Software of the Month Club 1997 April/Software of the Month Club 1997 April.iso/pc/dos/sri/edu/graphmat/graphmat.doc

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                               Graphmatica

                      Version 3.50b  Sat 07 Dec 1996


                             by Keith Hertzer
                      Copyright (C) 1996 kSoft, Inc.







                          System Requirements:
                          --------------------

IBM personal computer or compatible equipped with CGA, EGA, VGA, or VESA
SVGA graphics, at least 256 KB of RAM, running MS-DOS version 2.0 or
higher. Epson dot-matrix and HP DeskJet/LaserJet and compatible printers
are supported. Microsoft and compatible mice are supported but not
required for most functions. If you have a mouse driver installed,
Graphmatica will automatically detect and make use of your mouse. See
Section I for details on "mouse strokes."

----------------CGA users (if there still are any) NOTE:----------------
If you have a CGA monitor and notice that the menus and dialog boxes
are missing borders, you need to run the TSR program GRAFTABL.COM that
comes with DOS 3.0 and later before starting Graphmatica. This will
load the block graphics character generator which is sometimes not
implemented in graphics modes in the CGA hardware.






                 Graphmatica version 3.50b  GRAPHMAT.DOC

======================================================================
                         C  O  N  T  E  N  T  S
======================================================================


Section i: Introduction and Release Notes............................3

Section I: Using Graphmatica.........................................5

Section II: Using the Graph function.................................7

Section III: Extended graph functions: polar, parametric, dif eq....16

Section IV: Using AutoRedraw and the Redraw Queue...................20

Section V: View: Changing the appearance of the screen..............23

Section VI: Labels: Adding Explanatory Text to a Graph..............26

Section VII: Options: Viewing and setting special options...........28

Section VIII: Using a printer with Graphmatica......................30

Section IX: Point menu: evaluating a graph at specific points.......32

Section X: Calculus menu: Derivatives, Tangents and Integrals.......33

Appendix A: Graphmatica's Command Line Arguments....................35

Appendix B: Editing equation lists and GRAPHMAT.INI.................36

Appendix C: Upgrade Notes: Changes from 3.30........................38

Appendix D: Shareware / Order Form..................................40



----------------------Note: GRAPHMATICA FOR WINDOWS---------------------

Graphmatica for Windows, version 1.5 is now available for Windows 3.1
users. If Graphmatica for Windows isn't available from the source you
got this file from, download it from CompuServe (in the math/science
forum), FTP it from CICA, or send me $5 and I will send you a disk.

-----------------------------------------------------------------------



Section                                       ==========================
ii                                            | Introduction and Notes |
                                              ==========================
 i
 i
iiio                                                                   3
------------------------------------------------------------------------

                              INTRODUCTION

Graphmatica is an interactive algebraic equation grapher that can be
used as an aide to plotting mathematical curves. While it is designed to
be extremely simple to use, its advanced features may not be readily
apparent to the first-time user. Please take a moment to acquaint
yourself with them:

1. The Redraw Queue. Graphmatica remembers the last 25 equations you
   typed in or loaded from a file. You can save your work for use in a
   later session or with any text editor or Graphmatica 1.5 for Windows.

2. Automatic functions. Graphmatica will automatically
   + determine the type of graph you are entering based on the
     variables used,
   + recognize an equation's domain if you include one,
   + alter the sampling rate dynamically while graphing to make sure
     steep graphs like y=tan x are tracked correctly,
   + adjust the x/y ratio when you reset the range so proper aspect
     ratio is maintained,
   + redraw the most recently-entered equation(s) when you change the
     size or shape of the grid by any means, and
   + restore the grid and special options settings when you load an
     equation list that has them.
   You don't have to do anything to use these functions, but the
   Options menu still gives you complete control over them.

3. Advanced equation parser follows mathematical rules--not the
   computer's. You can use implied multiplication, a complete library of
   math functions (including trig), and even leave out those annoying
   parentheses in appropriate places. Forget about isolating variables
   before graphing! As long as there is only one instance of the
   dependent variable in the equation, Graphmatica will isolate it for
   you, and even graph relations. You also get the power of 6 styles of
   graphing: regular Cartesian, polar, parametric, and slope-field and
   initial-value approximations for up to fourth-order ODEs (and fourth-
   order linear systems as well), all detected automatically. Cartesian
   inequalities are supported as well.

4. Convenient controls: the status bar displays relevant information
   about the selected equation, and help or error messages, and the
   Redraw Queue combobox lets you select any equation in memory
   to graph, delete, or edit to form a new equation.

5. Pause and Print tables options let you see the coordinates of points
   on your graphs... as they are drawn. Print tables shows values at
   whole-number intervals so you can practice sketching curves yourself.



i. Introduction and Notes                                              4
------------------------------------------------------------------------

6. Convenient mouse-oriented operations. You can use the mouse to select
   a new range or view the coordinates of a point, select the initial
   value for an ODE, and even find the tangent line of a curve or
   integrate a function without pressing a single key.

7. Flexible graph paper. Choose between regular graph paper and paper
   appropriate for trig, polar, and logarithmic functions, at four
   levels of detail.

8. Lots of output options. Graphmatica lets you export your graphs to a
   standard PC-Paintbrush (.PCX) file for inclusion into other documents,
   or print directly to any Epson-compatible dot-matrix or HP DeskJet,
   LaserJet, or compatible inkjet or laser printer.

9. Every automatic option is also user-settable to give you absolute
   control over your graphs. And the Save Setup Info command lets you
   save your preferences so they are automatically restored whenever you
   run Graphmatica.

10. Instructive help topics explain the basics of each type of graph,
   online help lets you brush up on most features without exiting the
   program, and included demo files show you examples of each form of
   equation. See the section below for a list of these files.

                          DEMO EQUATION FILES

Eight prewritten equation list files are included with Graphmatica to
help demonstrate the kinds of graphs you can create and how to create
them. Try loading each of them up, manipulating the view of the
equations, redrawing, and modifying some equations. I suggest clearing
the screen (if not the redraw queue as well) before loading a new file
so the screen doesn't become too cluttered. Each file contains a
different sort of graphs that you can draw with Graphmatica.

DIFEQ.GR    - sample differential equation approximations
GRAPHMAT.GR - draws Graphmatica's name using parametric equations
INEQUAL.GR  - sample inequalities
LOGLOG.GR   - sample of logarithmic graph paper
ODE2.GR     - sample of second-order differential equations
POLAR.GR    - polar coordinates graphs
TRIG.GR     - trigonometric graphs
XYDEMO.GR   - Cartesian equations, quadratic equations, relations
Read section IV (page 19) to learn how to use these demo files.


-------------------------Note: ONLINE HELP FILES------------------------

Please do not edit the files GRMAT1.HLP - GRMAT7.HLP They are formatted
especially for use by the Graphmatica help view function. Altering these
files may prevent effective use of the help text. These files are a
subset of the information contained in GRAPHMAT.DOC, so if you want
printed documentation, you don't have to print the .HLP files.



Section                                            =====================
III                                                | Using Graphmatica |
 I                                                 =====================
 I
 I
IIIo                                                                   5
------------------------------------------------------------------------

                              USER INTERFACE

I've expended a significant effort to make the interface to Graphmatica
for DOS as close to that of Graphmatica for Windows as possible,
including writing my own menu and listbox routines. In general, I think
the interface is pretty intuitive and consistent between the two.

                       The EQUATION QUEUE COMBOBOX

The main feature that makes Graphmatica unique is the equation queue
combobox. Before you enter any equations, all you will see is the
equation editing line at the top of the screen directly below the menu.
Once you enter an equation, however, a down arrow will appear to the
right of the edit line. Clicking on this arrow with the mouse or
pressing the down arrow key brings down the listbox of the combobox,
which shows you several of your equations at a time. Use the arrow keys
to scroll up and down through the listbox; Graphmatica displays the
status of each equation in the color it is drawn in as you highlight it.
You can also move the highlight line by clicking on the up and down
arrows on the scroll bar (when it appears). Click on an equation or
press enter to select an equation. The listbox will close and the
selected equation will appear in the editing area. You can close the
listbox without selecting an equation by pressing Escape, clicking on
the edit line, or raising the menu as detailed in the section below.
    The equation queue is "home base": after completing any operation,
you will be returned without delay to the edit line with a copy of the
current equation, ready to create a new equation and graph it.
    Graphmatica operates on the object-action interaction model: i.e.,
first select an object (equation) to act on, and then select an action
to perform on it. You must select an equation from the combobox BEFORE
you choose a menu item to apply to it; the program will not prompt you
for which equation to redraw, evaluate a point from, or delete.

                POP-UP PROMPTS and the Graphmatica EDITOR

Most data input in Graphmatica is done through pop-up "dialog boxes"
which appear in the middle of the screen. (Of course, the equation input
takes place in the combobox at the top of the screen.) If there are
additional instructions, they will replace the menu bar on the top line.
The input editor makes full use of the standard editing keys: Insert,
Delete, Backspace, Left and Right arrow keys, Home, and End. Escape
clears the entire field. The flashing cursor shows the current position
and is either an underscore or a full block to indicate normal
(overwrite) or insert modes. The editor can handle lines longer than the
actual display space. When you move the cursor near the edge of the
field, for equations and titles, it will automatically scroll to show
you the rest of the line.

Usually if you choose a function by mistake and it presents you a pop-up
prompt, pressing enter or ESC on the blank field will return you to the
equation editor as if you had never selected the option. Note that this
is not true for the Range function; if you press ESC at all 4 prompts
the current range will be preserved, BUT the screen will clear.



I. Using Graphmatica                                                   6
------------------------------------------------------------------------

                                  MENUS

All features of Graphmatica are available through drop-down menus, which
operate in a manner pretty close to Windows. (The one exception is that
they are not technically "pull-down" like the Macintosh's; you must
release the mouse button before the drop-down menu will appear.)

You have a choice of several ways to select menu items. If you have a
mouse, just point and click. You can also use the keyboard in several
ways. Note that to activate the menu for the first two ways you must
first press the Alt key or F10. (Pressing the Alt key or Escape exits the
menu, in case you decide you actually want to continue editing.)

1. Cursor keys: As you navigate through the menus with the cursor keys,
an explanation of the highlighted option appears on the bottom line of
the screen. Highlight the option you want by using the left and right
cursor keys. Select the highlighted option in the top bar by pressing
enter. Most top-level items call up a secondary drop-down menu.
Highlight the item you want using the up and down cursor keys. Select
the highlighted item by pressing enter again. To cancel the menu and
return to the top bar, press Escape. Or, use the left/right cursor keys
to move to the next item on the top bar.

2. Alphabetic/mnemonic keys: Each menu item also has associated with it
an alphabetic accelerator key that is shown in a different color when
the menu is activated.  Pressing this letter selects that menu item.
Most (but not all) of these letters are the first in the word, so after
a while they are easy to remember. Thus, to select the menu item _P_aper
in the _V_iew menu, where underlines indicate highlighting, you would
press Alt, V, P. Holding down Alt is OK, too.

3. Function keys: This is not the most intuitive way around, but I left
it in for old times' sake. If you prefer, you can use function keys;
they select the options in the current menu in the order they appear on
the screen. (i.e. in the main menu, F1 selects File, F2 Redraw, etc., and
F7 selects Help. In the View menu, F1 would select Clear, F2 Paper, and
so on.) Sorry, function key labels aren't available, but there are not
many options to learn. If you use function keys, though, you do NOT need
to press Alt or Enter to select items.


                              CHOICE BOXES

For prompts where you must choose between several options, (i.e. yes or
no questions) Graphmatica displays a dialog box with instructions and
two or more pushbuttons. You can respond to these either by clicking on
the desired button with the mouse, or by pressing the highlighted letter
of your choice. In addition, you can use the left and right arrow keys
to highlight your choice with the double border, then press Enter or the
space bar to select it. To cancel the dialog box, click the right mouse
button or press Escape. This does whatever is appropriate at that point
to evade the question and return you to the equation editor. If you
don't know how to answer a prompt, Escape will never hurt.



I. Using Graphmatica                                                   7
------------------------------------------------------------------------

                          HINTS FOR MOUSE USERS

Graphmatica has internal support for Microsoft (R) and compatible mice.
If a mouse driver is installed when you run Graphmatica, it will
automatically be detected and the usual mouse arrow will appear. To use
the mouse on the menus, you simply "point and click." However, there is
much more you can do with the mouse in Graphmatica:

 1. The left button, as usual, is always used to select an item or
    emulate a keystroke. Whenever you see a prompt that says "Press
    any key...", clicking the left button will have the same effect
    as hitting a key.
 2. The right button has the same effect as pressing the ESC key
    whenever pressing ESC is a valid option. To get out of a
    menu, you can click the right button.
 3. Whenever you are using Graphmatica's input editor, the mouse can
    be used to do two things: Pressing the right button, as usual,
    causes the same effect as pressing the ESC key. You can also
    position the cursor inside the edit field by clicking the left
    button on the space you want to move to.
 4. In the equation combobox and file selector, clicking on the up or
    down arrows on the scroll bar to the right of the listbox is
    equivalent to pressing those keys, and moves the highlight up or
    down one line. However, the rest of the scrollbar is display only;
    you cannot click on the track or drag the thumb to change the
    contents of the listbox.
 5. To use the mouse to scroll through a help file, click on the key
    name (highlighted in red on the bottom line of the screen which
    corresponds to the action you want to perform. In addition,
    clicking the right mouse button corresponds to the escape key.
 6. The mouse is not available during the actual graphing loop, to
    prevent the pointer and the graph drawing routine from fighting
    over the screen. To pause the graph, you must use the keyboard.
    The mouse can be used to restart or cancel the graph at this
    prompt though.
 7. To select a graph into the equation editor, simply position the
    pointer over any part of the curve and click the left button.
 8. To zoom in to a region on the screen, drag the mouse over the grid
    to highlight the area you want. The selection is constrained to keep
    a constant aspect ratio, so vertical movement is basically ignored.
    Just move the pointer up or down slightly and then mark out the area
    by moving across the screen. Then choose Range from the View menu to
    set a new range.
 9. To use the coordinate cursor to get a quick look at the coordinates
    of a point without using the menu, hold down the right mouse button
    while the pointer is over the grid.



Section                              ===================================
III III                              | Entering and graphing equations |
 I   I                               ===================================
 I   I
 I   I
III IIIo                                                               8
------------------------------------------------------------------------

                      GRAPHING CARTESIAN FUNCTIONS

So now that the cursor is blinking in the equation editor, what do you
do? Just type in an equation and press enter. If you've used a
programming language, spreadsheet, or math package on your computer
before, you probably already know the basic format by heart. Just take a
look at the sample files to make sure you know what everything means. If
you're confused about how to specify division or exponents, see the
operator table on the next page.

Here are the ground rules for normal Cartesian equations:
+ Always use exactly one "=" per equation
+ Always use exactly one "x" or "y"; you cannot use more than one of
  both variables.
+ Always put some sort of expression on both sides of the "="
+ An expression consists of any mathematically meaningful combination of
  decimal numbers, binary operations like + and *, parentheses,
  functions, and variables or constants.
+ Spaces are completely optional, except where they serve to separate
  alphabetic identifiers.
+ The order of operations is the standard algebraic left to right of:
    Functions
    Parentheses
    Exponents
    Multiplication and division
    Addition and subtraction

Graphmatica supports implied multiplication of variables and constants
as in "3x" or "5(2x+3)", but variables and other alphabetic
identifiers such as functions and constants MUST be separated by a space
or they will not be correctly identified. Thus  "xx", "xcos(x)", and
"xpi" are not recognized, while "x*x" and "x pi" are OK. (Implied
multiplication of the two variables x and y [e.g. "xy=1"] IS supported
as a special case, however. I still recommend including the times sign
in these cases for clarity.) Although I have greatly refined it in
recent releases, the parser may still misinterpret some complex
expressions for no apparent reason. Keep trying! I suggest liberal use
of parentheses: if you are not sure whether something will be
interpreted correctly, go back and put parentheses around it.


                            CALLING FUNCTIONS

The argument to a function is determined by a simple rule which lets you
avoid using parentheses in most cases: if the next non-blank character
after the function name is not "(" then Graphmatica picks out the next
complete term as the argument. A term is defined as the product of any
number of factors, each possibly raised to a power. Thus the following
expressions are all equivalent:

        sin 6x^2 + 2        sin 2*3*x^2 + 4        sin (6x^2) + 4



II. Entering and graphing equations                                    9
------------------------------------------------------------------------

                   OPERATORS, FUNCTIONS, AND VARIABLES

Graphmatica uses an operator set almost identical to BASIC's, with a few
additions. The supported operators and functions are as follows:

   Operator            Meaning
   =================   ==============================================
   +, -, *, /          Add, subtract, multiply, divide
   ^                   Exponentiation
   [( )]               Parentheses: may be nested to any extent,
                       brackets are provided for ease of reading but
                       parser WON'T differentiate between "(" and "["
   ; (semicolon)       Separate halves of a parametric equation
   ' (single quote)    Make rest of the equation a comment
   { m, n }            Specify domain, where 'm' is the start of the
                       domain and 'n' is the end. Either end may be
                       left open by omitting 'm' or 'n'.

   Function            Meaning
   =================   ==============================================
   abs                 absolute value
   acos, asin, atan    arc cosine, arc sine, and arc tangent
   asec, acsc, acot    arc secant, arc cosecant, and arc cotangent
   cos, cosh           cosine and hyperbolic cosine
   cot                 cotangent (1/tan)
   csc                 cosecant (1/sin)
   exp                 Euler's number to the specified power
   int                 greatest integer ([x] is not supported)
   ln, log             natural logarithm, logarithm base 10
   sec                 secant (1/cos)
   sin, sinh           sine, hyperbolic sine
   sqrt (or sqr)       square root
   tan, tanh           tangent, hyperbolic tangent

   Variables           Usage
   =================   ==============================================
   x, y                rectangular coordinates
   r, t                r and theta in polar coordinates
   x, y, t             x and y as functions of t in parametric form
   x, t, dx            dif-eq mode, solves first order o.d.e.
     [or y, x, dy]     dx is actually dx/dt in dx/dt = f(x,t)
   d2x, d3x, d4x       second, third, fourth order differentials
   t,x,y,z,w,dx...dw   solves up to 4th order linear system of ODEs
   a                   user-settable free variable; can specify range
                       and step rate to calculate several values

   Constant            Value
   =================   ==============================================
   b, c                user-settable free variables (constant)
   d                   converts degrees to radians = pi/180
   e                   Euler's number = 2.718...
   pi (or p)           pi = 3.14159...

Note: by default, all trig functions work in radians, not degrees. You
can convert using the constant d: e.g. sin(45d) = sin (pi/4)
cos (x*d) = cosine of x, in degrees (you will need to change the range
of x to 0 to 360 to get the full graph)



II. Entering and graphing equations                                   10
------------------------------------------------------------------------

                           GRAPHING RELATIONS

Graphmatica's equation parser will automatically isolate the variable
"y" wherever it is in the equation. It will graph some relations, like
circles ("x^2+y^2=36") and ellipses ("x^2/3 + y^2/4 = 20"), as well as
hyperbolas and many other conic sections. (Consult a good Algebra
textbook for help on their formulas.) It will also graph equations that
are functions of y, not x (i.e. there are 0 or >1 occurrences of "y"
but only one of "x" found). However, Graphmatica cannot perform the
factoring needed to isolate a variable when it occurs more than once so
implicit functions or relations like "x^2 - x = y^2 +4y" cannot be
graphed. The relation graphing module (for graphs which may have more
than one y value for a given x value) works like this: if in isolating
the "y" in an equation Graphmatica finds an even power of it (i.e.
"y^2"), it solves for both the positive and negative roots. This method
by no means covers all possible relations, but it is adequate for the
most common.

                          GRAPHING INEQUALITIES

You can graph rectangular-coordinate inequalities by replacing the '='
with '<' or '>' for many simple functions and relations. This feature is
presently only available for cartesian graphs. (Also, since the free
variable 'a' implies that more than one curve will be drawn, equations
containing 'a' are incompatible with inequalities, and the parser will
not accept any inequality that uses 'a'.)

The region that solves the inequality is hatched in with the graph
color. (The curve itself is not actually dotted to indicate a strict
inequality, however, so '<' is effectively "less than or equal to".) In
most cases, asymptotes are detected and a boundary added there as
appropriate, so graphs like "y < tan x" and "xy > 1" are drawn
correctly. In addition, the valid domain of the function being graphed
is detected automatically, so "y > log x", for instance, shades only the
first and fourth quadrants.

To accommodate intersecting regions, the graphing routine alternates
between '\' and '/' hatching. Best results will be obtained when you
graph no more than two inequalities on the same screen.  Not only does
this make it easier to interpret the graph, but it can also prevent
conflicts between graphs that cause the inequality to shade incorrectly.
The graphing routine first defines a region by solid lines in the
current graph color plus the edges of the screen. It then "seeds" this
region, which may be discontinuous, in one or more places, and flood-
fills it. Since the flood-fill operation does not know anything about
the actual curves, but rather stops as soon as it finds the boundary
color, the region MAY SHADE INCORRECTLY if another graph of the same
color intersects it.

Unfortunately, in 2-color modes (CGA and MCGA) every feature of the grid
is the same color as the graphs. Because the fill routine is not very
robust, even graph paper interferes with the shading if a hatched
pattern is used. Therefore, a 50% gray pattern is used instead to graph
inequalities accurately in monochrome modes. Sorry if this wears down
your printer ribbon, but there is nothing I can do about it without
writing my own fill routine. Also note that this method does not
accommodate intersecting regions gracefully.



II. Entering and graphing equations                                   11
------------------------------------------------------------------------

                          SPECIFYING THE DOMAIN

Graphmatica allows you to specify the domain of each equation
independently. This allows you to draw only a particular part of a graph
or change the domain without using the Range or T range functions to
change the default domain. To specify a domain for an equation, type
anywhere on the line the expression
                                { m, n }
where 'm' is the start of the domain and 'n' is the end. If you want the
domain to start at the default start, leave 'm' out. Then, whatever you
change the start of the default domain to, that will always be where the
equations starts graphing. To leave the end of the domain open, leave
out 'n'. So if the range on-screen is (-10,10), specifying a domain of
"{ ,5}" will graph from -10 to 5, and one of "{-4, }"  from -4 to 10.

To graph a parametric equation, you MUST specify a domain that is closed
(i.e. one that has neither number left out).

The domain is parsed like any other expression, so you can use all of
the arithmetic operators and functions available in the rest of the
equations, as well as numbers and the constants d, e, and pi. The only
restriction is that you may not use a variable (see table, page 7) in
your domain, since this would yield ambiguous results.


                          USING FREE VARIABLES

In any equation, in addition to the pre-programmed constants such as e
and pi, you can also include the free variables 'b' and 'c' which you
can define yourself. These parameters are effectively symbolic constants
that make it easier to "play around" with the exact shape of the curve
without editing the equation extensively.

The values of parameters used in each equation must be specified along
with that equation, but if you don't type them in explicitly,
Graphmatica will prompt you for the value(s) and insert this
information in the equation for you. If you want, you can type in a
value manually using a format similar to the domain specifier described
above, for example:
         {b: 1}                {c: -1/2}            {b: -pi/4}
As usual, enclose the "domain" in curly braces. You must specify which
parameter you want to set by its name ('b' or 'c') and a colon, followed
by a value, which can be any expression that evaluates to a constant.
You may not use other parameters either, since this could become self-
referential.

You can change the value of a free variable after you have typed in
equations and Graphmatica will automatically update and redraw all of
the graphs using it with the new value. See section .... for details.



II. Entering and graphing equations                                   12
------------------------------------------------------------------------

                     GRAPHING FAMILIES OF FUNCTIONS

The third free variable, 'a', is fundamentally different because you can
specify not just a single value, but a range of possible values that it
can take. This allows you to graph families of functions or level curves
of a 3-D surface easily. For instance, "y = a*cos(x)" will graph cosine
curves of varying amplitudes, and "x^2+y^2 = a" will draw level curves
of the surface "f(x,y) = x^2+y^2".

You don't even need to know the syntax described below to use this
feature, since Graphmatica will prompt for the needed values and insert
them in the equation for you. If you don't specify a range for 'a',
Graphmatica will ask you for three values: the start of the range, the
end of the range, and the amount to step by. Graphmatica starts by
graphing the function with 'a' set to the start of its range, and then
increments 'a' by the step value and draws another graph until 'a'
exceeds the end of its range. (You can also specify a negative step
value as long as the end of the range is less than the start.)

To type this information in on the command-line, add the domain
specifier "{a: start, end, step}" to your equation, replacing "start",
"end", and "step" with the desired values. For example, "y=a*cos(x)
{a: 1,6,2}" will draw graphs of y=cos x, y=3cos x, and y=5 cos x.

Although the program does not put any limit on the number of curves in
the "family" you can graph, be aware that this feature uses memory
rapidly. In any case, the screen will likely become too cluttered to be
useful if more than ten or so graphs are drawn, so try to make your step
rate proportionate to the size of the range.

                            DRAWING THE GRAPH

Once you have entered an equation that parses successfully and pressed
enter, Graphmatica will proceed to graph it. To interrupt a graph when
the computer is in the process of drawing it, hit any key and the
program will display on the bottom line the message:
   "PAUSE at x=#, y=#. Press ESC to quit, any other key to restart.. "
where # indicates the x and y coordinates you stopped it at. If you
mistyped the equation and want to fix it, just press ESC.

Be patient! Graphmatica may need a while to produce a quality graph on a
slower machine. To speed up the graphing, you may want to select a lower
Fineness value.

When the graph for your equation is complete, the blinking cursor will
reappear in the equation editor. If you'd like to start  over with a
completely different equation, press ESC to clear the input field. If
you'd rather modify the last equation, go right ahead; it's already
stored safe and sound in the redraw queue (for more information see
Section IV: Using AutoRedraw and the Redraw Queue). Or you can modify
any previously entered equation by pressing the down arrow key (or
clicking on the down arrow next to the editing field) to drop down the
equation queue listbox and then selecting an equation from the redraw
queue. You can then redraw this equation by pressing enter, or modify it
to create a new equation. To access Graphmatica's other features, press
Alt or hit escape on a blank line to raise the menu, or click on the
desired menu item with the mouse.



II. Entering and graphing equations                                   13
------------------------------------------------------------------------

                             ERROR MESSAGES

A number of error messages may be encountered when graphing (apart from
messages ingrained in the library functions which I cannot control).
Many of them are fatal; the equation cannot be graphed and you must
edit it. They will cause the computer to beep so you know there is a
problem. The other five apply only to specific point(s) for which a y-
value cannot be generated. They will not appear unless you ask for them
using the Warnings option and then they appear silently.

[Please note that all warning messages which refer to the variables 'x'
or 'y' will actually be 't' or 'r' when you are dealing with a polar
equation.]

"Please type an equation (or select from the listbox); then press <-'
to graph."
    You pressed enter on the graph line without selecting or typing in
    an equation.

"One or both sides of equation evaluated to nothing. Please edit your
equation."
    Make sure there is some sort of expression on each side of the
    equation. Obviously, an entry like "y=" can't produce a meaningful
    graph.

"No equals sign or more than one found. Press any key to edit the
equation."
    To be a valid and graphable, your equation must include exactly
    one equals sign ['=']. If you get this error, you either left out
    the '=' or accidentally typed two or more of them. For parametric
    equations, there must be an '=' on each side of the dividing ';'.

"No dependent variable or more than one 'y' found. Please edit your
equation."
    Although Graphmatica can isolate ONE 'y' or 'x' variable and graph
    some relations, it cannot perform the factoring needed to isolate the
    a variable when both 'x' and 'y' occur more than once. It also
    cannot plot polar graphs as a function of r instead of t or isolate
    'r' when more than one instance is found. If you can adjust the
    equation so it uses only one instance of the dependent variable, do
    so; otherwise it can't be graphed. In parametric graphing, this
    message may also indicate that no 'x' variable was found in the x(t)
    equation.

"Inequalities only supported for rectangular equations. Replace < or >
with =."
    Inequalities cannot presently be evaluated for polar, parametric, or
    differential equations. You may still be able to draw the graph if
    you can express the inequality in rectangular form.

"Found unknown variable or function. Press any key to edit the
equation."
    Unfortunately, the evaluator isn't set up to return what caused
    the error, so you'll have to look for it yourself. Check that your
    equation contains only valid identifiers (x, y, pi, d, e, r, t, a,
    b, and the functions listed above) and that you separated each of
    them with an operator, space, or some other punctuation.



II. Entering and graphing equations                                   14
------------------------------------------------------------------------

                        ERROR MESSAGES, continued

"Found bad operation or mismatched parentheses. Press any key to
edit..."
    You either left out a paren somewhere, left out one or both
    of the operands for a binary operation or the argument for a
    function, or typed some other weird thing the parser and evaluator
    couldn't digest. Examine your equation carefully and fix whatever
    seems to be the problem.

"Missing argument to a binary operation or function. Please edit your
 equation."
    The parser couldn't find any identifiers or expressions to use as
    one of the operands to a binary operation (like '+') or as the
    argument to a function, like cos(x). All of Graphmatica's
    functions require one argument following the function name.

"Can't find the inverse of this function of y. Press a key to edit
equation."
    You tried to graph an equation like "int(y)=x" for which y cannot be
    isolated by taking the inverse of the function. The functions which
    cannot be isolated are "cosh", "sinh", "tanh", and "int". If you
    can't adjust the equation so this error does not occur, it is not
    graphable.

"Parametric equation requires that you specify domain! See 'Graph' help
file."
    You typed in a parametric equation (or accidentally hit the
    semicolon) and neglected to include a closed domain [like {1,6}].
    Because the diversity of parametric equations makes it hard to pick
    a default domain, you have to include one with each parametric
    graph. See above in this section for help on giving the domain and
    below in Section III for help on parametrics.

"Domain did not evaluate to a constant value. Press a key to edit
equation."
    The domain you entered either could not be parsed, or was found to
    contain a non-constant identifier, like x or y. Valid domains must
    have at least one side of the range defined and can't contain
    variables, although any other expression that evaluates to a
    constant is OK.

"Not enough initial values to draw graph. Make sure there are # and try
again."
    Although first-order ODEs may be graphed as a slope-field without
    providing an initial value, second and higher order equations
    require that you provide initial values for t, x, dx..., up to the
    derivative one order less than the highest one in the equation. Make
    sure that you have specified the right number of valid IVs.

"Cannot accurately draw the graph of a discontinuous function (like
y=w^x, w<0)"
    The correct graph of this class of functions has singularities all
    over the place and thus cannot be drawn accurately using the
    algorithms Graphmatica normally uses on smooth curves.



II. Entering and graphing equations                                   15
------------------------------------------------------------------------


                         WARNING ERROR MESSAGES

"Overflow at x=#.##."
    Some function or operation generated a number too large to fit
    into a 8-byte floating point variable. The point at x=#.## was not
    graphed. This error is not fatal, so the graphing process is
    continued, but if the message is repeated and no image is graphed,
    you may need to abort graphing and look at your equation again.

"Division by zero at x=#.##."
    At x=#.## your equation included division by zero so that point
    was skipped. Unless you get this error repeatedly, there's no real
    problem.

"Can't raise a negative number to a fractional power. [x=#.##]"
    Due to the possibility of getting an even root of a negative
    number (like -16^(1/2) which actually equals the square root of
    -16), the C Library pow() function refuses to process any
    arguments like these. This is not a fatal error, and the portion
    of your graph (if any) where the base is not negative or the power
    is not fractional should be graphed perfectly. This error also
    occurs when you try to take the square root of a negative number
    with the "sqr" function.

"Can't find the logarithm of a negative number. [x=#.##]"
    The natural logarithm (ln) and base 10 logarithm (log) functions
    are defined only on x greater than zero.

"Inverse of abs() not defined on negative numbers. [x=#.##]"
    This reminder warning occurs when you graph a curve like abs(y)=x
    without restricting the domain of the expression equal to abs(y) to
    be positive.

"Domain error: asin/acos functions defined only on -1<=x<=1. [x=#.##]"
    The arcsine (asin) and arc cosine (acos) functions are only
    defined between -1 and 1 (the range of the sin and cos functions).



Section                                     ============================
III III III                                 | Extended Graph Functions |
 I   I   I                                  ============================
 I   I   I
 I   I   I
III III IIIo                                                          16
------------------------------------------------------------------------

                      INTRODUCTION TO POLAR GRAPHS

Polar coordinates are a fundamentally different approach to representing
curves in two-dimensional space. The concept is pretty easy to grasp
graphically, but if you have never used polar coordinates and want to
understand them, you should probably read the section below.

The traditional Cartesian method relies on an x and a y coordinate to
mark how far a point is from the axes in two perpendicular directions;
polar coordinates plot the location of a point by one coordinate
represented by the greek letter theta (looks like an O with a line
through the middle--one of these methods should print it: Θ or O-) which
is simplified to "t" in Graphmatica and another called "r". The "t"
tells what direction to go in from the origin, and the "r" tells how far
to go out in that direction to reach the point. The direction is
measured in radians as an angle starting from the positive side of the
x-axis and turning around counter-clockwise (like measuring the angle
the hand on a clock has traveled starting at the 3 o'clock position and
going backwards). There are 2pi radians in a complete circle,
corresponding to 360 of the degrees you're familiar with. To put a polar
coordinate into Cartesian terms in order to graph it, we use the
equations: x = r cos (theta)  and  y = r sin (theta).

To make a graph using polar coordinates, we let theta be the independent
variable and calculate a distance to plot out from the origin as we let
the angle sweep around in the positive direction. The domain for the
graphing is 0 to 2pi (the first complete circle in the positive
direction), but you can easily change these values using the "T range"
function in the View menu [see Section V]. Polar graphs can be entered
in the equation editor just like normal graphs. The only difference in
what you type, and the way Graphmatica detects a polar graph, is that
you must use the variables "t" and "r" instead of "x" and "y". The
restrictions are still the same: you can have one and only one instance
of the dependent variable "r," although it can be located almost
anywhere in the equation. You can embed the "r" in a term like "r^2" to
graph functions that cannot be simplified by normal means and
Graphmatica will evaluate both positive and negative roots automat-
ically. You should watch as your graph is drawn, because often the
direction it is going is almost as important as the figure it draws.
(When you have a "double" equation with "r^2" in it, though, note that
the positive roots are drawn first and then the negative roots are
drawn: theoretically they should be drawn simultaneously, but this is
not practical.)

Please note that the x and y coordinate ranges and the range for the
variable theta function completely independently; in normal Cartesian
graphing, theta's value is irrelevant, and in polar graphing, theta
controls the domain of the graph, but the x and y ranges still control
the physical screen you see. If you want to change your view of a polar
graph, you use the scale or range functions just as you would normally.



III. Extended graph functions                                         17
------------------------------------------------------------------------

                    INTRODUCTION TO PARAMETRIC GRAPHS

Parametric graphing, like polar graphing, uses a different method of
calculating points on the plane to come up with curves that may be
difficult to compute using normal rectangular coordinates. They are
unique in that the cartesian x and y coordinates are calculated based on
a third variable (the "parameter" of x and y) which is traditionally
called 't' (not to be confused with the 't' used by Graphmatica to
represent theta). T is allowed to increase from the start of the domain
you specify to the end. At each value, the functions x(t) and y(t) are
calculated to give an (x,y) coordinate which is graphed. Graphmatica
then connects these points to form a smooth curve--if something you
graph begins to look jagged, you probably need to adjust the fineness.
(Parametric graph fineness is linked to the same fineness control as
Cartesian and polar graphing, and should be decent at the default
fineness value, but if you need to, you can increase or decrease this
value. Be aware that this will affect the fineness of non-parametric
graphs as well. See Section VII "Adjusting the Fineness" for details.)

To enter a parametric graph, you need to remember four basic parts: the
x(t) and y(t) functions, the semicolon between them (this is how
Graphmatica knows it's a parametric graph), and the domain for t.
                      semicolon
                x-function |   y-function    domain
                    |      |        |           |
                  x =  2t  ;   y =  2t^2   {-10, 10}

Although as in all other Graphmatica equations you don't need to solve
for x and y (i.e. t=5x would be OK), only one x and one y can appear in
the whole equation, and "double" equations like "x^2=t" where
Graphmatica would normally solve for both the positive and negative
roots are NOT supported (you can enter them but only the positive root
will be found; each equation in the redraw queue is allowed enough space
to hold two computable expressions, and either a double equation or an
x/y pair of parametric equations will fit, but not both). You can type
the x and y equations in in either order, as long as they are separated
by a semicolon, and the domain will be recognized anywhere on the line.
You MUST specify a domain for each parametric equation! The variety of
curves that can be drawn with parametric equations is great and makes
choosing an appropriate default domain impossible. Some curves (like
those including the circular functions sine and cosine) tend to work
best over a {0,2pi} domain, like polar graphs. Others will match up
better with the default domain of the normal graphs, the size of the
viewing area. Some have a very compact domain, between say 0 and 1,
where they will appear on the screen. If you over- or under-estimate the
domain, you can always abort the graph and edit your equation.



III. Extended graph functions                                         18
------------------------------------------------------------------------

               COMPARISON OF THE THREE GRAPHING TECHNIQUES

    While some curves can be drawn by cartesian relations, polar
coordinates, and parametric functions, each technique is better suited
for some graphs than for others. For instance, a circle with radius 5
around the origin which can be produced by the equation
                             x^2 + y^2 = 25
can be drawn faster by the parametric equations
                 x = 5 cos (t) ; y = 5 sin (t)  {0, 2p}
and can be drawn faster and much more simply by the polar graph
                                 r = 5.


            GRAPHING APPROXIMATIONS OF DIFFERENTIAL EQUATIONS

Graphmatica also provides the ability to approximate the solutions of up
to fourth-order ordinary differential equations. [I will not provide
background material on this function because if you need to use it, you
probably know more about differential equations than I do.] To let the
parser know you want to graph a differential equation, you must include
the differential "dx" (which actually represents dx/dt) as one of your
variables. If you specify an equation as dx = f(x,t) where f(x,t) is
some combination of the variables x and t (such as "x^3 + t" or "t * x")
and do NOT include the domain operator "{ , }", the program will draw a
slope field for dx/dt = f(x,t), with t as the horizontal axis and x the
vertical. Note that you can also use the notation dy = f(y,x) if you
prefer; both sets of variables are automatically recognized as
differential equations.

If you do include a "domain" {m, n}, however, it will not be interpreted
as a domain but will instead indicate that you want to graph a specific
solution to the initial-value problem x(m)=n by Runge-Kutta approximation.

Graphmatica will also solve second, third, and fourth order initial-
value problems using a Runge-Kutta method for linear systems. To specify
a second or higher order derivative, use the variables d2x, d3x, or d4x.
Remember that for an nth order equation, you must also specify n+1
initial values. You can type these into the equation using the "domain"
notation described above; the order of values is t, x, dx, d2x, d3x.
Thus d2x + x=0 {0,0,1} graphs a sine curve as the solution to
"d^2x/dt^2 + x = 0" for x = 0 and dx/dt = 1 at t=0.

In addition, you can specify an initial value by moving the crosshairs
around with the arrow keys or mouse. Type in an equation to add an IV to
or select an equation to modify and choose the Set IV option from the
Point menu. Move the cursor to the desired location and press enter or
click the left mouse button to add the initial value point to the
selected equation. For second-order or higher ODEs, you can also specify
the first derivative using the mouse or arrow keys. After you select the
IV point, Graphmatica displays a "rubber band" line from this point to
the current position until you click/press enter on a second point. The
coordinates of the first point are then taken to be the t and x values
for the IV, and dx/dt is initially set to the slope of the line between
the two points. If required, further initial values will be requested
via dialog boxes.



III. Extended graph functions                                         19
------------------------------------------------------------------------

                    SYSTEMS OF DIFFERENTIAL EQUATIONS

Graphmatica can also approximate the solutions of up to fourth-order
systems of ordinary differential equations. [Refer to your local
textbook for more background information.] You can pose your system in
either of two sets of variables: { t, x, dx, y, dy, z, dz, w, dw} or
{ t, x1, dx1, x2, dx2, x3, dx3, x4, dx4 }. In each case, t is the
independent variable, and the variables dX are the derivatives dX/dt.
You specify the complete system in a manner similar to parametric
equations, namely (for a fourth-order system):

dx=f1(x,y,z,w,t); dy=f2(..); dz=f3(..); dw=f4(..) {t0,x0,y0,z0,w0}

where f1...f4 are some functions of the variables x,y,z,w, and t, and
x0...w0 are the initial values of the system at t0. (You need not
isolate the differentials on the left of the equals sign, but you can only
use one per equation.) Separate equations with semi-colons, and enclose
the initial values in curly braces.

If you want to graph a third- or second-order system, just leave out the
equations for dw, or for dz and dw, and use the appropriate number of
initial values instead.

Note that to simplify the process of parsing the system and associating
each equation in the system with a graph, the order of equations dx,
dy,dz,dw (or dx1, dx2, dx3, dx4) is required, and the program will not
graph a system whose equations are listed in the wrong order.

The values for each curve in the system are calculated simultaneously,
but only the curve for x(t) is drawn as it is calculated; the others are
displayed as soon as calculation is finished. Each curve plots in a
different color, which corresponds to the color of its equation in the
system when it is listed in the status bar.

You may choose to plot an alternate coordinate space using the "vars"
domain: for instance, adding "{vars: x,y}" to an ODE system will plot
the solution for x against y over the default interval of t.

The algorithm used to compute all of the ODE approximations described
above is the 4th order Runge-Kutta routine with adaptive step size
documented in chapter 15 of "Numerical Recipes in C".

You can also choose the initial value points using the mouse. The
procedure is the same as the one for a single ODE, except for systems of
ODEs, this function allows you to click on 2, 3, or 4 initial value
points, one for each equation in the system. Since all IVs are required
to be at the same coordinate for t, after selecting the first IV you are
restricted to selecting points on the vertical line defined by the t-
coordinate of your first click. Graphmatica marks each IV as you select
it so that you will not forget how many IVs you have placed already and
how many you have left.



Section                        =========================================
III VVV VVV                    | Using AutoRedraw and the Redraw Queue |
 I   V   V                     =========================================
 I   V   V
 I    V V
III    V o                                                            20
------------------------------------------------------------------------

                            USING AutoRedraw

Whenever you change the scale to look at a graph in greater detail or
from farther back, or you shift the range of the axes so that the graph
you just drew will be centered, the screen must be cleared so that a new
coordinate grid can be drawn. So you're left with the viewing angle you
wanted for that graph, but the screen is blank. You shouldn't have to
solve this problem by retyping the whole equation again...and you don't.
You can redraw the equation much more easily by selecting Pick graph in
the Redraw menu. But with AutoRedraw ON, you don't even have to do that.

As its name suggests, AutoRedraw will redraw the last equation(s) you
typed in automatically when you change the scale or range. If you don't
want a graph redrawn, you can abort it by pressing ESC twice and you
will be returned to the equation editor. If you want to turn off
AutoRedraw completely, just select AutoRedraw under the Options menu and
answer "n" when asked if you want AutoRedraw on.

To change the number of graphs redrawn automatically, you should also
select AutoRedraw from the menu. After responding "y" when asked if you
want it on, enter the number of graphs (1-25) that you would like to be
redrawn each time AutoRedraw is invoked. (If there aren't that many in
the queue, Graphmatica will just redraw them all.)

AutoRedraw is ON by default. You can change the default to OFF by
turning it off and using the Save Setup command to record this change in
the GRAPHMAT.INI file.

                            THE REDRAW QUEUE

Every time you type in an equation, the character string you typed and
the program's internal representation of that equation get stored in the
redraw queue. From this queue, or list, you can call up any of the last
25 equations you typed to graph again with fewer keystrokes than
retyping it. Or you can redraw all of them or the just the last one you
typed with even fewer keystrokes. The equations in the queue are stored
in an order that puts the most-recently-used at the beginning and the
least-recently-used at the end. Hopefully, when the queue fills up, the
equations that are bumped off the queue at the end will be the ones you
won't miss very much.

As explained above, the screen clears whenever you execute Range or a
successful Scale (if you abort Scale the screen remains as it was). If
you execute either of these functions or clear the screen accidentally
with the Clear menu option, you can redraw the last equation (if
AutoRedraw hasn't already) by selecting Last graph under Redraw, or you
can redraw all of the equations in the queue using Redraw All under the
Redraw menu. Note that the redraw function does not keep track of which
graphs are on the screen at any point in time so "redrawing all" may put
more (if you plotted many equations on the screen before the last one)
or fewer (if you cleared the queue) equations on the screen than there
were before the screen was cleared.



IV. Using AutoRedraw and the Redraw Queue                             21
------------------------------------------------------------------------


                           REDRAWING MANUALLY

The Pick graph option in the Redraw menu allows you to graph any
equation currently in the redraw queue. First choose an equation from
the combobox by clicking on it with the mouse or using the down key to
highlight it and pressing enter. Then, choose Pick graph from the menu.
(If you do not select an equation, Graphmatica redraws the first one in
the queue.) When you pick a graph, it is re-inserted at the top of the
redraw queue, as you'll likely be working with it again in the future.
So if you scale the grid again, that graph will be Auto-Redrawn, and if
you redraw the last graph, that graph will be shown.

You can redraw a single equation more efficiently within the equation
editor by selecting the equation as above and then simply pressing enter
(do not modify the equation) while it is in the equation editor. The
only advantage of the menu item is that it allows you to pick and redraw
one-handed with a mouse.


                          DELETING AN EQUATION

To remove an equation that is no longer of interest from the queue,
simply select that equation from the combobox (see Section I for
details) and then choose the Delete Eqn item from the Redraw menu. The
equation is then vaporized from the queue, although its graph will
remain on the screen until you clear the screen.


                        CLEARING THE REDRAW QUEUE

Selecting Clear All from the Redraw menu essentially starts a new
session, except the options settings and grid size remain the same.
All of the equations in the queue are deleted, and the screen and labels
(including annotations) are cleared as well.



IV. Using AutoRedraw and the Redraw Queue                             22
------------------------------------------------------------------------

                    SAVING AND LOADING EQUATION LISTS

To save the list of equations you're working on, as well as the title
and y-axis labels if you've entered them, simply select Save List from
the File menu and enter a filename. This file will be saved in the
current directory with an extension of ".GR" (for GRaphmatica) unless
you specify differently. To save to diskette, you simply have to insert
"A:" or "B:" before your filename. Longer directories may not fit. When
asked if you want to include setup information, answer "y" if you want
to record the current size of the grid and options settings in the
equation list file. Otherwise, only the text labels and equations in the
queue are saved in the file.

To reload your list of equations or load up one of the demo files
provided with Graphmatica, you have more options. You can automatically
load a file when you run Graphmatica by typing its name (with or without
a file path or the ".GR" extension) as a command-line argument. Then
those equations will be loaded up (and graphed) immediately after the
opening screen, unless the file includes the autoredraw=off statement
(see Appendix B: editing equation lists). You can also load an equation
list at any time by selecting Open List from the File menu. You will
be presented with a list of files available (if there are any) which can
be selected using the up/down arrow keys or mouse. This list includes
two special options, "[Cancel]" and "[New path]". Select cancel to
return to the equation editor without loading a file, and new path to
change the current directory to look for files elsewhere.

After you select a file to load, if the redraw queue is not empty, you
must decide whether to add the new file's equations to the existing
queue or replace its current contents. Answer "y" to the prompt to clear
the queue before loading the new file, or "n" to keep it.

When the file has been loaded, if AutoRedraw is on Graphmatica will
proceed to draw all of the equations just loaded (but not the ones that
were previously in the queue, if you didn't clear it). You can abort
these by pressing ESC twice on each if you are impatient. The graph
title and Y-axis labels (if the file had any) that were loaded won't be
displayed until you print or do a Print Preview (see Section VIII).

If the file was saved with the Save setup option on, the grid and
special options settings recorded in the file will be loaded and set
automatically.



Section                  ===============================================
VVV VVV                  | View: Changing the appearance of the screen |
 V   V                   ===============================================
 V   V
  V V
   V o                                                                23
------------------------------------------------------------------------

                          CLEARING THE SCREEN

Select Clear in the View menu to clear the screen. Everything will be
erased and the grid will then be redrawn. Remember, every equation you
have typed is still stored and can be redrawn later even after you have
cleared the screen. If an error has garbled the whole screen, here's a
hint to help regenerate it: first press ESC until the top menu bar is
refreshed, and then use Clear to redraw the rest of the screen.


                  CHANGING THE BACKGROUND GRAPH PAPER

Graphmatica has three additional types of graph paper that may be more
appropriate than the regular rectangular grid in some situations: polar
paper and trig paper. Polar paper, instead of a grid of squares,
consists of concentric circles marking the distance r, and radiating
lines marking the angle theta at intervals of pi/6 radians. Trig paper,
intended to ease interpretation of trigonometric functions, uses the
regular rectangular grid but the x-axis is labelled in multiples of pi
instead of integers. Logarithmic paper allows you to scale one or both
axes logarithmically, so that, for instance, 0.1, 1, and 10 are equally
spaced.

You can choose from three levels of detail for the grid: none, reference
dots, and full gridlines. Note that polar and rectangular paper look the
same when you turn the grid off. Also, you can choose between five
settings for the appearance of the x and y axes: no axes, plain axes
with no decorations, axes with arrows on the positive end, axes labelled
with "x" and "y", and axes with both arrows and labels. Note that even
with "No axes" selected, the legends may still be displayed along the
invisible axes unless you turn them off independently.

You can change the graph paper type, grid detail, and axis decorations
by selecting the Paper option from the View menu and responding to the
prompts, "Select type of graph paper:", "Select grid detail level:", and
finally "Select axis decorations:" appropriately.

If you select Logarithmic paper, you will also be asked to specify a
base and choose between semi-log (y axis only) and log-log paper (both
axes logarithmic). If you choose base 10, you will see 9 lines per
decade just like real graph paper if there is enough space to display
them (generally when less than four decades are shown). Otherwise, the
graph paper will look like normal rectangular, but with gridlines at
integer powers of the base. Note that when you switch from or to log
paper, the graphs that are on-screen will have to be recalculated, since
the shape of the curves is distorted when you take the logarithm.

Your choice of graph paper type, like all of Graphmatica's options, is
recorded in your equation list files when you choose to save setup
information, so when you load up an equation list later the correct
type of graph paper will automatically be displayed. See the demo files
TRIG.GR, POLAR.GR, and DIFEQ.GR for examples of this.



V. View: Changing the Appearance of the Screen                        24
------------------------------------------------------------------------

                           CHANGING THE SCALE

The scale function allows you to change the scale of the graph you have
on-screen while keeping the center of the display the same (the origin
may move but if (2,2) is at the middle of the screen, it will stay
there). The scale is relative to the size of the grid that is presently
on the screen, so you need not make any comparison with the default
grid. To rescale the grid, select Scale from the View menu. Then you
must provide a "scale factor" to tell the program how much to change the
size of the grid.

Enter a positive number to zoom in, or a negative number to pan out. (In
the Camcorder terminology, a positive number acts as a telephoto and a
negative as a wide-angle lens.) The number of units across the screen
will be divided or multiplied by that number accordingly. Numbers
between -1 and 1, inclusive, are not allowed because they produce
meaningless values. Decimals greater than 1 or less than -1 are valid
though.

If you enter an invalid number, you will be asked to enter the scale
again. To abort rescaling, leave the field blank and just press enter.

The program remembers the value you type in and presents it as a default
the next time you Scale the grid. The original default is "2".


                           ADJUSTING THE RANGE

Graphmatica allows you to adjust the start and end of the x- and y-
ranges independently, so as to create a perfect fit for whatever
function you are graphing. Or, you can constrain one or more of the
coordinates to produce a graph with a square aspect ratio without
actually figuring out all of the values by hand.

Choose the Range item from the View menu to modify the grid. To create a
"custom" grid, estimate the top-, bottom-, left- and rightmost
extremities of the graph, and respond with the proper values to the 4
Range prompts. The new grid may be somewhat expanded or compressed
depending on the aspect ratio, but it will show the part of the graph
you're interested in the best possible detail.

To create a "square" grid where both x and y axis have the same scale,
just type "auto" for any of the four coordinates and it will
automatically be scaled properly to fit the other three. If you pick the
least significant coordinate to AutoScale, you can match the other three
exactly, framing your graph almost as well as a custom range, but with a
more natural aspect ratio. If you want to show the same amount above as
below the axis, type "auto" at both of the y-coordinate prompts and the
top and bottom of the grid will be set to properly scaled equal and
opposite values. (AutoScaling of both x-coordinates based on the height
of the grid is not supported.)

All range prompts allow you to input any valid expression, not just
decimal numbers. You can easily enter very large or small numbers using
scientific notation: e.g. 1.2*10^-8 for 0.000000012 or 3.4*10^7 for 34
million.



V. View: Changing the Appearance of the Screen                        25
------------------------------------------------------------------------

                       CHANGING THE RANGE OF THETA

Because the independent variable (theta) in polar coordinates is
fundamentally different from the 'x' of Cartesian coordinates, the
Cartesian x/y ranges can only be used to determine the size of the
screen and not the domain of the equation graphed for polar graphs.
Although the default 0 to 2pi range is the typical range of theta used
for most graphs that go on forever (like spirals) and some closed graphs
(like circles), other graphs cannot be completely drawn in this range of
theta. For instance, the figure-8-shaped "r^2=64cos(2t)", because it is
undefined where the right half is less than zero, is missing a couple of
spots unless theta's range is extended to -2pi to 2pi. To allow the
greatest flexibility, Graphmatica allows theta's range to be changed
independently of all other options.

To change the range, select "T range" from the View menu and enter
the start and end of the range you want (or press enter on the blank
line to keep the current range). You can type in any expression,
even using fractions and functions, as long as it evaluates to a
constant. In particular, although Graphmatica works exclusively with
radians, you can enter the range in degrees by typing in a number times
the constant 'd', which converts degrees to radians, as in "45d" for 45
degrees. You can also easily specify a radian measure as a multiple of
pi, as in "2p" for 2 pi.

If you would rather not change the default domain but simply change the
domain for a single graph, see Section II "Specifying a Domain".

                      CHANGING THE BACKGROUND COLOR
Although I've hand picked the default colors for good contrast, if you
find them dissatisfactory you have two alternate choices to the default
charcoal gray grid: a white or black background. The graph colors stay
the same, except for a few light colors which are replaced by darker
ones on the white background. Hopefully, at least one of these color
schemes will work for you.

To change the color scheme, select Colors from the View menu, and select
Gray, Black, or White. To make your choice of color "stick," select
Save Setup from the File menu and it will be written to your
graphmat.ini file. Note that color selection is not saved with .GR
files, only in your personal setup, since it has more to do with your
hardware than the equations you are working with.



Section                   ==============================================
VVV VVV III               | Labels: Adding Explanatory Text to a Graph |
 V   V   I                ==============================================
 V   V   I
  V V    I
   V    IIIo                                                          26
------------------------------------------------------------------------


                              AXIS LEGENDS

By default, Graphmatica labels the hatch marks across the graph with the
number of their coordinate so that you can more easily locate points or
find the coordinates of a graphed point. In some cases, though,
especially when doing graphs near the axes, the numbers can be confusing
and you may want to turn them off. To do this, select the Legends option
in the Labels menu. In the choice box, select the None button.

If you find that the interval that is automatically chosen for the grid
spacing is not well-suited to your task, select the Custom spacing
button and you will be prompted for the increments you prefer for the X
and Y axes. Next, you can choose between two behaviors when the grid
size changes: 1) Lock the legend spacing to the values you selected, or
2) Scale the grid spacing along with the grid itself. Unfortunately,
neither option will be likely to satisfy you if you change the grid
spacing by a non-integer factor or by too much, so you may have to reset
the spacing to more suitable increments again manually.

To go back to the default behavior, select the Automatic spacing button
in the Axis Legends dialog.


                      ADDING TITLES TO YOUR GRAPHS

If you want to add a title line and/or labels along the sides to your
graph, select Titles from the Labels menu. You can type in a new title,
edit one you typed in previously, or delete the title by pressing enter
on the blank line. For the y-axis labels, the length of the string you
can use depends on the number of rows you monitor has: CGA and EGA can
hold 23 letters and VGA can hold 28. The program presents you with the
previous label which you can edit, discard by pressing ESC and typing a
new string, or turn off completely by pressing ESC and then enter on the
blank line.

When you print (or select Print Preview from the menu), the title is
written across the top of the screen where the equation editor normally
is, and it is automatically centered. Do not try to center the title
manually when you are typing it in or it will interfere with the
automatic centering. When you are graphing and the menu is active, the
title line is not shown.  Like the title, the y-axis labels are
automatically centered (vertically) on the graph. Graphmatica does not
draw these labels immediately, since they are intended mainly to enhance
hardcopy printing, but they are also displayed if you perform the
Preview option under the File menu.

These two features were previously found in the View menu.



VI. Labels: Adding Explanatory Text to a Graph                        27
------------------------------------------------------------------------


                               ANNOTATIONS

Annotations are short text phrases which you can place anywhere on the
graph surface to label intersections, zero-crossings, and other
important points on your graphs. Graphmatica maintains a list of up to
ten 30-character annotations.

To add an annotation to the graph, select Annotate from the Labels menu.
Type in the text you wish to use to label the graph and press enter. Now
a crosshair cursor will appear. Move it with the keyboard or mouse to
the position where you want the upper-left corner of the text. Then
click the left button or press enter to paste it down. The text will
appear on the screen immediately. Remember that it is associated with a
logical x,y coordinate on the grid, not with the physical point you
clicked on or with any particular equation. Thus, when you rescale the
graph or change the range, the annotation will reappear at the same
logical but not always the same physical position. Also note that
annotations that would run off the screen cannot be drawn. You must zoom
out so they are fully on the screen before they will appear.

To edit a previously entered annotation, select the Edit Annote menu
item from the Labels menu. Next choose the annotation to edit from the
listbox. You can then choose the delete the annotation, change its text,
or change its position. Pressing Esc at this prompt instead leaves the
annotation unchanged.

All of the annotations you have entered will be saved with any equation
list you save, and will automatically be displayed again when you reload
it. Like the graph title, the current annotations are cleared before
loading a new file and when you Clear all of the equations from the
redraw queue.



Section                          =======================================
VVV VVV III III                  | Viewing and Setting Special Options |
 V   V   I   I                   =======================================
 V   V   I   I
  V V    I   I
   V    III IIIo                                                      28
------------------------------------------------------------------------

                          THE DEFAULT SETTINGS

When you run Graphmatica without a GRAPHMAT.INI file, the following
settings are in effect:

    Start of range:  (-8.0, *)
      End of range:  (8.0, *)
          Fineness:  1.0 **
           Legends:  ON              Warnings:  OFF
           Printer:  OFF           AutoRedraw:  ON
           AutoNum:  1            Graph paper:  Rectangular, Grid
       Theta Range:  0 to 6.28 (2pi)

*  varies depending on video mode; always equal and opposite and scaled
   to provide a square aspect ratio.
** on CGA systems fineness is automatically set down to 0.75 to speed up
   graphing.
This provides a basic rectangular grid of decent size with the origin
centered, a square aspect ratio (a 1x1 square on the grid really LOOKS
square), and a good resolution graph.

                          VIEWING THE SETTINGS

The current settings are almost always displayed whenever you change one
of them. Every item under Options (unless aborted) will bring up the
settings screen. You can also look at them manually by selecting
Settings in the Options menu.

                        SAVING SETUP INFORMATION

You can save your preferred grid size and settings of other user options
so that whenever you run Graphmatica again they will automatically be
restored. You can do this at any time by selecting the Save Setup item
in the File menu. The options will be saved in the GRAPHMAT.INI file in
the current directory. If this is not where the program files are
stored, Graphmatica will not be able to find it again, so if you change
the directory using the load or save list commands, be sure to reset it
before you save the setup.

The setup file is just a special equation list that is loaded
automatically when you start Graphmatica. It follows the same format as
a normal equation list (described in Editing Equation Lists) except that
when you save it Graphmatica leaves out the labels and the equations
(the important parts of normal equation lists).

When Graphmatica saves your options, to simplify things it only records
those options that are different from its own internal defaults. (See
Default Settings for a list of these.) If your settings are close to the
defaults, your setup file will be very short.



VII. Viewing and Setting Special Options                              29
------------------------------------------------------------------------

                         ADJUSTING THE FINENESS

The fineness factor determines the resolution of the graph, and in
effect, the amount of time it takes to draw it. The fineness factor of 1
is quite adequate in most circumstances. Graphmatica's SmartFineness
feature dynamically adjusts the actual step rate, making the manual
fineness adjustment almost obsolete, but I left it in to accommodate the
need for especially sharp graphs or for especially quick renderings.
SmartFineness automatically decreases the fineness when the point being
graphed is not on the screen to speed up blank areas, then recovers
automatically and backtracks to the spot where the graph comes on-screen
again. It also varies the fineness with the slope of the equation being
graphed--less when the slope is near-horizontal, greater when the slope
tends toward vertical, so steep graphs are tracked more effectively. As
the fineness is increased, the curve will become smoother--to a point.
Remember that for any increase in the fineness factor, there will be a
proportional increase in the time it takes to draw a graph.

If you have CGA adapter, Graphmatica assumes you are using an 8088-class
PC and reduces the fineness to 0.75 to speed up graphing; the difference
in resolution is usually not noticeable, but the difference in speed is.

Fineness is also linked to the rate at which the angle is allowed to
change in polar graphs, and Cartesian and Polar graphs will be of
comparable quality at the same fineness factor. Fineness is also linked
to the step rate of parametric graphs and differential equations;
because they vary so much, it is harder to insure that all parametrics
and dif-eqs will graph well at the default fineness, but those I have
tested look fine.

To change the fineness factor, select Fineness from the Options menu.
Any value greater than zero is valid, but I would not recommend going
below 0.25, as the image quality suffers and the graphs begin to look
like modern art. Also, a factor greater than 5, in addition to being
slow, can overflow the list of plotted points kept for quick redraws,
which may actually cause the graph to display incorrectly if you change
the graph paper or another option that forces the screen to be redrawn.

                           WARNING MESSAGES

By default, the error messages which do not require the equation to be
edited are suppressed, because they slow down the graphing process,
cover up the equation, and are somewhat annoying when you know the graph
shouldn't produce any values in some area. (Also, it saves you the
effort of specifying the domain for each equation.) If something goes
wrong (e.g. the graph doesn't show up on screen when it should and the
reason isn't readily apparent), you can turn on the warning messages
(using the Warnings function of the Options menu) and redraw it to see
what the problem is.

                        CHANGING THE DRAWING MODE

Graphmatica can render graphs in two ways: by drawing dots to represent
the points it has calculated or by drawing lines to "connect the dots."
By default, graphs are drawn using lines, because at a minimal expense
of time, this makes the graphs much more visible. If you want, however,
you can change the drawing mode to "dots" using the Draw Mode command in
the Options menu.



Section                             ====================================
VVV VVV III III III                 | Using a Printer with Graphmatica |
 V   V   I   I   I                  ====================================
 V   V   I   I   I
  V V    I   I   I
   V    III III IIIo                                                  30
------------------------------------------------------------------------


                    PRINTING PICTURES OF YOUR GRAPHS

Graphmatica includes two internal printing functions which work with CGA,
EGA, and VGA and many common printers. To print, select the Print
command from the File menu and then choose the printer type closest to
your own. If you have an IBM, Epson, or compatible dot-matrix graphics-
capable printer or a laser printer that doesn't support HP PCL but has
Epson emulation, select Epson dot-matrix. If you have a Hewlett-Packard
DeskJet, LaserJet, or other laser or ink jet printer compatible with
the HP Printer Control Language Level III, choose HP LaserJet/DeskJet.
If you didn't mean to choose the print command, press ESC to abort.

Graphmatica does not check the printer's status, so you must have it on
before you try to print.

All the graphs on the screen will be printed on the printer, with the
top of the screen at the top of the paper. At the bottom of the page is
a list of the equations for the graphs Graphmatica thinks are presently
on the screen. All graphs you have drawn since the last time the screen
was cleared, except ones that you aborted by pressing ESC, are included
in this list.

If you want to abort the printing process, press ESC. Don't turn off or
reset the printer until the "Printing..." message at the bottom of the
screen is gone or DOS may print an error message and mess up the screen.

The output is as close to what you see on the screen as I can make it,
but since none of the supported display resolutions makes a very good
match with the printer's resolution (aside from VGA on a LaserJet), I had
to do some stretching and squeezing to make the printouts the right
shape. Thus the size of the printout will vary between different monitor
types because to make the grid square, I had to expand the x axis
different amounts to make up for the different vertical resolutions.

On dot matrix printers, for CGA mode, the x direction is expanded by 33%
and the y by 100%. For EGA, the x is expanded by 20% and the y remains
as it was read off the screen. For VGA, the x is expanded 50% and the y
is compressed about 9% by merging pairs of adjacent lines. For super-
VGA (800x600), the x direction is expanded by 20% and the y is
compressed by 27% by merging adjacent lines. For this reason, dot matrix
printouts at high resolutions may not be as sharp as EGA resolution

On laser printers, Graphmatica produces crisper, more compact images due
to the superior resolution. The VGA image will print the best with a
laser printer: every point is mapped on a one-to-one basis due to the
square aspect ratios of both the screen and printer. EGA and CGA images
are scaled accordingly so the output isn't squashed, and they print
graphics that are not as sharp but still much better than dot-matrix
output.



VIII. Using a Printer with Graphmatica                                31
------------------------------------------------------------------------

                            NOTES ON PRINTING

Graphmatica assumes your printer resides at LPT1:. If your printer is
not connected to the parallel port but to the serial port instead, you
must use the DOS "mode" command to redirect the output if you want to
use the printer. For most people, the command would be
    C:\>mode lpt1 = com1
You need to use this command only once before you run Graphmatica; DOS
will remember the redirection until you shut off the computer or reboot.

If you have a color printer and a paint program that can print in
color, you may want to export the grid to a PCX file and use the paint
program to print. (Graphmatica for DOS can only print in black & white.)


                              PRINT PREVIEW

To see how the screen will look when it is printed, simply select the
Preview option from the Print menu. In EGA and VGA modes, you can choose
between color and monochrome preview. Super-VGA does not have any
corresponding monochrome mode available, unfortunately.

The menu bar and help line will clear, and the graph title and y-axis
labels (if any) will be drawn on the screen. If you have a color printer
that works better with an external print screen function, this is the
perfect time to press the Print Screen button (if you have a mouse, be
sure to hide the arrow at the bottom of the screen first). In any case,
you can look at it as long as you need to. When you want to return to
work, just press any key or mouse button and the menu will reappear.
(The side labels are not erased, however; they must be cleared manually.


                          EXPORTING .PCX FILES

If you want to include a picture of your graphs in a word processor
document, or modify the default appearance of the grid before printing
it, you can export the current grid to a Zsoft PC-Paintbrush (PCX) file
by choosing the Export PCX option in the file menu. You can choose
between color (in EGA/VGA modes) and two monochrome (black on white and
white on black) palettes for the bitmap. Then enter a file name for the
output (you need not include the .pcx extension). Note that this process
may take a few seconds on 386 and slower computers, as I have not
optimized the assembly code for this rather intensive operation.


                             PRINTING VALUES

This feature allows you to print out a table of coordinates as the
program is drawing your graph. This option is available to aid the
drawing of graphs manually (if you happen to be cheating on your
homework) and provide a reference for labeling the axes if you print the
graph without the legends on.

By default, the print tables option is OFF; most people don't leave
their printer on at all times and the process slows down graphing. You
can turn it on by selecting Tables in the Options menu and responding
"y" to the prompt.



Section                       ==========================================
III XX    XX                  | Point menu: evaluating specific points |
 I    X  X                    ==========================================
 I     XX
 I    X  X
III XX    XX o                                                        32
------------------------------------------------------------------------

Graphmatica provides two features to help you get numerical results
from your equations. Both are located in the Point menu.

                     NUMERICALLY EVALUATING A POINT

The Evaluate command in the point menu allows you to quickly find the
value of any equation you've entered at a specific point. First choose
an equation from the queue by selecting it in the equation combobox. (If
you do not choose an equation, Graphmatica will select the last one
graphed for you.) Then choose the Evaluate command from the Point menu.
You will be asked to enter a value for the selected equation's
independent variable. Any expression that evaluates to a constant is OK,
including fractions and constants. (e.g. "3/4" or "pi")

Graphmatica will calculate the value of the dependent variable(s) at
this point and display the result. For function families that use the
parameter 'a', only the starting value of 'a' will be used, since only
one result can be displayed. (Note that if an error occurs when
evaluating at the point, a message will be displayed on the bottom line
regardless of the state of the Warning Messages option.) Press any key
to type in another point to evaluate. When you want to stop evaluating
points, press ESC to return to the menu.

(Point Evaluate cannot be used for slope-field ODE approximations, since
they are not really functions of a single variable.)

                         THE COORDINATE CURSOR

If the point you're interested in is on-screen, you can use the
coordinate cursor to estimate its coordinates. Simply select Cursor from
the Point menu. (If you have a mouse, you can also just hold down the
right mouse button.) The crosshairs will appear at the current position
of the mouse pointer and will respond to both the mouse and the arrow
keys. As you move the crosshairs, the coordinates are displayed at the
bottom of the screen. To exit the coordinate cursor, click either mouse
button or press ESC. When you use the keyboard to position the pointer,
it moves about 20 pixels with each keypress, so you can move rapidly
from one side of the screen to the other. To move the crosshair one
pixel at a time, hold down the Ctrl key and then press the arrow keys.

When you move the cursor close enough to one of the curves on-screen, it
will automatically "lock on" to make it easier to trace the path of the
curve. When this happens, the equation of the curve being traced is also
displayed in the status bar. The crosshair will stick to the curve you
chose, even if it intersects other graphs, as long as you move the
cursor gently in the general direction of the curve. To release the
cursor from a curve, move perpendicularly to the slope of the curve at
that point, or just move quickly in any direction.

Remember that the values reported by the coordinate cursor are limited
by the resolution of your screen. At the default grid size, values may
be off by 0.01 or more. For best results, zoom in on the section of the
graph you are interested in first, or use the Evaluate command.



Section           ======================================================
XX    XX          | Calculus menu: Derivatives, Tangents and Integrals |
  X  X            ======================================================
   XX
  X  X
XX    XX o                                                            33
------------------------------------------------------------------------

                  FINDING THE DERIVATIVE OF A FUNCTION

Graphmatica is able to perform symbolic differentiation on most common
functions and to display the derivative of a given curve in both text
and graphical formats. To differentiate a function, select it in the
queue and then Choose Find Derivative from the Calculus menu. (If the
function you want to use is not in the queue, you must graph it first.)
The program will then manipulate its internal representation of the
equation to produce its derivative, add the resulting equation to the
queue, and immediately graph it.

Note that while the curve produced by this process will always be
correct, the equation of the curve may not be very well simplified,
especially for complex equations. Therefore, the best way to check a
derivative you found by hand is to overlay its graph with the program's.

Finding the derivative is only supported for Cartesian, polar, and
parametric equations. For relations such as equations containing y^2,
the derivative is only found for the function with the positive root.
For equations with more than one y, the derivative is found with repsect
to y instead of x. For parametric equations, the derivative is found by
differentiating the x and y functions separately with respect to t.

Equations containing the following functions are not differentiable:
   int     not a continuous function
   abs     not guaranteed to be a smooth function


                          DRAWING TANGENT LINES

At times you may be interested in knowing the slope of a curve at a
given point. Graphmatica will provide this information both numerically
and graphically about any curve on the screen with just a few clicks of
the mouse.

To calculate the slope of a curve and draw the tangent line at a
specific point, first select Draw Tangent from the Calculus menu. The
cursor will change to the crosshairs. Use the mouse or arrow keys to
move to a point on any curve on the screen, then click or press enter to
select it. The program will draw the tangent line and display the point
selected and the slope on the status line as well as in the Printout
window if it is on. The tangent line will be displayed only until you
hide or delete the equation it belongs to, clear the screen, or draw
another tangent line.

Currently, you can only find the tangent line for differentiable
Cartesian and polar equations (i.e. those which do not include the int()
or abs() functions). At this time, there is no efficient way to produce
accurate results for other curves (for parametric equations you will get
a rough approximation based on the slope between the two consecutive
points nearest where you clicked). Support for other equation types and
saving and restoring tangent lines along with equation lists will be
added to a future version of the program if enough people request it.



X. Calculus menu: Derivatives, Tangents and Integrals                 34
------------------------------------------------------------------------

               INTEGRATING TO FIND THE AREA UNDER A CURVE

Graphmatica can perform numerical integration to find the area under the
curve for any function on the screen. Just select Integrate from the
Calculus menu. The cursor will change to the crosshairs.

To select a curve and region to integrate over, do one of the following:
*  With the keyboard, use the arrow keys to move the crosshair to the
   starting point on the curve you want to integrate. Press enter, then
   move to the x-value of the end of the range to integrate. Press enter
   again.
*  With the mouse, click on the starting point on the curve and drag the
   mouse to the endpoint of the region to integrate. The interval you
   select is highlighted as you move the cursor.

The program will shade in the area that was found and display the
numerical result in the status bar,  as well as send it to the printer
if you have Print Tables on. If you select a region under the axis, or
highlight the region above the axis from right to left, you will get a
negative area. The shading will be cleared as soon as you hide or delete
the equation it belongs to, clear the screen, or perform another
integration.

You can choose from several methods of integration, as well as
specifying how many segments to divide the region into, to select more
accurate or faster computation. See below for details.

Any integral shading you create will appear on the grid when you print
or copy to a bitmap; however, it will not be listed with the rest of the
equations. Currently, there is no provision for saving or restoring
integration regions, or for showing more than one shaded region at a
time. This, along with finding the area between two curves, will be
implemented in a future version of the program.


                     SETTING INTEGRATION OPTIONS

You can choose between two methods of integration, as well as optionally
specify how many segments to divide the selected region into, to select
more accurate or faster computation. To modify the integration options,
select Int Options from the Calculus menu. Then choose between the
Trapezoidal and Simpson's Rule for the integration method. Simpson's
Rule tends to produce more accurate results, as long as the curve is
smooth in the region you are integrating.

You can also set the number of segments to divide the region into when
calculating the area. This must be a positive integer, and if you use
Simpson's Rule, it must be an even number. The higher the number, the
more accurate the result will be, but the longer it will take to
calculate it. By default (and if you type "auto" instead of a number
here) Graphmatica chooses the number of segments automatically to be
half of the width of the area you selected in screen pixels.


Appendix                        ========================================
 AAAA                           | Graphmatica's Command Line Arguments |
A    A                          ========================================
AAAAAA
A    A
A    Ao                                                               35
------------------------------------------------------------------------

With the introduction of the GRAPHMAT.INI file, which will not only
save your preferred options automatically from inside Graphmatica but
will also reset them automatically the next time you run the program,
most of the command line arguments that GraphIt! version 2.80 supported
were removed in version 3.0. [Although you shouldn't need to edit the
setup file yourself at all, its format is described in Appendix B.]

Arguments are NOT case sensitive (upper-case and lower-case letters
are treated exactly the same). Arguments must be separated by one or
more spaces on the command line. Excluding filenames, the other
argument must begin with the standard MS-DOS "switch" character ("/").
UNIX-style switches (starting with "-") are NOT supported.


/G:xxxxxx        Graph Equation
-------------------------------
    If you want to start up Graphmatica by automatically graphing one or
    more equations, use the /G: option. The only limits on the number
    of equations you can include on the command line are the capacity
    of the redraw queue (25 equations) and, of course, the maximum
    length of the command line (I think it's around 128 characters).
    The equation(s) will be parsed and graphed in exactly the same way
    as the regular prompted input, with one exception: if you want to
    include any spaces in the equation, you must enclose the whole
    "/G:xxxx" string in quotation marks or the spaces will be
    interpreted as delimiters between arguments.  As the Graphmatica
    editor is far superior to the DOS command line editor, this option
    has limited usefulness, but it is a good interface for execution
    from a batch file.

/V       Use VESA 800x600 mode
------------------------------
    Because the signals that high-resolution video cards produce may
    physically damage monitors that are not capable of displaying them,
    Graphmatica does not attempt to detect super-VGA cards. If you know
    your monitor is capable of supporting this mode and you have a VESA-
    compatible video card (or a TSR VESA driver), you can select 800x600
    resolution with this switch. kSoft ASSUMES NO LIABILITY FOR ANY
    DAMAGE CAUSED BY IMPROPER USE OF THIS SETTING. If your card is not
    compatible with this mode, Graphmatica will select standard VGA.
    The Microsoft mouse driver, version 8.20, does not fully support
    these modes. (The mouse cursor does not display.) A newer version of
    this or another mouse driver may.

filename[.ext]
--------------
    Specifying the name of a valid equation list file (with or without
    the ".GR" extension) loads that file upon startup. Unless
    AUTOREDRAW=OFF is specified in the [options] section of the file
    loaded (assuming it has one), the graphs are drawn as soon as the
    opening screen is gone. Note that the settings saved in GRAPHMAT.INI
    are loaded before any equation list files you specify.



Appendix                     ===========================================
BBBB                         | Editing Equation Lists and GRAPHMAT.INI |
B   B                        ===========================================
BBBB
B   B
BBBB o                                                                36
------------------------------------------------------------------------

Graphmatica does not really provide facilities for maintaining equation
lists; although you can edit an equation and change the order of
equations by redrawing selected graphs, determining the precise order of
equations in the list is difficult since they are dynamically shuffled
each time an equation is entered or redrawn to make Graphmatica more
interactive. However, the equation list is just a text file whose
structure is virtually identical to the win.ini file, you can create or
modify one with virtually any editor. The structure goes like this:
(sections in parentheses are explanations; don't type them in.)

  [saved-setup]   (if present, resets all options to defaults before
                  load to ensure the saved setup is restored correctly)
  [labels]
  title = xxxx    (any character string, up to 76 chars.)
  left = xxxx
  right = xxxx

  [grid]
  left = #.##            (left side of grid)     (any decimal number)
  right = #.##           (right side of grid)    (or "auto" to autoscale)
  top = #.##             (top of grid)
  bottom = #.##          (bottom of grid)

  [options]
  paper = polar, trig, rect, slog or llog  (controls graph paper type)
  detail = none, axes, dots, or grid       (controls paper detail level)
  autoredraw = on or off  (controls AutoRedraw)          ("on" or "off")
  autonum = ##            (number of equations to A.R.)  (any integer 0-25)
  legends = on or off     (controls axis legends (numbers))
 _color = 1,2, or mono   _
| hogcpu = on or off      |
| widelines = on or off   |   valid under Graphmatica for Windows ONLY
| autosquare = on or off  |  Graphmatica for DOS ignores these 4 settings
|_labels = on or off     _|
  tables = on or off      (controls print tables feature)
  warnings = on or off    (controls warning error messages)
  defscale = #.##         (default for "Scale" prompt - any decimal number)
  fineness = #.##         (default fineness value)
  tstart = #.##           (default start of theta's domain)
  tstop = #.##            (default end of theta's domain)

  [equations]
  (up to 25 equations, each up to 200 characters, each on a separate line)

The [labels] section specifies the title and y-axis labels. The [grid]
section gives the left, right, top and bottom coordinates of the grid.
Type "auto" for any one to auto-scale that coordinate based on the other
three. Type auto for both the top and bottom to auto-scale the y-axis so
the top and bottom coordinates are equal and opposite.



B. Editing equation lists and GRAPHMAT.INI                            37
------------------------------------------------------------------------

You can omit any item or even a complete section if you don't want to
change the default. Type exactly as specified above, including the
brackets "[ ]" around the section headings, but replacing the values on
the right side of the equals signs with your own values. If you don't
know or don't care what the value is for an option, just leave the line
out. Don't worry about capitalization, extra spaces, or even extra blank
lines. (However, each entry must be on a separate line or some may be
ignored.) You can put the options and sections in any order ... the only
requirement is that the [equations] section must come last, if there is
one. Invalid options are also ignored.

If you want to create a file with no titles or options, simply type the
"[equations]" header and then type all of your equations. Each equation
must be on a separate line. There is one catch, however: since
Graphmatica graphs the most recently entered equation first when it
redraws, when the list is loaded and graphed, the equation listed LAST
will be graphed FIRST. (It is certainly possible to correct this
situation by graphing the equations as they come in, but then every
subsequent "redraw all" command would draw the list backwards anyway.)
If you load an equation list and immediately save it without changing
anything, the net effect will be that the order of the equations in the
file is reversed.

The GRAPHMAT.INI file, which is loaded whenever you run Graphmatica
without specifying an equation list to load, uses the exact same
format... when you choose the Save Setup command, Graphmatica simply
saves the setup file as it would an equation list, except the [labels]
and [equations] sections are omitted, since you probably don't want to
have the same graph title and equations load up automatically forever
and ever.

                      CONVERTING OLD EQUATION LISTS

Anybody who upgraded from GraphIt! for DOS version 2.80 who wants to
convert equation list files from the original format to one compatible
with Graphmatica for DOS 3.5, please contact me and I will explain the
process.



Appendix                            ====================================
 CCCC                               | Upgrade Notes: Changes from 3.30 |
C                                   ====================================
C
C
 CCCC o                                                               38
------------------------------------------------------------------------

Version 3.5 is a major update which adds a number of features previously
only available in Graphmatica for Windows, as well as most of the new
features of version 1.5 for Windows. Please take a minute to acquaint
yourself with them:

The following features were enhanced:
 1. Incorrect "-" associativity in parser of version 3.30 fixed.
 2. Long equations (> 128 characters) that were occasionally truncated
    should no longer be molested.
 3. Exponents of the form x^(2/3) are graphed correctly (positive for
    all x). Exponents of the form x^(1/3) are graphed correctly (same
    sign as x). All other non-integer exponents are graphed
    conservatively, so they may cause a domain error warning, but NOT an
    incorrect graph.
 4. Steep functions that go off screen but should reappear again at
    larger values of x should no longer be in danger of missing lobes
    (except in the most extreme cases).
 5. The parser has been totally rewritten from scratch for better
    reliability and expansion. Now, function arguments are defined
    precisely to be the first term following the function name, or
    everything in parentheses following the function name.
 6. AutoRedraw draws all graphs on the screen by default now; set the
    "number of equations to redraw" to limit it to fewer.
 7. Delete Equation removes the graph from the screen as well as from
    the queue if it is present, to reduce clutter.
 8. Increasing the fineness factor now forces recalculation of graphs
    on the screen immediately.
 9. Parametric graphs offer smarter dynamic fineness based on absolute
    distance between consecutive points on the screen to allow fast
    graphing over any domain without worrying about curves becoming
    jagged. Also, a new option allows you to select the step rate
    manually by including it as a third parameter in the domain.
10. ODE approximation initial values no longer need appear on the
    screen. Also, the approximation is run until the independent value
    reaches the edge of the screen, not just until the curve goes off
    screen, so curves that return on screen will be drawn in entirety.
11. Coordinate cursor automatically "locks on" to curves when moved
    close enough, making it easier to get accurate readings.
12. Equation picking using the mouse. Clicking on a curve will now
    select the curve in the equation listbox so you can see its equation
    on the status line and then edit it, delete it, etc.
13. Added mouse-oriented features found in the Windows version: select
    a region and then choose Range from the View menu to zoom in to the
    displayed range; hold down right mouse button for coordinate cursor.
14. Refined menu and scrollbar mouse support allow true Mac-type pulldown
    menus and continuous scrolling in listboxes.



C. Upgrade Notes: Changes from 3.30                                   39
------------------------------------------------------------------------

The following new features were added:
 1. New Calculus menu offers the following useful features:
    + Symbolic differentiation using the "Find Derivative" option. Finds
    and graphs the derivative of the currently-selected equation. Works
    on most Cartesian and polar graphs.
    + Draw Tangent Line option allows you to click on a specific point
    on a curve to find the slope at that point, as well as draw the
    tangent line so you can see it graphically as well.
    + Integrate function lets you select a curve and an area to
    integrate with two mouse clicks and performs numerical integration
    to find the area under the curve by trapezoidal or Simpson's methods
    with selectable number of segments.
 2. Support for up to 4th order systems of ODEs added. Use the variables
    t,x,y,z,w and the corresponding differentials (dx, dy, etc.) to set
    up a system of equations separated by ';' much like parametrics. You
    can also use t,x1,x2,x3,x4. All curves in the system are graphed in
    different colors so you can tell them apart. You can even select all
    of the required initial values with the mouse using the Set Initial
    Value menu item.
 3. Added support for exporting to color or monochrome PCX files, so you
    can include graphs in other documents.
 4. New Hide Graph function removes the graph of an equation from the
    screen without deleting it.
 5. Spacing between axes legends can now be specified manually for times
    when the automatic spacing adjustment isn't good enough. Select
    Legends from the Labels menu to try it out.
 6. New user-settable axis decorations add to the diversity of graph
    paper choices. Now you can add arrows and labels to the axes, or
    turn them off independently of the gridlines. Look at the revised
    Paper item in the View menu.
 7. You can now select a domain using the mouse for Cartesian equations
    and ODEs.
 8. You can now plot Cartesian equations which are functions of y, not x.
 9. New floating-point error handler should prevent any crashes or
    unsightly messages from library routines due to complex equations
    or extremely large or small grid ranges.

I hope you find these new features beneficial. If you'd like to support
future versions with your comments and suggestions, see Appendix D or
the included REGISTER.TXT for the mail-in survey form.



Appendix                                      ==========================
DDDDD                                         | Shareware / Order Form |
D    D                                        ==========================
D     D
D    D
DDDDD  o                                                              40
------------------------------------------------------------------------

Shareware: Feel free to distribute copies of Graphmatica to your friends
and upload to BBS's where others can donwload it AS LONG AS 1) you
charge no fees for its use or distribution and 2) you do not modify the
program or documentation files in any way.

[Shareware catalogs have permission to distribute this product for no
more than $5 a copy. However, please contact me to make sure you have
the latest version before you do so.]

This program is to be taken AS IS with no warranties, express or implied.
Although I know of no serious bugs, kSoft will not be held liable for any
damage caused by use, misuse, or inability to use the program. However,
if you tell me about any bugs you may find, I will try to correct them
in the next release.

If, after using Graphmatica, you find that it is easy, helpful, and
convenient to use, please support the release of future versions by
filling out the registration form on the next page and sending your
contribution (payable to Keith Hertzer) to the address below:

                              kSoft, Inc.
                          345 Montecillo Dr.
                      Walnut Creek, CA 94595-2613

If you send $20 or more, you will be registered to receive a diskette
with the next versions of Graphmatica for DOS and Windows when they
become available. For releases after that, you will be informed that a
new version is available and how you can get it. Orders outside of North
America: please add $5 for shipping if you want to receive your upgrade
via air mail.

Even if you cannot send any money, please help me out by filling out the
response form below, especially if you have any suggestions about what I
should add to the next upgrade.

You can register Graphmatica via CompuServe's Shareware registration
database. The ID number is 111. Type GO SWREG to start the transaction.
The fee of $25 US will be added to your monthly bill.

I can also accept credit card orders through Public (software) Library.
Prices including handling are $27.50 (US & Canada) and $30 elsewhere.
Please see README.TXT for details.

WHAT'S NEXT? I am slowly proceeding with Graphmatica 3D, which may or
may not have a DOS version. If you still don't have Windows, and want to
use this program, please let me know. If there is sufficient user
support I will release DOS as well as Windows versions.

If you would like a copy of the source code for Graphmatica for DOS
(developed in C using Microsoft Visual C++ 1.5), please send me a check
for $25 and a short note telling why you would like it.

--G3.5r--------------------kSoft REGISTRATION FORM--------------------12/96--

Name &                                          E-mail
Company _______________________________________ Address _____________________


Address _____________________________________________________________________

                                        State/          Postal
City ________________________________  Country _______   Code _______________
  _
 |_|  Enclosed is my $25 or more contribution. Please register me to
      receive the next upgrade to Graphmatica for DOS.
  _
 |_|  Enclosed is my $25 or more contribution. I want Graphmatica for Windows
      NOW! Please use my upgrade credit to send me the current version.
  _
 |_|  Please send me the source code [for MSVC 1.5] to Graphmatica for DOS
      3.5. I have enclosed a note on why I'd like it and a check for $25.

------------(make checks in US funds payable to KEITH HERTZER)---------------

What size diskettes do you use?        5 1/4"           3 1/2"

Where did you get Graphmatica?  (name of BBS or catalog: ________________)

  From a friend     Local BBS     CompuServe      Internet      Catalog

What processor are you using to run Graphmatica for Windows?  (circle one)

     286        386        486        Pentium        Other (___________)

If you use a printer with Graphmatica, what type is it?

    dot-matrix         ink jet        laser          Other (___________)

What type of video card do you have?        Mono          CGA

     EGA             VGA           SVGA           Other (___________)


What DOS version do you usually run with Graphmatica?    ___________

What features do you like best about Graphmatica?

_____________________________________________________________________________

_____________________________________________________________________________

What problems have you experienced with Graphmatica?

_____________________________________________________________________________

_____________________________________________________________________________

Any other comments or suggestions for improvement in the next upgrade?
(Your comments are vital so I will know what to change or add. Please
feel free to use the back or another sheet of paper.)

_____________________________________________________________________________

_____________________________________________________________________________
Please send to: kSoft, Inc., 345 Montecillo Dr., Walnut Creek, CA 94595-2613.