Byoung Keum, August 25, 1988
Dept. of Math.
University of IL.
Urbana, IL 61801.


Welcome to Calculus and Differential Equations V 9.
===================================================

This disk contains programs to help college students in
Math, Engineering, or Science. Some of them are already used in
the Differential Equations classes in U. of Illinois. More likely,
you would not have is a printer drive for EGA screen, (We used a
program named "EGAEPSON.COM included in commercial program "MATLAB")
and a plotter (we used IBM COLOR PLOTTER). But even without these
only if you have EGA SCREEN (and preferably 8087), you'll find these
interesting.
This disk is a shareware (suggested registration fee is $ 30.00)
and for those who register I will send a diskett with latest and
customized versions for their particular request (if possible),
and some more document files.


*** All these require EGA (640x350, 16 Color).
*** They are both stack and heap intensive. (Large Memory maybe
necessary I tested them only on 640 K machine. For speed, I
supressed stack-check option at compilation, except for Euler3d.)
They will work best, at fresh boot-up, when the stack is
almost free.



Tips On Use
===========

0. These are so user friendly that most of the time you may need
help for Math, not for usage of these programs.

1. DE.COM is for Differential Equations, and CALCULUS.COM is for
Calculus. They are compiled in Turbo Pascal (TURBO.COM).
Although they are not as fast as STRING.EXE and EULER.EXE, they
will be very convenient to use as an "Integrated Software", in
relatively small size.

***************************************************************
Warning: In DE.COM and CALCULUS.COM, when you enter new
functions, DON'T use SPACES. And use multiple * instead of
power (like x*x rather than x^2). You don't have to do this
for the other .exe programs.
****************************************************************

When you are in the program you want (like in ODE of DE.COM),
just press ENTER several times to see the default setting.
During the animation, press ESC to quit.
Explore on your own, and if you have problems, let me know.

2. ODE.EXE:
This is an update of my previous program "EULER.EXE".
It allows the user to choose between Runge-Kutta method
and Euler's Method to get the solutions.
This draws some solutions to the system of linear ODE
dx/dt = F1(x,y), dy/dt = F2(x,y), for selected initial
points.
Functions to Try:
F1 = x+y,
F2 = x-y,

F1 = y,
F2 = -x-y,

F1 = y,
F2 = -sin(x),

F1 = y,
F2 = .5*(1-x^2)*y-x.

The default is for Runge-Kutta with step size 0.2.
To switch to Euler, it would be reasonable to reduce
the step size (down to 0.003, for example). You will
be surprised to see how accurate the Runge-Kutta method is.


3. EULER3D.EXE is a test program for 3-dim extension of ODE.EXE.
It is very similar to ODE.EXE. But, careful in choosing
functions, (they are more capricious in 3-D). You can
choose xy-view or yz-view or zx-view or oblique view
from view menu. The colors represent z-values. Try to
change window and initial conditions (try very small z
value like -10), without changing the function first.
The function set up as a default is nice. You can try
F1 = 0.1, F2 = -z, F3 = y, (Circular Helix) or any of the
examples in 3. above as F2 and F3, letting F1 = constant
for interesting results.


4. STRING.EXE
First, press ENTER a couple of times to see the demo.
Any time, press ENTER to interrupt the animation. Try to change
parameters. Make sure (vertical step size)*(number of steps) <= 1.
This is well known stability criterion. Try to violate it and see
the unstable case (well, don't carry on that too far, in fear of
overflow).

Initial Functions to try:
cos(25*x)-1.
sin(50*x).
Use abs() to use functions with vertices.
(Maximum length of function expression < 60.)
In this update, you can enter the initial velocity.


5. LP.EXE uses 3-dim graphics window to show graphical meaning of
simplex method for linear programming. This is a sample version
and the full interactive version is in progress.

Acknowledgements
================

I must say I owe lots of ideas from the Mathematical environment
of U. of IL.
More specifically, LP.EXE is an outcome of the Computer Geometry
course by Professor G. Francis, in which he suggested the need and
relevance of such a program. Also, we had a well-known program
"LINPROG.COM" by Professor Muller, who gave me advices, and his
program helped me to understand this subject.

Also, we already had a string vibration program written by Professor
Dornhoff using Fourier Series Method, which fascinated me so much that
I began to explore the possibility for interactive program. So, I
first developed a parser (which should be optimized, because it
usually goes into a loop), and as Professor G. Francis suggested,
tried to use Numerical Method, for speed. And, it worked fine (of
course there was hard work behind this program).

The Euler programs use well-known Euler method. We had a version
written in BASICA (new functions possible but was slow and we could
not print EGA Screen in BASICA). So, I used my parser which is very
optimized (compare it with my old parsers used in DE.COM and
CALCULUS.COM I guess this is more than five times faster, although
those were faster than the BASICA version), and developed a version
in Microsoft C Version 5.01.



ADVERTISEMENT.
=============

I have another disk in PC-SIG:

Disk1070: Particle Simulation.

Also, another disk is under screening process in PC-SIG:

Vibrating, Rotating, and Cooling Surfaces.

Also, I submitted a Microsoft Windows programs in Differential
Equations in WISC-WARE.

In addition to these, I have lots of Mathematical Graphics
programs either for IBM PC, or for Silicon Graphics Machines,
mostly on the theme of Differential Equations and Differential
Geometry (most of them contain my optimized parser for highest
level of interactive environment). Registered users will get
informations on further developments.


*** As of version 9, I implemented a Boolean function '&':

Value & Limit = 1, if Value > Limit,
0, otherwise.

With this '&', you can enter functions with vertices or with
several components of different formula.
For example, in STRING.EXE,
try x&.2 - 2*(x&.5) + x&.8 to use a function like

____
___ ___
0 ____

0 .2 .5 .8 1


Have fun!