The Modern Taylor Method ODEs Solver:
The Taylor Center
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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 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.
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 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
performing the evaluation of the derivatives and integration of 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 ODEs in
the standard format, 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 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 (with the only limitation that the
terms in the series do not exceed the maximum value of about 10^{4932}
implied by the Intel processor generic implementation of the real type extended
as 10-bytes with 63-bit mantissa);
- Perform
integration either "blindly" (observing only the numerical changes), or
graphically visualized. Perform integration for a given number of
steps, or until an independent variable reaches the terminal value, or
until a (former) dependent variable (now a new independent)
reaches a 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;
- 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;
- 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";
- 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 up to 99,
and then
integrate and explore the motion. In the case of n=99 there are
298
ODEs, 19404 auxiliary equations, compiled into over 132000 variables
and over
130000 AD processor's instructions: a "heavy duty" integration!
- 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 a fascinating example of the so called Choreography for the
Three Body
motion, an eight-shaped orbit, discovered just recently by Chenciner and Montgomery (2000) .
Click here to learn
more about the
Choreographies of N-body problem. You can "feed" the equations found there into the Taylor
Center,
integrate them, draw the curves and play the motion in the real-time
mode all
in the same place. Another recent fascinating example of the four body
non-planar trajectories inscribed in a cube discovered by Cris
Moore & Michael Nauenberg is incorporated too.
The Taylor Center is available both as a 32-bit and 64-bit
application
for Windows. The executable module is only 1 Mb (32-bit version) and
near 3 Mb
(64-bit). As a 32-bit application, the program can use no more than 4
Gb of
available memory for variables and their expansions - the limit far
exceeding
any practical needs. (The new 64-bit version does away with this 4 Gb
limitation).
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 41
ODEs,
45+90=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 via the polynomial
manager.
Generally for each system of ODEs there exists such a small value of
the
accuracy tolerance, that at this high accuracy the Taylor methods beats
any
fixed order method due to the unlimited order of approximation in the
Taylor
method (known since Moore in 1960s).
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) 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.
- A special
version will be developed to work with complex numbers, so that
integration along any pass in complex plane is possible in order to
study the solutions, and to locate and explore their singularities in
the complex space;
- 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;
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.
Similarly, it is one thing to see
the 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 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 with a huge
acceleration in big laps (as it follows from the complex analysis).
Here is an array of 5 Initial Value Problems integrated simultaneously
for the same systems of ODEs:
Similarly the real time animation is provided also in stereo display of
3D trajectories. 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 also display a field of
directions
for selected pairs of variables, or rather the field of elements i.e. curvy
strokes representing finite pieces of trajectories at points of a grid
obtained in one step integration. Due to the high order of the Taylor
method, these strokes are typically long enough to represent the near
continuous family of the solutions in the phase space like that below:
For pedagogical aspects of the Taylor Center look here: