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Welcome to Symbolix!
====================
Bright Software (now Synergetix) has developed several applications and utilities for the Apple IIGS in the late 80ies and early 90ies. Some of this software has been released as freeware, some larger programs have been sold commercially. Our freeware programs include Hermes (formerly ShadoWrite), MultiView II, Turbo Mandelbrot, and Oversampler. Commercial products include Symbolix, ShadowDial, The Gate (tm) and Spacefox (tm). (The latter two are still available from Seven Hills Software, 2310 Oxford Road, Tallahassee, FL 32304-3930, USA, sevenhills@aol.com. SpaceFox is an addicting shoot'em'up game, The Gate an award-winning action adventure.)
Due to our full-(over-)time jobs we cannot afford spending the necessary time on these products, so we decided to turn Symbolix into a shareware product. Nevertheless, this package is a mature products that has been available for years. The shareware version is not limited in any way.
ABOUT SYMBOLIX
Symbolix is a pretty advanced math application for Apple IIGS computers. It features symbolic math (derive, expand, sort, collect, simplify, rewrite), the best graph modules for Apple II computers (2D and 3D with lots of bells&whistles) and some other stuff that make it a real IIGS application. Almost all Symbolix users that have written me said that its user interface was awkward at a first glance, but as soon as you start working with it for extended periods, you'll suddenly notice that all windows and functions are where they belong to. I hope the initial "strangeness" won't drive you away - it's not that weird anyway, really. There *is* some logic behind it!
Symbolix supports some system extensions:
- Floating Point or Number Cruncher coprocessor cards accelerate math significantly
- The Manager (tm) from Seven Hills works *GREAT* with Symbolix. Symbolix incorporates extra code for such environments. You can even fine-tune Symbolix's interaction with The Manager. (The Manager is available from Seven Hills Software Corp.) Symbolix is the first application that actively supports The Manager.
- Extensions (=plug-ins) enhance the capabilities of Symbolix. You can write your own!
Symbolix has been written in assembly language. It has undergone several major revisions (especially 1.3, 1.5 and 1.6). There are so many source files that I somehow lost the overview after not touching it for about a year. There are about 200,000 lines of undocumented source code I guess. I haven't counted them.
Symbolix has been designed to be as fast as possible. Luckily, you don't see how things interact. The price for speed are a few dozen pretty complex, self-contained procedures. I'm glad I don't have to debug them anymore. I once conducted some tests running a IIGS against a Mac IIsi/Theorist, and Symbolix won hands down. Obviously, it was worth the efforts.
SHAREWARE
Symbolix is now being sold as a shareware product available from major ftp sites and from Shareware Solutions II, a magazine published by Joe Kohn. Don't ask me why, but Joe - a really cool guy - told me that I'm entitled to 50% of his profits originating from disk sales. However, since I'm already asking for a small shareware fee, I decided to pass my part on to Greenpeace (Switzerland). Your money can save many lifes!
Symbolix is shareware. If you like the program and sometimes dream of animated 3D graphs, it'd be nice if you sent me $15 in cash money. There's no special feature that needs to be unlocked. However, if you're using Symbolix regularly or if Symbolix has saved your life at least once (math homework) or if you just appreciate the giant efforts that went into this application, the really fair shareware fee is everything I'm asking for. Thank you very much!
BINGO!
Oh, I almost forgot that I have one printed manual left. So here's the deal: the 50th payer will get this thing for free! So send your shareware fee now. (Hint: if you pay the fee TWICE, your chances are higher! Hehehe...)
ABOUT THIS DOCUMENT
This document is a complete replica of the current Symbolix user's manual. I hope it will serve you well.
Unfortunately, the electronic version of the manual was unavailable when Symbolix went shareware. Thus I had to scan and OCR the manual which produced a "text stream" with lots of misread words and expression. I have carefully revised this manual, though, but please excuse if the one or the other typo made it into the final document. (The best text garbage was: "drug all the files", "click the calculator iron", "let Symbolix calculate the Orange automatically".) Finally, some of the lapses in this manual originate from my way limited knowledge in the English/American language. It's not my native language, thus some flexibility on your side is mandatory. :)
Since I do not intend to develop Symbolix any further, I can finally present you a list of great people who helped me make my dream come true:
Thanks to Joerg "Gemini" Kienzle for his support and friendship. Joerg co-design Gate's graphics, coauthored parts of Spacefox with me and filled me in in lots of techniques in math and computer stuff.
Thank you Yann for the same quality support. Yann is a real computer wiz. He knows things I doubt anyone else knows.
Thank you brave Symbolix users. I especially thank those early buyers who could afford this expensive program and trusted my promises despite A+/InCider's depressing "review". I hope that you won't be mad at me now that I turned Symbolix into shareware.
JOIN ME!
If you have questions regarding Symbolix, I'll gladly answer them. The most convenient place is to log on "my" home server ezinfo.vmsmail.ethz.ch (X.25 address: SET HOST 57413). You first have to order a (public) account. However, this account is absolutely free of charge and doesn't cost a single buck.
Once you got there, enter "MICS" (the chatter) and look for someone called "rat" or "ratte". That's me.
You can also send e-mails to gudath@ezinfo.vmsmail.ethz.ch.
For snail mail and shareware fees, contact:
Henrik Gudat
Pantelweg 2
4153 Reinach
Switzerland
or
Synergetix H. Gudat
Missionsstrasse 38
4055 Basel
Switzerland
Have fun!
- Henrik Gudat
Bright Software/Synergetix
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Chapter 1: Introduction
Wait!
Read this chapter and you'll find out
how you can become a Symbolix expert
in a flash
Introduction
Welcome to Symbolix, the first math program on the Apple IIGS that does not distinguish between pure numbers and algebraic expressions. Completely programmed in ultra-fast assembly language using the latest techniques in computer algebra, this unique program sets a new standard in math software for the Apple IIGS. An easy to use desktop interface allows you to concentrate on the math problem and not on the program itself. Enjoy Symbolix's simplicity, its power, its capabilities.
Some of the Highlights of Symbolix are:
desktop application � resources allow easy customization � written in assembly language � scientific RPN calculator � formula module with many symbolic commands � object oriented graph modules � integrated formula tables and scrapbook � powerful print commands � superb error handling � and much, much more...
This manual will help you master Symbolix in a very short period of time. It has three parts:
� In part one (Chapter 3), you'll learn some of Symbolix's functions and how the different modules are linked together.
� The second part (all remaining chapters) explains all of Symbolix's features in detail. You should read these chapters after you've completed Chapter 3.
� The last part (the appendices) summarizes commands, functions, and flags. Refer to this section if you have any specific questions.
This manual has been designed to familiarize you with all the necessary functions very quickly. However, it does not cover mathematic fundamentals. Symbolix is a collection of tools that will allow you to do things more easily and quickly that normally are tedious and time-consuming.
If You're in a Hurry...
You probably just got your copy of Symbolix and are eager to use it. However, it's crucial that you first read at least the following section "Things You Need to Know" and Chapter 3 "A Quick Tour" because you'll need at least a fundamental knowledge of the structure of this program before you get started. If you try to discover Symbolix on your own, you'll risk getting frustrated and may need even more time to understand how it works.
So: it pays to read the recommended chapters!
Chapter 2: Things You Need to Know
Symbolix - behind the scenes...
Things You Need to Know
Symbolix runs on standard Apple IIGS computers with at least 1.5 Mb of free memory (2 to 3 Mb recommended). One 3.5" disk drive is sufficient though we strongly recommend you use a hard disk.
Symbolix is compatible with system 5.x and later. Please note that menu icons and the built-in bell work only with GS/OS system 6.0 (or later).
Symbolix allows you to start it up even if the printer drivers are not correctly installed. If the print commands are disabled and can't be selected, make sure that you've configured your printer software properly.
2.1. Internal Precision and Data Types
Symbolix uses Apple's SANE (Standard Apple Numerics Environment) tool set for all computations. Instead of writing our own routines for Symbolix, we decided to make use of SANE because only this tool offers:
� extremely accurate results
� careful rounding (direction selectable)
� gradual underflow detection
� exception handling
In addition, Symbolix will profit from improvements on the SANE tool or the hardware. Future Apple IIGS models will hopefully already have a built-in floating-point coprocessor, which will increase Symbolix's performance dramatically.
We've chosen the extended data type for Symbolix because the precision of calculations has the highest priority in a math program. The following table describes the range and precision of this data type:
� Size: 10 bytes
� Significant precision: 64 bits or 19 decimal digits
� Maximum value: +/- 1.1E+4932
� Minimum value: +/- 1.9E-4951
The only problem in choosing such a precise data type is that you sacrifice speed. A few functions can be very slow (especially 3D Plot), and adding an accelerator board or a floating-point coprocessor (with SANE patch) to your system might be a good investment. However, we're convinced that in a math program like Symbolix accuracy is more important than speed, especially if future SANE upgrades will solve this problem.
Special Features
In addition to "normal" numbers, SANE is able to calculate with "NaNs" (NaN is an abbreviation for "Not a Number") and infinities. Although these special "numbers" do not represent a value, SANE allows you to calculate with them.
NaNs are produced when an operation can't be performed due to invalid arguments (example: LN(-1)). SANE distinguishes between 10 different NaNs, which are listed in the following table:
NaN(1) invalid square root
NaN(2) invalid addition
NaN(4) invalid division
NaN(8) invalid multiplication
NaN(9) invalid modulo
NaN(17) invalid ASCII string
NaN(33) invalid argument to trig routine
NaN(34) invalid argument to inverse trig routine
NaN(36) invalid argument to log routine
NaN(37) invalid argument to power routine
NaNs are always displayed - as shown in the table - with the NaN code in parentheses. This enables you to locate the invalid expression.
Infinities are used when an operation produces a mathematical infinity (example: 1/0). As with NaNs, INFs can be used like any other number and always give the correct result. You must, however, distinguish between two infinities, +INF and -INF. The first one stands for an infinite positive number, the second keyword for a infinite negative number.
For more information please refer to the "Apple Numerics Manual".
2.2. Conventions in this Manual
Sometimes it is quite difficult to make dear what button should be pushed or which menu has to be selected. Therefore, the following conventions are used throughout this manual:
FORMULA / PREFERENCES
..means that you should select the menu item PREFERENCES in the FORMULA menu.
[FORMULA / STACK / ENTER]
The brackets indicate that you should press or click on a control (i.e. a button, check box, or radio button). To make the control in question appear, choose STACK from the menu FORMULA. If the window "STACK" is already open (on the desktop), the command [STACK / ENTER] produces the same result. And, if it's clear from the context that the STACK window is meant, only [ENTER] may appear in the text.
{StdP}
The curly braces are used to specify symbolic constants. They are all placed on the left side of the window of constants and variables. Five constants can be accessed via buttons, all others are listed on a pop-up menu (the large, empty button). Clicking on this control opens the library.
<RETURN> ..tells you to press RETURN on the keyboard.
OA
..is an abbreviation for the Apple key (Open Apple). This form is often used when referring to the keyboard equivalent of a menu item. For example, when you read "..then select WINDOW / FORMULA (OA-1)", you can either select FORMULA from the WINDOW menu using the mouse, or hold down the Apple key and then press <1>. Both actions will have the same effect.
Clicking
..means that you should move the cursor to the indicated control or menu item and push the mouse button once. Don't move the mouse while you are holding down the button.
Pressing ...means the same as clicking.
Double-Clicking
...means you have to push the mouse button twice in rapid succession. You can change the interval in which the IIGS interprets two mouse clicks as a double-click by entering the control panel (OA-control-escape) and opening the submenu "options" (ROM 1.0) or "mouse" (ROM 3.0).
Figures
At the end of this manual you'll find several pages with figures (i.e. screen dumps). All "items" (controls) in a window as well as the windows themselves are individually numbered. Please remove these pages from the ring binder and have them ready when reading this manual since there are many references to these illustrations (for example -> 4.1).
Chapter 3: A Quick Tour
Now get ready...
Launching Symbolix:
� To start Symbolix from your working disk, boot into the Finder. Then insert Symbolix and double-click on the calculator icon.
� To start Symbolix from your hard disk, boot into the Finder. Create a new directory. Then insert Symbolix and drag all files into the folder you've just created. (Double-click on the disk icon "Symbolix", select all files, then drag them into the directory.) Double-click onto the file ''SYMBOLIX" in this folder to start the program.
Note: use the original program disk only for making working copies. After making copies, store the original disk in a safe place so if a problem develops with your working copy another can be made.
A Quick, Guided Tour
This chapter will give you an idea of Symbolix's capabilities. After having finished this guided tour you'll be familiar with most of its functions and will be able to work with Symbolix on your own. When you have any questions, refer to the corresponding Reference Section where you can find detailed descriptions.
To run Symbolix, start GS/OS(tm) and the Finder(tm). Then insert the Symbolix program disk and double-click on the calculator icon.
As Symbolix loads into your IIGS, a short animation will appear on the screen. After a while, the main screen replaces this picture and Symbolix opens three windows.
At the top of the screen you see the menu bar. Clicking on one of the words on this line opens a "menu". A menu is a group of related commands. To select an item from one of the six menus, click on a name in the menu bar and hold down the mouse button. Then move the cursor to the command you want to execute and release the button. If you want to learn more about the menu bar and the desktop interface, please refer to the Apple IIGS User's Manual.
Three windows are automatically placed onto the desktop as soon as Symbolix has been completely loaded into memory. The upper window (fig. 1) is called the formula window and holds the formula. The formula has to be entered a bit differently from what you might expect since the current version of Symbolix does not support the international mathematical notation of formulas. Instead, it uses another system found on many calculators and computers: a formula must be written like a text in a word-processor, i.e. as a string of characters. The formula window is thus like a little word processor: you can cut, copy, paste, clear, and edit any parts of the text that represent a formula. But more about that later.
The window to the right (e fig. 3) is a window with a lot of buttons in it. Each button represents a mathematical function (tangent, cosine etc.), and pressing one of them inserts the function in the formula at the current insertion point or, depending on the selection of the controls explained next, executes the function on the calculator.
Two other controls are located in the lower right corner of this window (fig. 3.1). They are called radio buttons. They retain and display an on-or-off setting and are organised into so-called "families". Only one radio button in a family can be on at a time. In this example, the family consists of only two buttons, "Formulas and "Stack". These two controls determine to which window variables or built-in functions and constants are linked. If you select "Formula" (the small circle is then filled out with black color), Symbolix links them to the formula window and puts the program in symbolic math mode. If "Stack" has been chosen, Symbolix will calculate like ordinary calculators. In this case, the numeric math mode is active and selecting either a constant, variable, or function stores a copy of the value of the constant or variable in the calculator window or executes the function. Please make sure that this switch is set to "Formula" before you continue, otherwise the following examples won't work properly.
The window at the bottom of the screen (fig. 4) is the window of constants and variables. To the left you can see six buttons (the large one is empty). They give you full access to the built-in library of constants. When you press one of these buttons, their names (like Pi, e or +INF) will be inserted at the insertion point of the formula window, very similar to pressing the function buttons in the tall window to the left.
To the right is the section of variables. Two buttons, "Delete" and "Define" (fig. 4.2, 4.3), let you add and remove variables.
If you add a variable, it will appear in the list to the right (fig. 4.1). Clicking on a variable in this list highlights or selects the variable. Some menu commands work only when you've selected one or two variables before executing the command.
Let's Get Started!
We'll begin with the following simple formula:
FN(x) = (a+b)^(2) + 5a^(2) - ba(2)
whereby a is a parameter (a=15) and b is a variable.
3.1. Entering a Formula
First enter this formula into the topmost window (e fig. 1):
� Click in the content region of the window
� Enter the formula In fact, there are two ways to enter this formula:
1) Type in all characters via the keyboard (without entering spaces):
(a+b)^(2)+5*a^(2)-b^(2)
Now select FORMULA / CHECK SYNTAX OF FORMULA (fig. 10.1). An alert box appears telling you that there is something wrong with the formula you just typed. CHECK SYNTAX OF FORMULA has highlighted the position where Symbolix stopped parsing the formula.
Obviously, something is wrong with the 'a'. Symbolix stopped there because it doesn't recognize 'a' as a variable. If you now choose FORMULA / ADD VARIABLE, Symbolix converts the highlighted text in the active window (in this case 'a') to a variable and adds it to the list of variables. Since you also use a second variable in the formula, repeat this procedure for 'b': click to the left of one of the b's in the formula, hold down the mouse button and move the cursor over the variable. Then release the mouse button and select FORMULA / ADD VARIABLE again. The list of variables (fig. 4.1) should now contain the variables a and b.
However, a does have a value (a=15) assigned to it. To set a to the correct value, click on its entry in the list of variables and push the button DEFINE (see 4.2) to the left of the list. Now enter '15' in the value field and press OK. That's it.
This is the other way to enter the same formula:
2) First define all variables you'll need in your formula. To do so, press DEFINE (see 4.2) in the lower window and enter a name (a) for the variable and a value (15). Then push OK. Follow the same steps to define the new variable b.
Now let's enter the formula:
� Push the [(..)]-button in the window of functions (fig. 3)
� Double-click (click the mouse button twice in rapid succession) on 'a' in the list of variables (fig. 4.1)
� Push the [+]-button in the window of functions
� Double-click on 'b' in the list of variables in the lower window
� Push the red arrow button in the window of functions
� Push the [y^(x)]-button in the window of functions
� Enter <2> (press "2" on the keyboard)
� Push the red arrow button in the window of functions
� Push the [+]-button in the window of functions
� Enter"5"
� Push the [*]-button in the window of functions
� Double-click on 'a' in the list of variables in the lower window
� Push the [y^(x)]-button in the window of functions
� Enter <2>
� Push the red arrow button in the window of functions
� Push the [-]-button in the window of functions
� Double-click on 'b' in the list of variables in the lower window
� Push the [y^(x)]-button in the window of functions
� Enter <2>
Finished! You see that you can enter a formula almost without using the keyboard. It's up to you to decide which way works better for you. Of course, you can also use a combination of both methods.
3.2.Manipulating the Formula
Perhaps you are a mathematician and notice immediately that our formula can be written in a much shorter form. Don't even think any longer about how this could be done, Symbolix will accomplish this task for you - within seconds.
Because there are some exponents in the formula, it's a good idea to expand everything first. Select RULES/EXPAND (fig. 9). The formula should now look like this:
a*a+a*b+b*a+b*b+5*a*a-b*b
After executing EXPAND you should normally sort the formula every time. Choose RULES / SORT and after a second you'll see...
a*a+a*b+a*b+b*b+5*a*a-b*b
Only a single term (b*a) has been changed. Once you work with longer and more complex formulae, you'll soon discover how useful this command can be.
Of course, our formula is not yet in its final form. Select RULES / COLLECT and watch how Symbolix collects the expression:
6*a^(2)+2*a*b+0*b^(2)
You think this formula could be further simplified? Then execute RULES / SIMPLIFY to get rid of the quite useless term 0*b^(2). The upper window should now contain...
6*a^(2)+2*a*b
which is equivalent to the original formula!
3.3. Displaying the Formula
Now we want to plot the graph of 6a^2+2ab:
� Highlight the independent variable (in our example b) by clicking on it in the list of variables (a 4.1)
� Select FORMULA / PLOT 2D GRAPH (fig. 10.2)
A new window appears. The interval is set to [-6;6] and grid lines and labels are turned on. Press [Autoscaling] (fig. 6.1) to accept these settings and to let Symbolix calculate the y-range automatically. After a while the graph in the interval [-6;6] will be displayed in another window (fig. 6.6).
Now we want to take a closer look at the graph by zooming in, let's say from -1 to 2. You can do this by dragging a selection box (fig. 6.5) in the system of coordinates:
� Move the cursor to the x-coordinate -1. (First scroll down the window, then orientate yourself by the labels below the system of coordinates.)
� Click the mouse button and hold it down
� Move the cursor to the x-coordinate 2
� Release the button
Note: the height of the selection box doesn't matter since Symbolix will calculate the vertical display range.
Now move the cursor to the yellow information bar and hold down the mouse button. A pop-up menu (fig. 6.6) appears offering you several operations on graphs. Select the menu item ZOOM WITH AUTOSCALING (fig. 6.8).
The screen is cleared and the new graph is drawn after a few seconds.
Now we want to know the slope at x=1. Move the cursor to that point on the screen (the vertical position does not matter) and click the mouse button. Open the pop-up menu again and choose SHOW SLOPE (fig. 6.7). Symbolix displays the slope at x=1 in the information bar
Slope= 3.00000000086e+1
-> Because Symbolix can give you only an approximation of the exact result, this value is very close but probably not equal to 30 (which is the correct result).
To learn more about the many other functions of the plot module turn to the corresponding chapter in the Tutorial.
3.4. Finding Roots
Many mathematical problems can only be solved if you know roots (i.e. points where the formula crosses or touches the xaxis) of a formula. Symbolix has already built-in a fast and flexible algorithm that finds any root, provided you specify the appropriate start value. In our example, though, it doesn't matter what value you enter.
� Select FORMULA / SOLVE (fig. 10)
� Press SOLVE several times and watch the value shown above this button.
The value should eventually balance out at 45 (=4.5el). This means that Symbolix can't find a better approximation of this root. Then press CANCEL to dose the solver window.
Tip: the 2D module we have used above also features a solver but does not show intermediate results.
3.5.Calculating Integrals or Areas
Suppose you want to calculate the area between the graph and the x-axis in the interval from -50 to 43, then multiply the result by 1.5 and finally subtract the sine of Pi/6. The result is to be stored in 'P. Or in more mathematical terms
P = ( Area of 6a^2+2*ab from -50 to 43 ) * 1.5 - SIN( Pi / 6)
First you need to calculate the area:
� Select FORMULA / INTEGRATE (fig. 10)
� Enter -50 in the topmost edit line ("From") and 43 in the box below.
� The field [Initial Accuracy] is set to 20 and indicates that 120 data points will be evaluated. You can leave this field as it is.
� Press AREA. Symbolix now calculates the area between the graph and the x-axis. Symbolix first calculates an approximation of the area, then increments INITIAL ACCURACY by one and calculates the area again. The program repeats this loop until you hold down the Open Apple (OA) key.
� After the first loop has been terminated and Symbolix inputs the results into the blank space at the bottom of the window, press Open Apple to stop this loop. Since we need this value for further calculations, move the area to the calculator by pressing TO STACK.
-> If you do not stop Symbolix by pressing the Apple key after the first loop has been terminated, results of other calculations in the rest of this chapter may differ (+/-0.000001).
Press CANCEL to close the window.
3.6.Calculating with Symbolix
Until now you've seen only three windows: one for the formula, one that contains all built-in functions, and one that gives you access to constants and variables. However, there is a fourth and important window: the calculator in the stack window (fig. 2a).
-> It's a good idea to close the top window by pressing OA-1 before you continue because your desktop may otherwise get jammed.
First select WINDOWS/STACK, then WINDOWS/STACK COMMANDS. These two windows enable you to calculate "numerically", i.e. similar to pocket calculators. The stack window displays a few numbers (fig. 2.3) and an edit line (fig. 2.1a). The second window you've just opened (fig. 2.4) offers a couple of commands for manipulating, saving, or loading stacks.
A "stack" consists of 21 levels, or planes, and each level can hold exactly one number. How to calculate with the stack and the RPN system is explained later in this manual. For now, just follow the instructions and everything will work fine.
The value at the bottom of the stack window is the area you have calculated earlier in this chapter. The next step to take is to multiply this value by 1.5:
� Enter '1.5' in the edit line (or 2.1a)
� Press <Enter> or <Return> on the keyboard
The area we calculated before has just been moved one level higher. '1.5' is now located in the first level of the stack.
How can we multiply these two values?
� Activate the numeric math mode by clicking on the lower radio button ("Stack") of the module link switch in the middle window (fig. 3.1).
� Press [*] in the same window
(If you had not set the module link switch to "Stack", Symbolix would have inserted the '*' into the formula.)
The result of this multiplication is 652.5, but the value you see on the stack will probably differ slightly from this mathematically correct result. That.'s because Symbolix has returned only an approximation of the area (about 435) figured earlier in this chapter.
Finally we have to subtract the sine of Pi/6:
� Push the button with the Pi constant in the lower left hand corner of the screen (fig. 4). A numeric approximation of this symbolic constant is pushed onto the stack.
� Enter '6' in the edit line (fig. 2.1a) and press <Return>.
� Click on [/] in the window of functions.
� Push [SIN] in the same window to get the sine of .523598...
� Press [-] again in the same window.
Our last step is to store the final result in P. a yet undefined variable:
� Press DEFINE in the bottom window (fig. 4.2)
� Name the new variable 'P'
� Press [OK]
� Highlight P in the list of variables (fig. 4.1) by clicking on it once
� Execute [STO] in the stack commands window (fig. 2.4) to move the result to P
� Double-click on P the list of variables to verify that the variable has been set correctly. The result should appear in level 1 of the calculator.
That's it!
We hope this chapter has given you a good idea of Symbolix's capabilities. Some important functions (Derive, 3D graphs) have not been mentioned in this quick tour but the following chapters will give you a complete and detailed explanation of all features.
Chapter 4: The Desktop
Let's scratch the surface...
Chapter 4: The Desktop
Before you can efficiently work with Symbolix you need to be familiar with the desktop interface. This chapter explains the different types of controls, the menu bar, and the various windows.
4.1. Controls
Window (fig. 1)
A window is a rectangular area that displays information on the desktop. It can be closed or moved around the desktop. Some windows in Symbolix have a variable size and a scroll bar to the right.
Check box (fig. 6.2)
A small box associated with an option in a dialog box or window. Checking a box usually activates a certain function. Clicking the check box a second time turns it off again.
Radio buttons (fig. 3.1)
Small circles organised into families. Clicking on a radio button turns off all others of this family.
Lists (fig. 4.1)
Scrollable, vertical arrangements of similar items (variables, for example) on the screen. You can select one item by clicking on it. Using the Apple key and shift, it's possible to highlight sovoral variahlos at anon
� Holding down the Apple key and clicking on a variable selects / deselects it without clearing other selected variables
� Range selection: clicking on a variable, then holding down shift and clicking on a second one selects all variables between these two.
Some functions in Symbolix require that you select or highlight a variable before executing it. If you highlight more variables than the function needs to work properly, Symbolix takes the top variables in the list first.
Buttons (fig. 6.1)
Controls that execute a certain routine in a program when pressed (by clicking on it). Normally they are used to confirm, cancel, or designate an action.
Pop-Up Menus (fig. 1.2, 6.6, 7.4)
Symbolix features several "hidden" menus which are opened when clicking in certain regions. For example, holding down the mouse button in the yellow information bar of the formula window opens such a menu. To select a command, hold down the mouse button, highlight the desired item, then release the button.
4.2. Menu Bar
The menu bar is organised into seven sections: File, Edit, Rules, Formula, Window, and Extensions. The menu on the far left is the Apple menu and appears in every desktop application. It gives you access to desk accessories like the Control Panel NDA and - in Symbolix - activates the online help.
-> The menu icons appear only if you are using GS/OS system 6.0 or later.
The File Menu (fig. 8)
When you click on this menu, you should see a list of choices like in the figure. You can load or save a formula, mix lists of variables, insert formulas into others, print screens and formulas, or quit Symbolix.
Tile Edit Menu
This menu contains a few edit functions (cut, copy, paste, clear, select all), a command which undoes the last symbolic operation, and two functions that transfer data from the symbolic to the numeric module and vice versa.
The Rules Menu (fig. 9)
This is the core of Symbolix. Several extremely powerful commands that can manipulate formulas are hidden behind this menu. "Bell" lets you turn on or off an audio signal which sounds as soon as Symbolix has finished a complicated calculation. The last item tells Symbolix to always sort a formula before simplifying it.
-> The bell option is turned on only when using GS/OS system 6.0 or later. Also note that you must have assigned a sound to the "Task Complete" event (in the Sound Control Device of the Control Panel desk accessory).
The Formula Menu (fig. 10)
Some versatile tools are located in the Formula menu. They allow you to evaluate, solve, integrate a formula, plot functions, change the formula's font, and add variables to the list "on the fly". The item PREFERENCES opens a window which enables you to customise Symbolix (fig. 13).
The Window Menu (fig. 11)
Selecting "Window" opens a menu that lets you show or hide most of the windows.
The Extensions Menu (fig. 12)
The items in this menu depend on you to add even more functions to the main program. Each command activates a stand-alone program which is able to communicate with Symbolix.
4.3. Windows
The Formula Window (fig. 1)
This window holds a formula and works like a word-processor. Symbolic operations manipulate the formula directly in this window, as if you had changed it by hand. Note that this window holds the currently active formula. It's possible to have up to eight formulae in memory, but normally you can work with only one formula at a time.
The Stack Window (fig. 2a/2b)
This is Symbolix's built-in scientific calculator. Although it doesn't use the ordinary algebraic enter system, it's basically a simple calculator, with some program specific improvements.
The Window of Functions (fig. 3)
This window is made up of two parts: standard buttons and two radio buttons. The former represent all built-in functions (like logarithms and trigonometric functions), the two radio buttons determine to what module these functions are actually linked. Setting this switch to "Formula" inserts the corresponding function into the formula when pressing a button, while setting it to "Stack" executes the function on the calculator.
The Window of Constants and Variables (fig. 4)
This gives you access to all variables and to the built-in library of constants. It also helps you to add to or delete from the list of variables (DEFINE and DELETE). What exactly happens when you select a constant or double-click on a variable depends, like in the window explained above, on the setting of the switch in the window of functions (fig. 3.1). If the radio buttons are set to "FORMULA", Symbolix inserts the symbolic constant or variable in the formula, otherwise it pushes its numeric counterpart onto the calculator's stack.
Mathematics, Geometry, Elements (WINDOWS menu)
These windows show you an alphabetically sorted list of all elements of the periodic system. Or a table containing all important formulas. By copying values/formulas to the calculator/ formula you can use them for your own calculations.
The Scrapbook (WINDOWS menu)
This is an area where you can write anything you like. Store a note, an entire formula, an important result - everything is allowed. And best of all, Symbolix saves the scrapbook each time you quit the program, and loads it automatically when you launch Symbolix again.
The EVALUATE Window (fig.5)
This window allows you to evaluate a formula. It offers several as to where to store the result.
The SOLVER Window (FORMULA menu)
Activates the SOLVER routine which scans for roots in the current formula.
The INTEGRATE Window (FORMULA menu)
Opens a dialog box which lets you calculate integrals and areas of a formula.
Trigonometric Conversions (RULES menu)
Open this window when you want to replace one trigonometric function (SIN, TAN, COS, COT) with another one.
Preferences (fig. 13)
This window enables you to turn on and off some of the internal settings.
2D Control Window (fig. 6)
Displays a window with the current settings of the 3D module.
3D Control Window (fig. 7)
Displays a window with the current settings of the 3D module.
2D / 3D Display Windaws (fig. 6.4/WINDOWS menu)
These two windows display graphs. Both feature a "pop-up menu" that can be accessed by clicking in the yellow information bar in the 2D Display Window or in the content region of the 3D Display Window. Note: both menus are not accessible when the windows are empty.
Chapter 5: Numeric Math
INF * O = NaN
Chapter 5: Numeric Math
Numeric Math
The core of the numeric math module is Symbolix's built-in scientific RPN (Reverse Polish Notation) calculator. It offers:
� 15 stack commands
� 39 functions
� 21 levels (6 visible)
� complex numbers
� 39 numeric constants
� access to up to 100 (complex) variables
� full control over SANE environment and display format
� full integration into other modules
We recommend that you have Symbolix running while you're working through the following chapters. If you've just started Symbolix, close the top window (select WINDOW/HIDE FORMULA), otherwise the desktop gets too cluttered. The function of the formula window will be further explained in the next chapter.
An entire window, the "stack" window, is dedicated to the calculator. You can bring it to the front by pressing OA-4 or by selecting WINDOW / SHOW STACK from the menu bar. Other windows that should be open now are:
� WINDOW / FUNCTIONS (OA-2)
� WINDOW / CONSTANTS AND VARIABLES (OA-3)
� WINDOW / STACK COMMANDS (no equivalent)
-> Before you can perform numeric calculations with built-in functions, constants or variables, you must tell Symbolix that you want to use the numeric math module. To do this, set the module link switch in the window of functions (OA-2) to "Stack" (fig. 3.1). If you skip this step, some examples in this chapter won't work.
5.1. The Stack Window
What you see in the stack window (fig. 2a) is the "level" display. The six numbers you see there (fig. 2.3) represent the "head" of the stack. The remaining but invisible eleven levels can be found on top of this window. There's also an edit line at the bottom where you can enter values (fig. 2.1a).
The display format of the numbers is, by default, the auto format. Symbolix supports several display formats (scientific, fixed, engineering, and auto mode). See chapter 12 ("Display Modes") for a detailed explanation.
5.1.1. How the Calculator Works
The calculator uses a special input system commonly called Reverse Polish Notation (RPN). This system is much more efficient than the ordinary algebraic system because there is no need for parentheses. The RPN is therefore easier and faster to use when entering very long formulae. If you aren't yet familiar with this system, it'll take some time to get used to it. But you'll soon realize how advantageous this system is.
5.1.2. The Stack
The stack is a group of 21 levels, where each level holds exactly one number. In the stack window you can only see the first six levels of the calculator's stack, all others are located "on top" of the window (fig. 14).
The stack follows the simple rule of "last in, first out" (LIFO). When you enter a value, the previous value moves up to level 2 and no longer has the highest priority. Similary, if you delete level 1, all levels above move down one position.
The last value you entered will be used for the next calculation. If a certain action requires two arguments (like adding two values), the second level will also be used.
Since level 1 is always the target level and it may happen that you need a value located somewhere else, Symbolix offers a couple of stack commands which let you change the order and contents of the stack. They are located in the small window "Stack Commands" (fig. 2.4).
Only one stack can be active at a time, but you have the opportunity to work with several stacks by saving or loading entire stacks to or from a disk.
To make it even more difficult, newer releases of Symbolix even support complex numbers on the stack. You'll learn at the end of this chapter how to enter complex numbers and calculate with them
5.2. Using the Stack for Calculations
Calculations are performed by
� first entering all arguments that are used by the command or function (for example, SIN(1.5) has one argument, namely 1.5),
� then executing a command or function (SIN in this example)
(How to enter values and execute commands or functions is shown in the examples below.)
The basic concept of the "stack" is the following:
1) Commands (like sine, +, or FLOOR) take their arguments from the stack.
2) The arguments of such a command are removed from the stack when the command is being executed.
3) Results are pushed onto the stack and can be used for further calculations
5.2.1 . One-Argument Commands
Commands that need only one argument to operate take their argument from level 1 of the stack and replace it with the result. Diagram 14.1 shows how the stack changes while executing a one-argument command.
Here is a list of all one-argument functions:
SIN COS TAN ASIN ACOS ATAN SINH COSH TANH ASINH ACOSH ATANH SEC CSEC COT LN BLOG LOG EXP ALOG SQ CEILING FLOOR FAC INV NEG ABS IP FP SIGN STO PICK RE IM
An example for an one-argument function:
Calculate LOG(0.00001)
� Push 0.00001 onto the stack:
Write "0.00001" into the edit line (fig. 2.1a). Press <Return> to copy the value from the edit line to the stack. You should now see the value in level 1 of the stack (fig. 2.3).
� Press on the [LOG] button in the window of functions. After you've executed this function, level 1 is set to -5.
5.2.2. Two-Argument Commands (Dyadic Operators)
As the name already implies this second group of functions takes two arguments (instead of one) from the stack. When the command is being executed, both arguments are removed from the stack. The levels above (level 3 and up) move down one level. The result, which is always a single value, is stored in level 1. (For a diagram see fig 14.2)
The only two-argument functions are:
XROOT Y^X MOD + - /
An example:
Calculate (Pi)A(5)
� Click on [Pi] in the window of constants/variables. Level 1 is set to the numeric approximation of Pi.
� Go to the edit line in the stack window and type "5"
� Press <Return>.
After you have entered all the arguments onto the stack, execute the appropriate function:
� Push [YAX] to calculate 3.14159A5
The result (about 306.02 or 3.0602e2) is located in level 1 of the stack.
The "No-Argument Function
Yes, there is also a function that doesn't take any argument from the stack. Pressing [RAN] stores a random value between 1 and 2^(31)-2 in level 1. Before putting the result onto the stack, [RAN] scrolls the entire stack up one level.
-> From now on, "entering values on the stack" means that you should activate the stack window (fig. 2a/b) and type the given number into the edit line (fig. 2.1a/b, 2.2). Then press <Return> on the keyboard.
5.3. Stack Commands
There are 15 stack commands which help you manipulate the stack (fig. 2.4):
Last
When executing a function (sine, +, SQ and so on), Symbolix stores the argument in level 1 in a separate register called LAST. So if you've executed a function by mistake, or if you want to use the same argument for another function, press [LAST] to get a copy of the last argument. Note that the name of this button is replaced with the number the LAST register holds as soon as you press a function. So you always know to which value this register is currently set.
Drop
Clears level 1 by moving all levels above down one level.
Clear all
Clears the entire stack by setting all levels to 0. Because this is a highly destructive command it works only when you double-click on this button.
DUP (fig. 14ff.3)
Duplicates level 1 by scrolling the entire stack up one level and then copying level 2 to level 1.
Example: to calculate 3+3, enter 3, press [DUP], and then [+].
DUP2 (fig. 14ff.4)
Duplicates levels 1 and 2. It moves the entire stack up two levels and then copies levels 3 and 4 to level 1 and 2, respectively.
Swap (fig. 14ff.5)
Switches levels 1 and 2. This command is quite helpful when you want to execute a two-argument function and both arguments are already on the stack but in the wrong order.
Over (fig. 14ff.6)
Moves the entire stack up one level and copies level 3 to level 1. This command is very similar to [Duplicate] but duplicates level 2 and not level 1.
Up/Down
These two commands scroll the stack up and down. When you scroll up, level 21 (the highest level) wraps to level 1. When you scroll down, level 1 moves to level 21.
Pick
Retrieves the contents of a level specified in level 1. For example, if level 1 is set to 5, then executing [PICK] would replace the "5" with the value that is stored in level 5.
STO (store in variable)
Stores level 1 in a variable that must be selected before you execute this command. See "Variables" below for more information
RCL (recall variable)
Levels 1-20 are moved to the next higher levels and the contents of the selected variable is copied to level 1. See "Variables" below for more details.
Save
Stores a copy of the current stack in a stack file.
Load
Loads a stack file from a disk. The currently active stack will then be replaced.
ReoIm
Swaps the imaginary and real parts of the complex number in level 1. How to work with complex stacks is discussed at the end of this chapter.
Copy to edit line
It's possible to copy a value from one of the six visible levels to the edit line - just move the cursor over the number and click the mouse button.
5.4. Variables and Symbolic Constants
The window of variables and constants (OA-3) lets you access built-in constants and already defined variables. When Symbolix is in the numeric math mode, selecting a constant (fig. 4.4) pushes its numeric approximation onto the stack. Double-clicking on a variable in the list of variables to the right (fig. 4.1) copies the contents of that variable to level 1.
5.4.1. Defining New Variables
Press [Define] (fig. 4.2) in the variable window to define new variables or to change the value of an existing variable. As soon as you push this button, a dialog box appears (fig. 4.5) asking you for a value and a name.
!! Symbolix does not distinguish between variables and parameters. If you want to define a parameter, enter a name and the associated value. When defining a new variable, however, you need not enter a value: just leave the value field as it is. !!
A name for the variable is obligatory and can be up to 15 characters long. Please be very carefull when choosing a name: do not start a name with numbers or other characters that might confuse you and/or Symbolix's parser.
!! In case a variable has exactly the same name as a built-in constant, the variable overrides the constant. You can therefore no longer use this constant in a formula. Delete the variable if you intend to use the constant. !!
5.4.2. Deleting Variables
[Delete] (fig. 4.3) removes one or more variables. First highlight all variables you want to delete, then press [DELETE]. Please refer to chapter 4 ("List control") for information on how to select one or more variables.
5.5. Chain Calculations
The stack is especially helpful when working with long formulas. You can store the result of an operation on the stack and use it as an argument for the next calculation.
Instead of working through a formula from left to right (as you do with a 'normal', algebraic calculator) always start with the innermost operation.
For example, calculate the formula "FN(X)= 5 * ( Pi + ( 4 / 2 ) ^3 )" as shown below:
Calculate 4 /2:
� Enter 4,2 (Remember: type '4' <Return> '2' <Return>)
� Click on [/ l in the window of functions
Calculate 2 ^ 3:
� Enter 3
� [y^x]
Calculate Pi + 8:
� Press [Pi] in window of constants and variables
� [+]
Calculate 5 * 11.1416
� Enter 5
�1*1
You see, only nine steps were necessary to enter this formula. With an ordinary calculator it would have taken 13.
Here are some examples of chain calculations and how to enter them:
5 + ( 3 * NA )
Enter 3 {NA} (car 4.4) [*] Enter 5 [+]
3 - ( ( 6 - 4) ^ 6 )
Enter 6,6,4 [-] [Swap] [yAx] Enter 3 [Swap] [-] or: Enter 3,6,4 [-] Enter 6 [y^x] [-]
> You see, starting with the innermost argument is not always the fastest way, but certainly the safest. If you try to nest too many operations you'll soon loose control over the stack. <
( c - SIN ( Pi / 3 ) - 1 / StdP )
Enter 15 {c} LPi] Enter 3 l/ 1 [-] {StdP} [INV] [-] [y^x]
5.6.Complex Numbers
If you want to calculate with complex numbers on the RPN calculator, you must first set a flag that tells Symbolix to allow such numbers. Open the window "Preferences" (in the FORMULA menu) and mark the check box "allow complex numbers" (fig. 13).
Symbolix then resizes the stack window automatically. To the right of the old stack you'll see a second stack as well as another edit line (fig. 2b, 2.2).
Entering complex numbers
You enter complex numbers almost the same way as you enter real values. First go to the right edit line (fig. 2.1b) and enter the real part of the complex number. Then proceed with the imaginary part by typing it into the left text box (fig. 2.2).
Calculating with complex numbers
For you there's no difference when working with complex numbers. For Symbolix, however, several functions work quite differently.
Let's take a look at the cosine. For real arguments, this function outputs - you guessed it - the cosine. If the argument is complex (x+iy), then Symbolix calculates COS(x)*COSH(y)iSIN(x)*SINH(y).
Three functions need to be explained in detail:
ABS returns the hypotenuse (or radius) of z. (SQRT(x*x+y*y)). NEG returns -x-iy.
SIGN returns the unit vector (r=1) in the direction of its argument.
> Also note that non-elementary functions affect the real part only. By using the [Re<>Im] button you can manipulate both parts individually. <
The imaginary stack in real mode
Though the imaginary stack cannot be seen when complex numbers are disabled, the stack commands (DUP, Drop etc.) will change it anyway. Only scientific functions are restricted to the real stack.
Symbolix Page 5-11
Chaster 5: Numeric Math
5.7. More Chain Calculations
Here are (more tricky) examples of chain calculations:
3 - ( ( 6 - 4) ^ 6 )
Enter 6,6,4 [-] [SwapJ [y^x] Enter 3 [Swap] [-] or Enter 3,6,4 [-] Enter 6 [y^x] [-] Result: -61
15 ^ ( vm - SIN( Pi / 3 ) - 1 / StdP )
Enter 15 {Vm} [Pi] Enter 3 [/] [-] {StdP} [INV] [-] [y^x] Result: 2.14e25
SQRT(Pi/(2a))*(COS(SQ(b)/a)-SIN(SQ(b)/a))/2
DEFINE variables aa" (a=0.4) and "b" (b=0.909) Enter 2 [highlight variable a] [RCL] [*] {Pi} [SWAP] [ / ] [SQRT] [highlight variable b] [RCL] [SQ] [variable a] [RCL] [ / ] [COS] [highlight variable b] [RCL] [SQ] [variable a] [RCL] [/] [SIN] [-] [*] Enter 2 [/ ] Result: -1.3425...
ABS(G^(.5)-3.3^(2))^(3/2)/3
{G} Enter .5 [y^x] Enter 3.3 Enter 2 [y^x} [-] [ABS] Enter 3, 2 [ / ] [y^x] Enter 3 [ / ] Result: 11.978986...
e^(-at/2)/(3a^2)*(e^(3at/2)-COS(SQRT(3)at/2)-SQRT(3)* SIN(SQRT(3)at/2))
DEFINE variables "a" (a=1.5) and "t" (t=4.88) {e} [highlight variable a] [RCL] [highlight variable t] [RCL] [*] Enter 2 [/] [NEG] [yAx] [highlight variable a] [RCL] [SQ] Enter 3 [*] [/ ] [highlight variable a] [RCL] [highlight variable t] [RCL] Enter 3 [*] [*] Enter 2 [/] {e} [SWAP] [yAx] [highlight variable a] [RCL] [highlight variable t] [RCL] Enter 3 [SQRT] [*] [*] Enter 2 [/] [CC)S] [-] [highlight variable a] [RCL] [highlight variable t] [RCL] Enter 3 [SQRT] [*] [*] Enter 2 [ / ] [SIN] Enter 3 [SQRT] [*] [-] [*] Result: 223.7297...
Chapter 6: Symbolic Math - Basics
How to work with formulae...
The symbolic math module is probably the most exciting part of Symbolix. With Symbolix you can:
� enter formulas of up to 48kB (or more than 17,500 characters) in length
� nest up to 127 parentheses
� nest any number of formulae
� use variables as well as symbolic and numeric constants in a formula
Before working through this chapter you should have started up Symbolix and opened the following windows:
� WINDOWS / FORMULA (OA-1)
� WINDOWS / FUNCTIONS (OA-2)
� WINDOWS / CONSTANTS AND VARIABLES (OA-3)
All other windows are not needed and can be closed.
If you intend to use constants, variables, or functions in a formula you must switch Symbolix to the symbolic mode. To do this now, click on the radio button [FUNCTIONS / FORMULA] (fig. 3.1). If you skip this step, the examples in this chapter won't work.
6.1. The Formula
The core of Symbolix is the formula window (OA-1). This mini-word processor holds exactly one formula and offers sophisticated editing capabilities (fig. 1).
6.1.1. "Objects" or the Structure of a Formula
A formula may consist of any of the following "objects":
� up to 715 numeric constants (i.e. numbers) - such as: 1. 8e10e, 1.87010E2001
� symbolic constants - such as: e, NA, alpha, Pi
� up to 100 (different) variables
� functions
� characters for formatting ("white space") - such as: return, tab, space
� calls to other formulas: <EquationFile>
� "i" when using complex formulae when entering equations
Examples of valid formulas are:
SIN( Pi ^( 15 / X ) )
-2 * INV( NA XROOT( SQ( X ) ) -.5 )
1 + - + 3 - - x (same as 1 - 3 + X)
FP( CEILING( RAN * 10 ) / 10 )
7 + 3 * i - 4 / i ^( 3 )
(X is a variable)
Of course, Symbolix supports equations as well:
3 = X
15 / 3 = 7 - i * X * SIN( X )
6.1.2. Complex Formulae
Symbolix supports complex formulae. A pseudo-constant, "i", has been added for "marking" expressions as complex. Symbolix manages this marker intelligently, which means that the program processes multiple occurrences of "i" correctly. Here are some examples:
7+3*i evaluates to 7+3i
7+3*i*i evaluates to 4+0i
7*i+3*i*i^(2) evaluates to -3+7i
As you can see, one single term may contain several "i"s.
Additionally, the power function may be used to raise "i" to any power.
The power function is the only function that can be applied to "i". Symbolix evaluates its argument and converts it to an integer. As a result, fractions in expressions such as i^(x) may occur but are ignored. Consequently, the two expressions 3*i^(3) and 3*i^(3.5) are exactly the same.
Symbolix also distinguishes between *i and /i. Take a look at
3/i*i
3/i^(2)*i*i
3*i^(5)/i
They all produce the same result, namely 3.
Functions for Complex Math
Two functions allow you to "mask" complex expressions: RE() and IM(). These functions "split" their argument and output either the real (RE) or imaginary (IM) part. The other part is set to zero. Both functions cannot be differentiated since they are not elementary.
Example: Result:
IM(3*i) 3i
RE(3*i) 0
IM(4i+5*i*i) 4i
RE(4i+5*i*i) -5
6.1.2. Entering a Formula
You enter a formula exactly as you would enter a text in a word processor. First make sure the formula window (press OA-1) is on the desktop (it need not be the foremost window) and then start entering the formula. However, some objects (functions, symbolic constants and variables) can be entered either directly using the keyboard or by clicking on a button or a list using the mouse:
� Symbolic constants can be inserted into the formula by selecting a constant from the built-in library.
� To insert functions, push the corresponding buttons in the window of functions.
� Double-click on a variable in the list of variables (window OA-3) to let Symbolix insert it into the formula at the insertion point. Of course, this works only if you've already defined the variable before (press [DEFINE] to add variables to the list of variables). See also "Defining Variables" below.
> When pushing a function button after having highlighted (selected) characters in the formula, Symbolix will put the highlighted text into the function's parentheses.
Although you may not immediately see the advantages of this alternative input method, you should consider using it because:
� Symbolix always inserts the correct number of parentheses into the formula (a frequent error when entering formulas directly via the keyboard),
� there is no chance to misspell functions, constants or variables, and, finally
� you can keep working with the mouse.
Symbolix is case sensitive, i.e. "SIN" and "sin" or "StdP" and "Stdp" are not the same. This rule applies to all objects.
All functions except "RAN" are immediately followed by an argument in parentheses. Consider the opened parentheses of these functions as part of the function and do not separate the keywords and the parentheses with spaces, returns, or tabs. For example, Symbolix does not recognize "COS ( Pi )" (note the space between "COS" and "(Pi)").
6.1.3. Defining Variables
Before Symbolix can work with algebraic expressions containing variables, vou must have already defined them.
Press [Define] (fig. 4.2) to define new variables or to change the value of an existing variable. As soon as you push this button, a dialog box appears asking you for a value and a name.
Symbolix does not distinguish between variables and parameters. If you define parameters, enter a name and an associated value. When creating a new variable, just ignore the "value" field and specify the name only.
The name of the variable can be up to 15 characters long. Please be very careful when choosing a name: do not start a name with numbers or other characters that might confuse you and/or Symbolix's parser. If the name you specified contains spaces, they're automatically replaced by underline characters ("_").
When you press [DEFINE] and a variable is highlighted, both edit lines ("value" and "name") are set to the name of that variable and its corresponding value. This is useful when you want to
� change the value of an existing variable (change the value and press [OK])
� duplicate a variable (change the name and press [OK]).
A variable appears only once in the list of variables. If you try to [DEFINE] a variable twice, the old one will be deleted first.
6.1 .4. Checking the Syntax of a Formula
After you've entered an expression you'll most likely want to know whether Symbolix accepts it in that form. Select FORMULA / CHECK SYNTAX OF FORMULA to let Symbolix check the formula. One of the following situations will happen:
Symbolix shows a syntax error message (fig. 15ff.8).
This means that Symbolix doesn't know what to do with the highlighted item. Common causes: undefined variables or misspelled functions, variables, or constants.
Do not forget to put the second argument of the three functions MOD, XROOT and ^ in parentheses.
Example: Cause: Correct:
SINE(0.5) unknown keyword SIN(0.5)
12^3 Symbolix does not know 12^(3)
the function "^"
(but "^(")
FLOOR (.5) space between "FLOOR" FLOOR(.5)
and "(.5)"
3 XROOT 9 missing parentheses 3 XROOT(9)
Symbolixgivesyou a Sense of expression not clear message
The formula you entered makes no sense. Common causes: forgotten items (example: 5+SIN()-3), items inserted by mistake, or incomplete expressions.
There are situations where multiplication signs normally don't need to be written (for example: 4*a*b can be written as 4ab). Since this is, strictly speaking, against the rules, Symbolix is not able to insert them in all cases. In some circumstances you may get a Sense of expression not clear error message when this happens.
Symbolix shows a Number of open and closed brackets does not match error message (fig. 15.1).
In this case, Symbolix can't assign a closed parentheses to each opened one. Example: SIN(COS(TAN(3)
Symbolix does nothing ([CHECK SYNTAX REWRITES FORMULA] is cleared, fig. 13).
This means that the formula has been entered correctly and has been accepted.
Symbolix rewrites (reformats) the formula [CHECK SYNTAX REWRITES FORMULA] is set, fig. 13).
This also means that the formula has been accepted. If the check box [INSERT SPACES WHEN REWRITING FORMULA] is set, Symbolix inserts a space between two items.
Symbolix outputs a buffer error
"Table of numeric constants full": you tried to exceed the limit of 715 numeric constants
"Buffer too small to hold entire formula" (fig. 15.2 or 15.6): Symbolix's internal buffers are too small to hold the formula
An alert box bad der() expression appears
You've used the der() pseudo-function incorrectly, i.e. the expression where Symbolix stopped does not match der( variable: expression ).
When you've entered nested formulas some error messages display the additional information "Last file included was xyz" (whereby xyz is the name of a file). This means that the error occurred somewhere after the <xyz> nest call. (Symbolix can't highlight the exact position and therefore gives you a "hint" as to where to look for errors.)
6.1.5. Editing a Formula
In the formula window you can enter any formula up to 48kB in length in a word processor-like environment. It supports a number of editing keys which are summarised below:
� Left arrow: moves the insertion point one character to the left
� OA + left arrow: jumps to the beginning of the previous word
� Option + left arrow: moves insertion point to the beginning of the line
� Shift + left arrow: extends the selection back by one character, word (if OA is held down), or line (if option is held down).
� The right arrow key works analogously to the left arrow key
� Up arrow: moves the insertion point up one line
� OA + up arrow: jumps to the beginning of the page
� Option + up arrow: jumps to the beginning of the formula
� Shift + up arrow: extends the selection up by a line or a page (if OA is held down), or to the beginning of the formula Xif option is held down)
� The down arrow key works analogously to the up arrow key
� Delete: deletes the character to the left of the insertion point, or removes a selection if text has been highlighted
� Clear: clears the current selection
� Control-F: deletes the character to the right of the insertion point, or removes the current selection of text that has been highlighted
� Control-Y: clears all characters from the insertion point to the end of the line
� Control-X: cuts any selection and puts it onto the clipboard
� Control-Y: copies a selection and puts it onto the clipboard
� Control-V: inserts the contents of the clipboard at the insertion point, or replaces a selection with the contents of the clipboard if there is a selection
� Double-click: highlights a word
� Triple-click: highlights an entire line
You can also use the edit menu to cut, copy, paste, or clear a selection. These four commands are equivalent to the keystrokes control-X, control-Y, control-V and <Clear>.
Pressing OA-A or choosing EDIT / SELECT ALL highlights the entire formula.
Cut, copy, paste, and clear work in the active window only.
The font of the formula is set to Shaston 8 point by default, but you can load another font at any time. It is, however, not recommended to change the font when working with the formula because the system font, Shaston 8 point, can be displayed much faster than any other font. When you load another font, commands such as Expand, Collect, and Sort will be delayed significantly.
6.2. Switching Formulae
Though you can enter and view only one formula at a time in the window, Symbolix can manage up to eight formulae simultaneously. Those you can't edit in the formula window are kept in a safe place in memory.
There are two commands for creating a new "document": FILE / NEW and OPEN. NEW creates an empty document. OPEN loads an existing formula.
When you are completely done with a formula, choose the FILE/CLOSE menu item. This deletes the formula (only in memory, not on the disk) and frees the memory that has been allocated for it. If there's also a graph linked to the formula (more about that in the chapter about 2D graphs), Symbolix erases it as well. But if you need the formula or graph later, don't forget to save them to disk before closing the document.
To switch to another formula in memory, move the cursor to the information bar (fig. 1.1). Then push the mouse button and hold it down.
Symbolix instantly opens a pop-up menu (fig. 1.2) which displays the names of all eight formulae. The checked formula is the one you can see in the formula window (fig. 1.3). All commands that manipulate, save, load, or graph a formula will use this and only this formula. Only the formula in the window can be changed.
The last item of the pop-up menu is RENAME (fig. 1.4). This command renames the active formula: it opens a dialog box where you can enter any name you want (max. 15 characters) as long as you don't save the formula. If you want to save it to disk, make sure that the name does not violate the naming rules of the operating system you're using!
6.3. Nesting Formulae
The nest command is a very powerful and versatile function. Although its function is fairly simple - it inserts a specified formula from a disk at a certain position in the current formula - you'll find this command very useful. Frequently used expressions can be stored in separate files and "induded" into other formulae at will, without modifying the original expression. You can even nest expressions in already nested formulae, or more strictly speaking, you can nest as many files as you want. This command not only frees you from typing the same expressions over and over again but also enables you to build libraries of frequently used expressions.
Do not nest a file that is already called by another formula. If you do so, the program will run in an endless loop and will not stop until the internal buffers are full.
The syntax for the nest command is <Filename> whereby "Filename" can also be a partial or full pathname.
Examples:
<1/Equation> nest file "equation" in the program directory
<Root> nest "Root" in the current folder
<:Math:Library:Degree> nest the file "Degree" in the folder "Library" on the disk "Math"
As soon as you select a menu item from the RULES menu or CHECK SYNTAX OF FORMULA, the nest command is replaced with the contents of the nested file.
If you want to keep the nest command when checking the syntax of your formula, make sure that you have turned off the check box [FORMULA / PREFERENCES / CHECK SYNTAX REWRITES FORMULA] before.
An example of a nest call
Let's assume you want to calculate the surface area of a sphere ( A = 4*Pi*r^2 ), and you have the following three files on the disk:
� the file "Circumference", containing the formula "2 * Pi * r",
� "Area", containing "<Circumference> * r / 2",
� "Sphere.Area", containing "4 * <Area>"
Load the file "Sphere.Area" and then [DEFINE] the variable "r". Now select FORMULA / CHECK SYNTAX OF FORMULA. (If nothing happens, mark the check box [FORMULA / PREFERENCES / CHECK SYNTAX REWRlTES FORMULA] and try again.) You'd now see in the formula window:
4 * 2 * Pi * r * r / 2
which is the same as 4 * Pi * r^2
Chapter 7: Numeric Math with Formulae
Numeric Math a little bit different...
This chapter shows in detail how to
� evaluate
� solve and
� numerically integrate a formula
All of the these functions can be used on a formula stored in the formula window (fig. 1). Most of them produce a result which can be copied to the calculator and used as an argument for further calculations.
!! The routines for integrating and solving formulae do not support complex numbers. You can, however, use them even when the formula produces a complex result although they will ignore the imaginary part.
7.1. Evaluating a Formula
FORMULA / EVALUATE (fig. 5) evaluates a formula and stores the result in one of three possible places:
� [DISPLAY RESULT ONLY] writes the result in the window,
� [TO STACK] displays the result and pushes it onto the calculator's stack,
� [TO VARIABLE] displays the result in the window and stores it in the selected (highlighted) variable.
The fourth button, [TO VARIABLE,LOOP], does the same as [TO VARIABLE] but continues evaluating the formula and storing the result in the highlighted variable until OA is held down. (If you press Option, Symbolix waits until you release the key again.) This command is especially helpful when working on recursion formulae. An example:
To find roots of a formula, Symbolix uses the algorithm x = x - FN(x) / FN'(x) (Newton)
By continually evaluating this formula and storing the result in x, Symbolix approximates x to a root. We'll now apply this algorithm to find a root of the function COS(X):
x = x - COS( x ) / NEG( SIN( x ) )
( NEG(SIN(x)) or -SIN(x) is the first derivative of COS(x) )
[Define] the variable "x" (set the value to 2) and enter the formula above into the formula window. Then choose [FORMULA / EVALUATE / TO VARIABLE,LOOP]. After a few seconds Symbolix will display the number 1.57079 at the bottom of the EVALUATE window (the correct result is Pi/2). Since this value doesn't change any more, you can be sure that Symbolix has found the best approximation.
7.2. The Solver
SOLVE helps you to find roots of a functions (i.e. points where the function touches or crosses the x-axis). If the current formula is an equation, the SOLVER scans for the closest intersection of the two expressions. It uses Newton's algorithm, an uncomplicated and straightforward method.
Highlight an independent variable in the list of variables. When you now select FORMULA/SOLVE, Syrnbolix opens a dialog box with the following items in it:
� an edit line where you can enter a starting value
� the [SOLVE] button which returns the next approximation
� the [LOOP] button which continually returns approximations until you press OA
� [TO STACK], which copies the current value to level 1 of the stack
� [CANCEL] which closes the window
Pushing [LOOP] is the same as pressing [SOLVE] many times. Normally you will use [LOOP], but if you want to write down intermediate steps then you'll probably prefer [SOLVE].
[TO STACK] serves as a link to the calculator. Pressing this button pushes the last approximation (i.e. the value in the edit line) onto the stack so that it can be used for further calculations.
The start value can be of any value, but the closer this value is to a root the faster it can be found.
> It's a good idea to take a look at the graph first, estimate a possible root and then use your own estimation as a starting value for the solver.
Interpretation of the Approximation
The following cases may occur:
� The value doesn't change any more after a while
Symbolix has found a root.
� The value changes quickly but is always between twofixed values
The function has a local maximum or minimum. Try another start value.
� The value increases or decreases permanently
The function approximates to the x-axis but neither touches nor crosses it.
Cases two and three haven't been successful searches and you'll have to try other starting values.
(Also take a look at the root finder routine in the 2D graph module which features automatic interpretation of the results.)
7.3. Numeric Integration of Functions
The last function discussed in this chapter is the integration module. It calculates not only integrals but also areas using an extremely powerful and fast algorithm. You can specify the precision of the calculation and use the result for further operations.
First highlight an independent variable in the list of variables and then choose FORMULA/INTEGRATE from the menu bar. A dialog box with the following items in it appears:
� "from", which is the lower limit of the integral/ area to calculate,
� "to", which is the upper limit,
� "initial accuracy", which indicates how many data points have to be calculated (data points = 6 * initial accuracy),
� [INTEGRATE], which integrates a formula in the given interval,
� [AREA], which calculates the area of the formula in the specified range,
� [TO STACK], which copies the last result to the calculator,
� and finally [CANCEL], which closes the window.
The first thing you should do is set the upper and lower limits of the integral/area to be calculated. You can set any limit between -1E4932 and +1E4932.
Then specify the accuracy of the calculation. You can enter into this edit line any positive integer (1,2,3...). The higher the initial accuracy is, the more precise results you'll get - and the longer you'll have to wait for them. This factor is set to 20 by default which is a happy medium between accuracy and execution speed.
> As a rule, "complex" formulae (in the sense of complicated) have to be calculated at higher accuracy factors than simpler ones.
Pressing either [INTEGRATE] or [AREA] starts the calculation of the integral or the area, respectively. Then Symbolix displays the result at the bottom of the window, increases "accuracy" by one and calculates the same integral/area again, though at a higher precision. You can stop this endless loop by holding down OA at any time.
[TO STACK] enables you to use the last result for further calculations. It copies it to the calculator from where it can be used as an argument for built-in functions, stored in a variable, or even transferred to the formula.
Press [CANCEL] to close the window.
Chapter 8: 2D Graphs
Draw Formulae - and more!
2D Plot Module
8.1. The 2D Control Window
2D Plot lets you graph a formula in any interval between -10^4090 and +10^4090. Before starting 2D Plot, highlight the independent variable in the list of variables (usually "X"), then select FORMULA/2D PLOT or press OA-G. After you've chosen this function, a new window with the following items will be opened (fig. 6):
� two edit lines to the left specifying the horizontal range of the graph to be plotted
� two edit lines to the right of the screen which set the lower and upper limit of the vertical range
� a check box named "lock" which lets you superimpose graphs
� another chedc box, "draw grid", tells Symbolix to draw thin grid lines, making it easier for you to locate points in the system of coordinates
� a third box titled "draw labels" which labels the x- and y-axes
� [PLOT], which clears the window and draws the graph
� [AUTOSCALING] which does nearly the same as [PLOT] except that it calculates the y-range automatically, ignoring any values in the y-range edit lines,
� [Choose Font], which allows you to change the font,
� "double height", a flag for improving quality of high-density printouts
� and last but not least three color palettes
Under normal circumstances you'll only change the x-range ("x start" and "x end") settings and then press [AUTOSCALING] (fig. 6.1) which analyzes the graph in the specified interval and scales it so that the graph fills the entire screen.
The autoscaling feature is normally the fastest and easiest way obtain a good-looking graph. But it doesn't always produce the desired result if the function has one or several extreme values that would distort the graph too much (i.e. lower the vertical resolution in the remaining part of the graph). In such instances you should set the y-range "manually" by entering appropriate values into the "y start" and "y end" edit lines and then pressing [PLOT] (instead of [AUTOSCALING]).
Symbolix now always draws both sides of an equation. Previous versions allowed you to draw the left side only but this is no longer possible (and not very useful, either).
Labels
When you press the box "draw labels", Symbolix will calculate appropriate units for the x and y axes separately and label them to the left and at the bottom of the system of coordinates. The units of the axes are always multiples of ten and divide the axes into 2 to 20 sections (adjustable in Symbolix 1.9).
Superimposing graphs
By checking the check box "lock graph" you can superimpose any number of graphs very easily. Instead of clearing the old graph first, Syrnbolix combines both graphs - the new and the old one - into one single picture.
You might find this option useful when comparing graphs and their derivatives.
Of course, you should draw the grid and labels only once because the picture otherwise gets too cluttered.
Coloring graphs
Symbolix displays three color palettes (fig. 6.3): one for the left side of an equation, one for the right side, and a third for the axes. By clicking on a color box you can set the colors for each of the three objects.
Choose font
Symbolix allows you to change the font of the labels. Previously, the program displayed them in Shaston 8 only.
Unfortunately, printing graphs with the "double vertical height" option (see "Print graphs" below) also doubles the font size of the labels. (Symbolix can't prevent the system software from doing so.) Consequently, the labels at the bottom often overlap which is quite an eye-sore. The new version therefore features a function for eliminating these justification problems. If you want to print a graph at "double vertical height", make sure that the check box "double height" (fig. 6.2) is also checked. Note that the labels in the 2D Display Window will be much smaller than the chosen font size. But the print routine will draw them correctly and even without overlapping them.
Graphs and Formulae
Symbolix manages up to eight formulae in memory. You can "connect" or link one graph to each formula. Consequently, Symbolix allows you to work with up to eight graphs simultanously.
It's very important to know that formulae are only loosely linked to formula documents. If you change a formula and want to zoom into a previously calculated graph, Symbolix can't remember the original formula and displays an error message. Of course, functions for exporting, purging, printing, and saving graphs will still work.
To switch to another graph, just switch to its formula. When Symbolix swaps the current formula and the to-be-activated formula, the graphs of the two formulae are also exchanged.
> If you want to copy a graph to another formula, just
� save the graph to disk
� switch to another formula
� execute FORMULA / LOAD 2D GRAPH and load the graph.
8.2. The Graph
As soon as you press [AUTOSCALING] or [PLOT], Symbolix calculates the graph(s). While evaluating the formula, it displays a "please wait" message in the menu bar.
When the complete graph has been calculated and drawn in memory, the 2D Display Window will open (if not already) and the graph appears (fig. 6ff.4). You can close, resize, or move this window without loosing the graph.
The window consists of two sections: the information bar and the content region.
The content region shows the system of coordinates. You can select with the cursor...
...either a single coordinate by clicking the mouse button once
...or an area (selection box) by clicking in the upper left corner of it, holding down the mouse button, moving the cursor to the lower right corner of the box and then releasing the button.
When holding down the mouse button in the yellow information bar, Symbolix opens a pop-up menu (fig. 6ff.6) with a couple of commands for analysing the current graph.
8.3. The graph menu
Finding roots
This item does exactly the same thing as the solver but is easier to use. The leftmost coordinate of the selection or the selected x-coordinate is taken as a starting value. Approximations and the status of the routine appear in the information bar and help you interpret the result:
�approximating means that Symbolix comes closer and closer to a root
�extrememum? says that Symbolix has found a local extremum. Try another starting value.
�bad guess means that a SANE error occurred (i.e. SANE returned a NaN)
�found signals that Symbolix has found a valid root or intersection
As with the SOLVER, this command searches for roots when the current formula is a simple algebraic expression. If it's an equation, it scans for the closest intersection.
> You can interrupt the routine at any time by pressing the mouse button.
Zoom
Magnifies the current selection. When zooming with autoscaling enabled (fig. 6ff.8), Symbolix will stretch or shrink the graph vertically so that it fits into the window.
!! The flags in the 2D Control Window (font, colors, etc.) have global character and affect the zoom function as well. Since this window is a modeless window, you can adjust the attributes (i.e. check boxes and font button) before zooming and Symbolix will then use the new settings.
Show Slope
Calculates the slope at the selected x-coordinate (fig. 6ff.7).
Show Coordinates
Displays the relative position of the cursor in the system of coordinates
Show range
Shows the display ranges.
Save Graph
Saves the graph. You can load a previously saved graph with the FILE/LOAD 2D GRAPH command.
Because of the object oriented structure of graph files, pictures will be smaller than 4kB though they can be printed at full 300dpi and more.
> The Symbolix package contains a System Extension (Symbolix plug-in) for exporting graphs in the Apple Preferred file format.
Copy to clipboard
Stores the graph in the clipboard file. Afterwards, the graph can be pasted into most desktop programs that support object oriented pictures.
Print graph
Prints the current graph. Before the graph is sent to the printer, Symbolix asks you for a title and font.
Mark the check box "double vertical size" to print the title and the graph in proper proportions.
Purge graph
Erases the graph from memory.
Chapter 9: 3D Graphs
Don't touch them - thev aren't real...
3D Plot Module
The 3D Graph module has been part of the Symbolix 1.6 update. Unlike 2D graphs, only one single 3D graph can be kept in memory. You'll loose a 3D graph as soon as you clear it or calculate a new one.
9.1. Plotting a Graph
The 3D module offers dozens of features. Don't worry if you don't understand everything at first glance! Here's how you get your 3D graph a hundred per cent:
1. First enter or load a formula.
The expression must be in the form of f ( x, y ) =z (for example, "COS( .1* ( x*x+y*y ) )" ). As you can see, two independent variables are involved which makes it a bit more complicated.
2. Now tell Symbolix which variables have to be linked to the x and y axes.
First select the "x" variable in the list of variables (i.e. the variable that will run along the x axis). Then open the 3D CONTROL WINDOW from the WINDOWS menu (fig. 7) and push [X Var:] (fig. 7.2) in the lower left corner.
Linking a variable to the y axis works analogously: select the variable and click on [Y Var:].
To the right of both buttons you should see the descriptors (the names) of the two variables.
!! Deleting variables disconnects the independent variables. You'll have to reassign them.
3. Enter display ranges
Enter the intervals of the independent variables. The x axis will run from x MIN to x MAX, the y axis from Y MIN to Y MAX, and the z axis - you guessed it - from z MIN to z MAX. However, the last range, the z interval, may be ignored provided that you let Symbolix calculate the upper and lower limits (see "autoscaling" below).
4. Specify the resolution
A very important factor is the graph's resolution. Symbolix allows you to change the resolution in discrete steps from 4x4 to 124x124 blocks (or "tiles"- the graph looks like a distorted chess-board.)
The lowest resolution is 4x4 blocks. This means that Symbolix calculates 16 blocks which produces only a rough sketch of the graph. However, it takes only a few seconds to calculate the image.
The highest resolution is 124x124 blocks, a rather theoretical limit. The program needs about 30 minutes, and the graph may be as large as 2MB (depending on its attributes). Fortunately this precision is so high that the graph is nothing but a black blotch, but if you want to magnify the graph several times (in another application), you might want to make use of it. But note: the enormous memory requirements will probably prevent you from doing so. (Symbolix detects out-of-memory errors and cancels the command immediately.)
A fair compromise between speed, memory requirements, and precision are resolutions from 14x14to 25x25.It depends on the formula mainly: some functions require less accuracy, others more. Try it!
5. Set the attributes
All other buttons, check boxes (fig. 7.1) and scroll bars in this window control the appearance (or "attributes" ) of a graph:
� HIDDEN LINES enables Symbolix's hidden line algorithm. If it's turned on, the program suppresses lines in the graph that are hidden behind others and aren't visible to you.
� COLOR FILL means that the blocks will be filled with a color. The actual color depends on the average "height" of such a block (similar to contour lines in a map).
!! This option works only if HIDDEN LINES is also checked.
� DRAW LINES shows (checked) or hides (not checked) the black lattice-work. Of course, this control is only meaningful if COLOR FILL and HIDDEN LINES are active.
� If DRAW AXES is marked, Symbolix draws x, y and z axes. Note that the origin of the three axes is not necessarily at (0/0/0) but pinned to the most negative coordinate of the graph.
� DRAW GRID: Symbolix calculates appropriate units for the x and y axes and draws thin, gray lines.
� The COLORS button opens a window where you can select one of four possible color sets.
� The PHI/THETA SCROLL BARS rotate the graph in discrete steps. The sample graph will be updated accordingly.
� The PERSPECTIVE scroll bar lets you change the perspective distortions.
Because of the way Symbolix calculates graphs and stores data in memory, you don't have to recalculate a graph each time you want to change an attribute. Instead of pressing PLOT or AUTOSCALING (explained below), just push APPLY and the program skips the time-consuming process of evaluating the formula many times.
7. Calculate the graph
And last but not least it's time to calculate the graph. You do this by pushing either PLOT or AUTOSCALING. The first button starts the 3D routines, assuming that you have entered all of the three intervals (x,y, and z}. The latter command, however, ignores the z range and scales the graph automatically by scanning for the most positive/negative z coordinates.
Symbolix now starts evaluating the formula many times. Because this may take a very long time, the program informs you of what it is doing right now. The messages are:
Blocks remaining to be calculated: xxxx
Phase 1: Symbolix evaluates the formula
xxxx indicates how many times Symbolix will evaluate the formula
Please wait, scaling graph...
Phase 2: Symbolix normalizes the graph
Afterwards Symbolix stretches or shrinks the graph so that it fits into the window
Please wait, processing data...
Phase 3: It records the picture elements
The program draws the picture and stores the picture in a QuickDraw picture data structure.
Please wait, rendering image in memory...
Phase 4: Symbolix updates the off-screen buffer
To accelerate update events, Symbolix then "plays" back the previously recorded picture and stores a 1:1 copy of the screen image in a separate buffer. This enables Symbolix to update the 3D window very quickly.
The more attributes you have enabled, the longer it'll take. A graph without hidden line algorithm (and consequently without color fill) requires only a few seconds. More complex graphs may occupy the computer for more than half an hour!
The most important factor is the resolution. You'll have to experiment with this scroll bar: some formulas produce acceptable results only on even resolutions, others need extremely high precision.
> The speed depends also on your system. An accelerator card is certainly a good investment.
9.2. Data Types / Precision
Symbolix uses different data types in order to calculate graphs as fast as possible.
Formulae are evaluated using SANE's extended data type (10 bytes). Before the result is stored in memory, it is converted to a single value (4 bytes) which saves a lot of memory.
When Symbolix scales a graph, the single values are retrieved, scaled, and put back as fixed values (4 bytes, fixed point). From now on, all calculations are done without using slow floatingpoint routines.
The resolution of a fixed point value is <1.E-5 which is more than enough for a super hires image. Symbolix preserves NaNs and infinities during conversions, thus you won't notice any difference in precision.
9.3. The Graph Menu
The 3D module has its own graph menu (fig. 7.4). It allows you to save, print, copy, and purge graphs.
To open this menu, activate the 3D Graph Display window (menu WINDOWS) and click anywhere in the content region:
Saving Graphs
You can save 3D graphs in any of three possible formats:
�Save Graph (Symbolix format) stores the graph in Symbolix's own format. Such graphs can be loaded with the FILE / OPEN 3D GRAPH command. Since the program does not save the picture itself but the normalized vectors of the graph, you can apply new attributes.
�Export (super-hires) exports a 3D graph in superhires (bitmap) format. Most paint, DTP and other desktop programs are compatible with this type.
�Export (object format) exports a 3D graph in object format. (It's a 1:1 copy of the QuickDraw Picture data structure.) Only very few programs support this type.
(Again: also take a look at the APF Extension that exports graphs ill the Apple Preferred piefureformat)
Copying Graphs to the Clipboard
Select copy Graph to Clipboard and Symbolix will store a copy of the current graph in the system clipboard. (See 2D graphs for more details.)
Print
Sends a graph to the printer. Because regular printouts have twice the vertical resolution of the screen image, there's a check box that - when marked - stretches the graph vertically. Only double-height printouts show text and graphs in proper proportions.
Chapter 10: Symbolic Operations on Formulae
Symbolic Operations on Formulae
Symbolix offers several extremely powerful commands that manipulate formulae:
� Expand
� Sort
� Collect
� Simplify
� Eliminate ((..))
� Trigonometric Conversions
� Derive (All)
As soon as you execute one of these functions, the indicator in the formula windows information bar shows you how much memory is used.
>> If you want to cancel the commands Expand, Collect, or Derive, press the keys Open Apple-Option-Shift-Control all at once and hold them down until an alert box appears.
10.1. Expand
Expand rewrites a formula by expanding products and powers.
Examples:
(a+b)^(2) EXPAND returns a*a+a*b+b*a+b*b
b^(4) EXPAND returns b*b*b*b
(a+b)*(c-d) EXPAND returns c*a+c*b-(d*a-d*b)
10.2. Sort
Sorts the individual components of a formula and puts them in the following order:
� Numeric constants
� Symbolic constants
� "i"
� Expressions containing constants
� Variables
� Expressions containing variables
� Other expressions
� RAN
Examples:
SIN(X)*e*3*X
c*a+c*b-d*a-d*b
SORT returns 3*e*X*SIN(X)
SORT returns a*c+b*c-a*d-b*d
10.3. Collect
Simplifies a formula by collecting like terms. It also evaluates numeric expressions (see "Symbolic and numeric results" below). It does the opposite of the EXPAND command (although it doesn't always return exactly the original expression).
Examples:
15+2*2.5 COLLECT returns 20
a+3*a COLLECT returns 4*a
SIN(Pi)*X^(RAN)-4*NA*X^(RAN) COLLECT returns (SIN(Pi)-4*NA)*X^(RAN)
a*a*a COLLECT returns a^(3)
It's highly recommended to sort a formula before executing COLLECT, otherwise it may not produce the expected result.
10.4. Simplify
Simplifies a formula by eliminating unnecessary expressions.
Examples:
3 +0 *S IN (3 )+5 SIMPLIFY returns 3+5
5+(SIN(4)) SIMPLIFY returns 5+SIN(4)
EXP(LN(X)) SIMPLIFY returns X
i^(5) SIMPLIFY returns i^(1)
i^(-1) SIMPLIFY returns i^(3)
ELIMINATE ((..)) is a subcommand of Simplify: it removes parentheses which do not affect the priority of operations. It does not rewrite expressions.
10.5. Trigonometric Conversions
Rewrites trigonometric expressions. Please use the help function for more information about the individual buttons.
Example:
SIN(X) TRIG CONVERSION returns TAN(SQRT(1 +TAN(X)^(2)))
10.6. Derive (All)
Syrnbolix 1.5 introduced a superfast algorithm for differentiating elementary functions. You can apply this command to either the complete formula or to marked expressions only.
The command DERIVE differentiates only expressions that have been put in "der()" pseudo-functions. The syntax is...
der( variable: simpex)
der( is the keyword, variable is a variable that has already been defined, and simpex is a simple algebraic expression.
Contrary to the second menu item (DERIVE ALL), you aren't restricted to a single independent variable in a formula. You can use different variables and even nest "der"-funtions at will.
Example: Result:
der(x: COS(x))-IP(x) NEG(SIN(x)>IP(x)
der(x: der(y: x*SQ(y)) 2*y
...and so on.
DERIVE ALL on the other hand is useful when you want to know the derivative of the entire formula. Before executing this command, select the independent variable in the list of variables.
Example: Result:
SIN(x)=3*x COS(x)=3
( x has been highlighted in the list)
Limitations of the Algorithm
Of course, Symbolix can differentiate elementary functions only (except for ABS'(x)=SIGN(x) ). When the program encounters other functions (MOD, IP, FP, FAC, CEILING, FLOOR, SIGN, RE, IM), it shows an alert box and cancels the command. If you want (for whatever reasons), you can use block protection brackets to skip invalid expressions (see below).
10.7. Symbolic and Numeric Results
Symbolix lets you use symbolic constants in your formulae. However, these constants represent strictly speaking nothing but pure numbers.
As already mentioned above, COLLECT evaluates numeric expressions and replaces them with the result. You can tell Symbolix
� to treat symbolic constants as numbers and convert them to numeric results when a COLLECT command is being executed,
� or to keep them in the formula with their name.
Both possibilities have their advantages and disadvantages. When you convert symbolic constants to numeric results you do not know what happened to them, and sometimes it's very useful to know where a constant has been moved to. On the other hand, formulae will be shortened much more if you let Symbolix convert them to numbers.
Examples:
You enter the formula e*e*e*e*e With numeric results enabled: COLLECT returns 148.4131591 With symbolic results enabled: COLLECT returns e^(5).
You can select the appropriate mode by setting one of the two radio buttons- in the window "Preferences" ([EVALUATE SYMBOLIC CONSTANTS TO NUMERIC/SYMBOLIC RESULTS]).
Note that Symbolix will convert symbolic constants to numbers only if they're part of an expression consisting of (numeric or symbolic) constants only. For example, COLLECT doesn't change the formula 3+Pi*SIN(X) since the sine function can't be converted to a numeric result. But if you change the expression to 3+Pi*SIN(e), COLLECT is able to solve the parentheses and returns 4.291.
10.8. Block Protection Brackets ("[...]")
This special kind of parentheses "locks" expressions in formulae. This means the commands Expand, Collect, Simplify, and Derive cannot manipulate expressions in such brackets. Note that they're treated like regular parentheses, thus any violations of the syntax will force Symbolix to abort the current command.
For example, protect expressions like the following:
...+ (a+b)^(9) +...
Now choosing EXPAND would occupy your computer for at least an hour. (Don't try it: you'll get an "out of memory" message anyway.) However, putting the dangerous expression in brackets solves the problem. Just rewrite the formula so that it looks like this:
...+ [ (a+b)^(9) ] +... and everything will work fine.
> It's possible that COLLECT collects two protected but completely identical expressions. An example:
You enter:
4*[(c-d)^(9)]+[(c-d)^(9)]
COLLECT returns: 5*[(c-d)^(9)]
Chapter 11: System Extensions
Copy & use them. Your very own Symbolix.
11. System Extensions
"System Extensions" is a new concept that enables assembly programmers to improve the "kernel" very easily. While the writing of such additional programs requires some skill in programming in assembly language, extensions are very easy to use.
11.1. Installing Extensions
Installing them is a as simple as copying files. Launch your favorite GS/OS copy program and put a copy of the files into the EXTENSIONS folder. When Symbolix is being loaded, the program scans this directory for system files and shows their names in the new menu to the right ("Extensions").
!! Only programs that specifically have been written for Symbolix may be copied to the EXTENSIONS folder. Putting other programs - even with the same file type - into this directory will almost certainly crash Symbolix.
11.2. Executing Extensions
If you want to execute an- extension, just open the EXTENSIONS menu and select it with the mouse. It's up to the extension what will happen next. If you are not sure what the program will do, go into the help mode (APPLE / HELP), then choose an extension. A well programmed extension now displays a help screen.
Most routines display a modal dialog box only. That's because previous versions of Symbolix did not support desktop items belonging to extensions. This has changed with Symbolix 1.7: an extension can now open its own windows just as if Symbolix had opened it.
Loaded or not loaded?
There are two kinds of extensions: temporary and memory-resident programs. The latter category is loaded when you select it for the first time and will be kept in memory until Symbolix is shut down. On the other hand, temporary extensions will be released immediately after they have passed control back to Symbolix.
This really doesn't affect you. The only reason why Symbolix supports temporary extensions is because they can be launched even on very small systems (since they need memory only while they are running).
You can easily determine whether an extension is kept in memory or not: if its menu item is checked, then this program is still in memory.
Problems with Extensions
If an extension causes problems, you can de-install it by executing EXTENSIONS / SHUTDOWN. This call closes all programs and frees their memory blocks.
11.3. Sample Programs
Symbolix comes with several sample programs:
Formula Status
...show some information about the formulae on the desktop.
FastVariable
...scans your formula for yet undefined variables. It frees you from defining variables and is therefore a quite useful utility. Note: all descriptors (i.e. the "names" of the variables) must consist of exactly one character (x, y, a,...).
Miscinfo
... changes the flags in the "Preferences" window and adds new variables to the list.
Skeleton
...is a simple extension for your own projects. It does nothing except for not crashing...
ExpandCollect
...selects the commands EXPAND, SORT, COLLECT, SIMPLIFY for you.
Taylor
...is probably the most useful extension. It calculates a Taylor/McLaurin approximation of the current formula. Although it could stand improvement, it works quite well as it is.
First enter the formula. Then select the independent variable, choose EXTENSIONS / TAYLOR and press a number between 1 and 9 (the red border indicates that Symbolix is waiting for a key to be pressed). Afterwards Symbolix will do some difficult calculations that may take some time. The Taylor approximation will be stored in a new formula document called "Taylor". The new independent variable consists of a "T" followed by the name of the original variable. For MacLaurin approximations, just set the old independent variable to zero.
3D Animator 1.1
...is the most exciting extension. It calculates animated sequences of 3D graphs. This routine actually does two things at once: it varies a third independent variable which is linked to the time axis and it also allows you to change the view-point while the animation is shown.
First prepare the 3D module as if you were to draw a single graph: enter the display ranges and set the attributes. Before pressing PLOT or AUTOSCALING, select a third independent variable in the list and start the animator.
A modal dialog box appears. The first two controls are edit lines which determine from where to where the third parameter will run.
Then enter the number of frames you wish to calculate. Of course, the more frames you calculate the longer it will take. 20 to 30 frames are good values. Be careful when entering large numbers: each frame requires 32kB of free memory! So first look at the RAM CDev in the Control Panel NDA and divide the free RAM by 32 kB. The result is upper limit of frames Symbolix can store in memory.
After that go to the "phi" and "theta" edit lines. These two controls enable you to change the view-point during the playback. The values must be in degrees.
Finally set or clear the autoscaling check box. If you have specified a z interval in the 3D Control Window, unmark the box, otherwise check it.
Now press CALC. Symbolix calculates and draws the individual frames and stores them in memory. The bottom line shows information as to how far the program is. If you want to interrupt the animator, just click the mouse button and Symbolix won't calculate the remaining pictures.
When the text "status: done" appears (or you have pushed the mouse button), you can adjust the speed at which the whole sequences will be played in the "delay" edit line. 0 means single step mode, 1 means as fast as possible, 2 and all integers greater than 2 say that Symbolix will add pauses between two frames. When the check box "loop" is checked, Symbolix will play the sequence again and again until you press the mouse button.
Press PLAY to view the animation. Provided that you have activated the single step mode (delay=0), press any key to advance to the next frame.
!! The buttons CALC and DONE erase a previously calculated animation from memory.
You can even export sequences with the SAVE button. The animator then stores the frames in the $C1 (super-hires) format. The program automatically adds an index to the file's name (name.000, name.001 etc.).
11.4. More About Extensions
The main idea behind this new concept was that anybody can easily improve the overall functionality of Symbolix. We've included a complete description of the strucuture of extensions in the PROGRAMMERS folder on the program disk. Also, most of the extensions come with fully documented source code. We hope that we can encourage you to write your own utilities.
Please give Synergetix a buzz if you get stuck - we'll be happy to answer your technical questions and provide you with tips and suggestions.
We'll be glad to help you distribute your own programs at no charge (under the condition that they are freeware). Please contact Synergetix if you have further questions.
Chapter 12: Display Modes
Show the results in the most favourable lifht...
The Display Format
The display modes controls the format in which Symbolix displays numbers. Except for some special places mentioned below, the program stores numbers - regardless of the current mode - always as a mantissa (with up to 19 significant digits) with a signed exponent.
To select a display mode, open the window EDIT / DISPLAY FORMAT. The program is divided into four groups: each group can have its own display mode. Note: the settings in the last group, "other", affect the remaining part of Symbolix (everything except the stack, formula, and graph windows).
Symbolix supports three display formats: fixed point numbers (FIX), scientific format (SCI), and engineering mode (ENG). A fourth type, AUTO, automatically selects the most suitable type. Both main modes, FIX and SCI, display x digits to the right of the radix mark, where x is any integer between 0 and 12. In addition, the scientific mode adds an exponent to the number.
Since the fixed format can display only a very limited range, Symbolix automatically switches to scientific mode (or engineering format when [ENG] is checked) when a value is too large or too small.
ENG is a slight modification of the scientific format: the program displays one to three digits to the left of the radix mark and an exponent that is a multiple of three.
AUTO is a combination of the above modes: the program selects the appropriate mode for you. Also, Symbolix does not show trailing zeros when in auto mode. Hence AUTO shows numbers always in the most convenient format.
Examples
value places FIX SCI ENG AUTO
1 4 1.0000 1.0000eO 1.0000eO 1
0.5 3 0.500 5.000e-1 500.000e-3 0.5
12300 2 12300. 1.23e4 12.30e3 12300
1.7x10^7 0 1.e7 1.e7 17.e6 1.e7 or 17.e6
infinity 4 INF INF INF INF
0 1 0.0 0.0e0 0.0e0 0
Important
In most cases, the display format does not affect the precision of the mantissa, i.e. digits that have been truncated so that a number fits a certain display mode are not actually eliminated.
However, there are exceptions. As a rule, the precision of those numbers that can be edited (in edit lines or the formula window) depends on the selected format. Only those digits that are visible will be preserved, all others are truncated. In such situations, you can switch to an appropriate mode display format (preferably AUTO) before manipulating values in edit boxes.
Appendix A: The Menu Bar
Mathematics a la carte...
This chapter gives you a detailed description of all menu choices. Some of them have already been discussed in previous chapters, others haven't.
A.1. The Apple Menu
� About Symbolix...
Displays a short animation, some information about the people behind Symbolix, and the folks at Bright Software/Synergetix.
� Help
Activates the help mode. When in help mode, a colored HELP! blinks in the upper right hand corner of the screen, signaling that Symbolix is waiting for a selection. To learn more about a certain item, select, highlight, or push it (depending on the kind of the item). Symbolix will then open a window and explain its function.
After you have read the help text, either press OK to leave the help mode or NEXT to get help on another item.
� System... Displays information about the version you're using.
A.2. The File Menu (fig. 8)
� New
Opens a new, blank document. If you've already opened eight formulae, this menu item is disabled.
� Open Formula
Loads an equation file with all variables.
Symbolix installs a new document and retrieves the chosen formula. Once you've already opened eight formulae, this menu item is disabled.
� Open 2D Graph
Loads a 2D graph into memory. Since Symbolix stores such graphs in the widely used QuickDraw picture format, you can actually open any picture of this type.
The graph is displayed in the 2D Display Window.
� Open 3D Graph
Loads a 3D graph into memory. Symbolix uses an internal format which preserves important characteristics only.
Note that loading a Symbolix 3D graph document doesn't change the graph in the 3D Display Window! It only loads internal parameters and the graph's coordinates into memory. You have to open the 3D Control Window, set the attributes and then press [APPLY] to calculate the picture. Please refer to the chapter "The 3D Plot Module" for more information.
� Close
Closes the current formula and erases its 2D graph (if any).
Note: at least one document must be open. If there's only one formula in memory, this menu item is disabled.
� Save (As)
Saves the current equation and the list of variables (under a name you specify).
� Merge Variables
Merges the list of variables and the variables stored in an equation file. When there are two variables of the same name, the one from the selected file replaces the variable in the current list.
� Insert Formula
Inserts a formula at the insertion point. The list of variables isn't affected by this command.
� Delete
Deletes a file from disk.
� Page Setup
Lets you specify paper size and other specifications.
� Print Formula
Prints the current formula like a document in a word processor. By using the menu FORMULA / SET FONT you can print the formula in any font you like.
> The formula will appear exactly as it is displayed on the desktop. Thus, if the window is not as wide as the screen, the right margin on the printed page will be set accordingly.
� Quit
Saves the scrapbook and returns to the calling program (normally the Finder).
A.3. The Edit Menu
� Undo
Cancels the last symbolic operation.
� Cut, Copy, Paste
Moves the highlighted text from a text window (i.e. from the formula window, the formula table, or the scrapbook) to the clipboard or from the clipboard to the formula or to the scrapbook. These commands work only when the target window is the foremost window.
� Clear
Clears any highlighted text in the foremost window.
� Select All
Highlights the entire formula.
� Selection To Stack
Converts any highlighted text (from the formula, the scrapbook or the formula table) to a number and stores it in level 1 of the calculator's stack.
� Stack To Formula
Inserts the value in level 1 of the calculator at the insertion point in the formula window.
If the complex mode is turned on (refer to appendix B), Symbolix copies the whole complex number to the formula.
� Diplay Format
Opens the Display Format window. Please turn to chapter 12 for more information.
A.4. The Rules Menu (fig. 9)
� Derive (All)
Calculates the symbolic derivatives of the marked expressions (DERIVE) or of the entire formula (DERIVE ALL). The latter commands assumes that you have highlighted the independent variahlo in the list
� Expand
Rewrites a formula by expanding products and powers.
� Sort
Sorts a formula.
� Collect
Simplifies a formula by collecting like terms.
� Simplify
Simplifies a formula by replacing certain expressions with shorter, equivalent expressions.
� Eliminate ((..))
Deletes parentheses that are not needed.
� Trigonometric Conversions
Rewrites trigonometric expressions.
� Bell
Turns the bell on or off to signal the end of a symbolic operation. You can install your own bell with the Sound CDev in the Control Panel NDA (assign the sound wave to the event "Task Complete" ).
> Symbolix disables the bell when it runs under system 5.x.
� Sort Before Simplify
When this menu item is checked Symbolix will always execute SORT (see above) before simplifying a formula.
A.5. The Formula Menu (fig. 10)
� Check Syntax of Formula
Validates the formula. If there are errors in the formula. Check Syntax shows exactly where and why an error occurred.
� Evaluate
Opens a window with several options for evaluating the formula
� Solve
Shows a dialog box which lets you find roots of a simple expression or intersections of a formula.
� Integrate
Calculates integrals and areas of a formula.
� Plot 2D / 3D Graph of Formula
Plots the graph of one or two formulas with one (2D) or two (3D) independent variables.
� Add Variable
This option can be used to quickly define new variables in formulas. Highlight the variable's name in the formula (or in another text window) and select Add Variable (OA-+). A new variable will be created with a value of 0. If the same variable already exists, Add Variable initializes it to zero.
� Preferences
Shows the Preferences window.
� Set Font
Changes the font in the formula window. The system font is Shaston 8, but you can use any other font as well.
A.6. The Window Menu (fig. 11 )
� Forrnula
Shows or hides the formula window.
� Functions
Shows or hides the window of functions.
� Constants/Variables
Shows or hides the window of constants and variables.
� Stack
Shows or hides the calculator.
� Stack Commands
Shows the stack commands.
� 2D/ 3D Display Window
These windows display the graphs that have been calculated
� Elements
Shows the library of elements.
� Mathematics
Shows the math formula table.
� Geometry
Shows the geometry formula table.
� Scrapbook
Shows the notepad.
� Close Front Window
Closes the active window
Appendix B: Preferences
Everything
you always wanted to know
about these check boxes...
The Window "Preferences
The window "Preferences" (fig. 13) contains many switches and check boxes that let you customize Symbolix to fit your individual needs. Following is a list with explanations of all controls:
� Check Syntax Rewrites Formula
Cleared: FORMULA/CHECK SYNTAX parses the formula only, but does not rewrite it.
Checked: Symbolix scans for errors in the formula and rewrites the formula if none were found. How the formula is displayed depends on the check box discussed next.
� Insert spaces when rewriting a formula
Cleared: When an item from the RULES menu or FORMULA/CHECK SYNTAX is executed, Symbolix clears the formula window and writes the formula again without separating items with a space.
Checked: Symbolix separates two items by inserting a space between the two. The result is a much easier to read formula.
� Update screen during symbolic operations
Cleared: When you select an item from the RULES menu, Symbolix shows the final result only.
Checked: Symbolix shows intermediate steps. Note that a symbolic operation takes much longer when this flag is set.
� Show alert windows
Cleared: Symbolix doesn't show the alert windows which tell you that you are about to delete something.
Checked: Inform you that you are about to do something dangerous (such as delete a variable).
� Allow complex numbers
Cleared: Symbolix allows real values only.
Checked: the program accepts complex values and resizes the stack window to make room for a second, imaginary stack. Formulae are evaluated to complex results.
� Auto-route keystrokes
Cleared: keystrokes that can't be processed by a control in the active window are ignored.
Checked: Symbolix activates either the Formula or the Stack window (depending on the state of switch 3.1) and tries to handle unknown key commands in the formula or stack module.
� Don't clear edit lines in stack window
Cleared: ENTER pushes the values in the edit lines (fig. 2.1, 2.2) onto the stack and then clears them.
Checked: ENTER pushes the values in the edit lines onto the stack but does not change them.
� Evaluate constants to symbolic/numeric results
To symbolic results: constants are kept in the formula.
To numeric results: constants are converted to numbers.
Please refer to Chapter 10 ("Symbolic and Numeric Results") for further information
� Rounding direction
This group of radio buttons gives you full control over the standard rounding direction:
To-nearest: The value closest to the infinitely precise result
Upward: The value closest to and no less than the infinitely precise result
Downward: The value closest to and no greater than the infinitely precise result
Toward-zero: The value closest to and no greater than the infinitely precise result
!! Note that these buttons affect the SANE environment and thus Symbolix's internal calculations, too. It is therefore highly recommended that you don't change the default settings unless you know what you're doing.
� Halts enabled
These switches tell Symbolix under which error conditions an execution should be interrupted. If a certain halt is enabled, and such an error occurs, Symbolix prints the condition that caused the halt and then continues as if nothing has happened. Possible sources are:
invalid operation: occurs when an operand is invalid for the operation to be performed
underflow: occurs when a floating-point result is both tiny (i.e. its magnitude is smaller than 1.7E-4932) and, therefore, inexact
overflow: occurs when the result of an operation is larger than 1. lE4932
divide-by-zero: occurs when an operation on finite operands produces an exact infinite result
inexact: occurs when the rounded result is not identical to the mathematical result
!! Enabling halts on inexact results is not recommended because some internal conversions may cause this error even though nothing is wrong.
Assendix C: Scientific Functions
Everything
you always wanted to know
about these buttons...
Scientific Functions
Symbolix knows the following functions:
Button: Function Syntax
+ addition Y + X
- subtraction Y - X