viewing and real time animation is implemented also in stereo mode. To view stereo images, you need anaglyphic Red/Blue glasses (Left - Red, Right - Blue). Such glasses may be ordered here: look for a Red/Blue type (not Red/Cyan).

If you have the anaglyphic glasses, click to view 3D stereo images. You may view non-planar curves also in conventional axonometry projection (where their spatial properties are lost). However, in order to improve axonometric view of such curves, this software can draw them as though tubes of a required thickness implementing the proper skew resolution at points of illusory intersections like in this family of curves representing the double Möbius surface outline:

And here is the Klein bottle
outline with a screw line:

You may also construct and display
an *enhanced
field of directions* or a *phase portrait* for selected pairs
of
variables built not with small vectors, but rather with* analytic
elements*
i.e. curvy strokes representing finite pieces of trajectories obtained
in
multi-step integration at points of a grid or at arbitrary points of
your
choice. Due to the high order of the Taylor method, these strokes are typically long
enough to
represent the near continuous family of the solutions or the phase
portrait
like that below for the system:

*x' = -y + *2*x*^{2}* +
xy - y*^{2}

*y' =
x + x*^{2}* -
3xy*

*x' =
*1

y' = y^{2}*
- x*

The image below depicts the two curves
of this
phase portrait near the separatrix. The initial values of these curves
differ
only in the last binary digit of their 64 bit mantissa:

Normally, in order to populate and
reach the
maximal accuracy in 64 bit mantissa, 19 or 20 significant decimal
digits
suffice. The routine converting real decimal numbers into this float
binary
format may take also extra decimal digits, yet the binary result will
be
rounded to 64 significant binary digits anyway. In this example the two
decimal
numbers are represented with 22 decimal digits, however the difference
between
their binary form is exactly one last bit of the mantissa (the goal set
in this
experiment).

This smallest difference in the initial values however was enough to
finally
direct the black curve upward yet the red curve downward, demonstrating
the
ultimate accuracy of integration for this 64-bit mantissa real type.
The
relative error tolerance in this case was set to 10^{-25} < 10^{-19} meaning that at every of 80 finite steps,
all 64 binary digits of the
result were true, so that the highest accuracy at every step (in a
format with
64 bit mantissa) was achieved.

For pedagogical aspects of the Taylor Center look here: