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Text File | 1991-04-06 | 13KB | 413 lines

                             Tutorial for Al-Jabr V1.03

                                  by Uchenna Ogbuji

          This tutorial is very easy to use. Print it out as you will not be
     able to view it on screen while Al-Jabr is running. You simply enter the
     commands you are asked to and read the commentary on the results. The
     commands are denoted by a ">" sign at the beginning of the line. This
     symbolises the Al-Jabr command prompt and you should not include it in your
     commands. Any text that doesn't have a ">" sign at the beginning of the
     line is a comment and should not be entered at the command prompt. You
     don't have to try alll the examples and you can skip over any material you
     feel you have a good grasp of. At any point you can enter "quit" at the
     command prompt and you will be returned to DOS.
          Please read readme.doc first as it gives information on Al-Jabr's
     minimum requirements and which file you should run.
     ------------------------------------------------------------


          Let us start simple. Al-Jabr has 26 variables you can use, each
     denoted by a single letter. You can assign a value to a variable. The
     simplest assignment, of course, is just to assign a number:

     >v = 3.678

     remember that the above line starts with a ">" sign so you should enter "v
     = 3.678" at the ">" prompt in Al-Jabr. When you do, Al-Jabr will respond
     with "= 3.678". This tells you that the number 3.678 has been successfully
     stored in the variable v.

     >u = v

     now Al-Jabr still reports "= 3.678". Perhaps you have figured it out by
     now. The variable v contains the value 3.678 and will always contain that
     value until you assign it another value.

     >u = v + 6

     now the value in v, 3.678, is added to 6 and placed in u. Al-Jabr reports
     that 9.678 is assigned to u.

     >f = v * 8.6e-3

     here v (still 3.678) is multiplied by 8.6e-3 (which is the computer
     exponential form for 8.6*10^-3 or .0086) and the result (.0316308) is
     placed in f. Al-Jabr has two special stored constants:

     >d = :p

     you will find 3.14159265 has been assigned to d. Some of you will recognise
     this number. It is the ratio pi. :p is a short form of this number for Al-
     Jabr. :e is the short form of e (2.718281828), the base of natural
     logarithms, also known as Naperian logarithms.

     >f = :e ^ 2  +  d

     will place 10.5306488 in f. There is one more stored constant you will
     encounter soon. Let us now explore the functions supported by Al-Jabr:

     >y = sin t

     should return 0 because since t hasn't been assigned a value yet, it is
     assumed to be 0, and the sine of 0 is 0.

     >g = 4! + abs (-10)

     should return 34. First the factorial of 4 (4*3*2*1) is computed and then
     added to the magnitude of -10 (10).

     >b = 2 * :p
        = 6.28318531

     >c = sin (cos b) + atan (log 54)

     should return 1.88875426, the sine of the cosine of 2 times pi plus the
     arctangent of the base ten logarithm of 54.

     >r = sqrt 16

     will return 4, the square root of 16. However if you execute

     >v = -16
        = -16

     >r = sqrt v

     you will encounter a domain error because Al-Jabr cannot return the square
     root of a negative number. For a detailed list of the functions supported
     by Al-Jabr, see part I of the manual.

          Al-Jabr can compute the sum or product of sequences. I will show you a
     few examples so you might figure out what is going on. To sum up all the
     integers from 1 to 10, you would enter:

     >s = [n=1,10,1,n]
        = 55

     first the sum is initialised to 0, then n is set to 1 and this is added to
     the sum. n is then increased by 1 and this is added to the sum. This keeps
     on going until n becomes greater than 10 which is the stop value. Then the
     loop ends and the sum is assigned to s. To compute the sum of the squares
     of the integers from 1 to 10 you enter:

     >c = [m=1,10,1,m^2]
        = 385

     To sum up the square roots of all the integers from 10 to 100 and then add
     50 to the sum enter:

     >h = [j=10,100,1,sqrt j] + 50
        = 702.156947

     Here sum is initialised to 0, j is set to 10 and then the square root is
     found and added to the sum. j is then increased by 1 and the process is
     repeated until j is greater than 100. Then the sum is added to 50 and
     assigned to h. To sum up all the odd numbers from 11 to 25 you could enter:
     >v = [l=11,25,2,l]
        = 144

     here l is increased by 2 at a time rather than 1 thus the above effectively
     computes 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25. Note that the start value,
     the end value and the increment do not have to be integers:

     >s = [f=1.5, 3.5, 0.5, log f]
        = 1.89625056

     adds log (1.5) + log (2.0) + log (2.5) + log (3.0) + log (3.5) and returns
     the sum to s. Finding the product of the sequence is similar except that
     you use curly brackets {} instead of square brackets. For instance

     >p = {f=1.5,3.5,0.5, log f}
        = 0.00547579472

     finds log (1.5) * log (2.0) * log (2.5) * log (3.0) * log (3.5) and assigns
     the product to p. See the manual for a more detailed coverage of the syntax
     and operation of Al-Jabr's sum and product of sequence functions. Now try
     something interesting. Enter

     >show all

     You should see a display of all your variables and their contents. This is
     useful when you need to know what the variables contain. You don't have to
     display them all to check the value of one, though, if you want to see just
     what's in p for instance, you would enter

     >show p
     p = 0.00547579472

     Now try this:

     >w = 45 * 2
        = 90

     >s = :e
        = 2.718281828

     note that each variable is automatically updated in the display as you
     alter it.

          Try the following sequence of commands:

     >x = 1
        = 1
     >y = cos (3*x/2) + sin (x/2)
        = 0.55016274
     >store
     cos (3*x/2) + sin (x/2) has been stored as expression #1
     >f = #1
        = 0.55016274

          You will find that f has been assigned the same thing as y. Now try:

     >x = :p
        = 3.14159265
     >z = #1
        = 1

     Let me explain the above. the store command stores the last expression that
     you worked with in memory. It calls the first expression it stores #1, the
     second, #2 etc. up to #9. Thus you can store up to 9 expressions in memory
     at a time. In the above sequence I stored "cos (3*x/2) + sin (x/2)" as
     expression #1. At the time x contained 1 so #1 was evaluated as "cos (3/2)
     + sin (1/2)" and f was assigned 0.5501627, the result. But when I changed x
     to pi, the value of #1 changed as well. You can incorporate stored
     expressions in to other expressions too. Eg.

     >q = #1 ^ 2 + #1 + 3.0
        = 5

     There is a short-cut for storing expressions:

     >store b = 6 * :e - ln 5
        = 14.7002531

     does exactly the same thing as

     >b = 6 * :e - ln 5

     >store

     You can display stored expressions just as you can stored variables:

     >show #1

     displays expression #1, while

     >show #all

     displays all the currently stored expressions. Here's a neat trick: try

     >store z = k^2
        = 0
     k^2 has been stored as expression #3.

     >y = {k=2,11,3,#3}
        = 774400

     does exactly the same thing as

     >y = {k=2,11,3,k^2}
        = 774400

     that is it finds 2^2 * 5^2 *8^2 * 11^2.

          If you have a printer connected enter

     >print

     as you can see the "print" command sends all the variables and stored
     expressions to the printer.

          Now we finally get to Al-Jabr's graphical capabilities. Let us say we
     want to plot the graph of y=4*x^2 + 2*x + 1 between x=-5 and x=5, and
     between y=-20 and y=20, we could enter

     >plot x;4*x^2 + 2*x + 1;-5,5,-100,100

          If we wanted to plot the graph of d=v*cos v  between x=-3 and x=5 and
     we didn't know the maximum and minimum values of d in this interval, yet we
     wanted to show the entire function, we could enter:

     >plot v;v*cos v;-3,5

     since we didn't specify the range of the vertical axis Al-Jabr
     automatically calculates it so that the entire function is shown. Note that
     Al-Jabr cannot always compute the maximum and minimum vertical axis values.
     For example if you enter

     >plot n;log -n;-2,2

     Al-Jabr will respond with an error when n becomes 0 and return you to the
     command prompt. This is because log 0 is an infinite quantity and Al-Jabr
     cannot display infinite quantities! To view the graph of this function you
     will have to specify the vertical axis range yourself. For example you
     might decide to see the portions of the graph that fall between -6 and 6 on
     the vertical axis:

     >plot n;log -n;-2,2,-6,6

          Note that each time you plotted a new graph the viewing area is
     cleared. You can plot a new graph uopn an existing one but you must use the
     "over" command. For instance, to show the graph of r=tan (t^2) over the
     last graph you plotted enter

     >over t;tan (t^2)

     and slect any color you would like for the new graph. Note that there are
     no axis specifications. Over automatically displays the graph in the same
     region as the last one and you shouldn't specify horizontal or vertical
     axis ranges. There are 4 colors available to help you differentiate between
     graphs. You can keep displaying graphs over and over each other in any
     combination of white, grey, cyan and yellow ad infinitum.

          Al-Jabr can calculate a polynomial of best statistical fit to a set of
     data and plot the graph of the polynomial. This is accomplished with the
     "cfit" command. There is a sample file called infile.dat that should have
     come with Al-Jabr. If it is in the current directory enter

     >cfit infile.dat

     You can view the contents of infile.dat at the MSDOS prompt by entering
     "type infile.dat". As you will see, infile.dat just consists of several
     sets of numbers. These numbers constitute the data that Al-Jabr attempts to
     fit a curve to. These data points are shown as small green circles on the
     graph. See the manual (part II, subheading "CFIT:") for a description of
     the syntax of the data files.
          One of Al-Jabr's most useful abilities is its ability to capture its
     graphical displays to a .PCX format file. If you have a graph on the screen
     right not, try capturing it to a file called sample.pcx with the command

     >capture sample.pcx

     You can later manipulate sample.pcx as any PCX format file and incorporate
     it into other programs.

          Al-Jabr's commands are more fully described and explained in the
     manual, part II.

          Now we shall explore some common error conditions and what might cause
     them.

     >hello
     Error 00: Unrecognized command.

     "hello" is not one of the commands that Al-Jabr understands.

     >plot 3*d;0,10
     Error 01: Syntax error in command.

     The independent variable wasn't specified and thus Al-Jabr couldn't plot
     the graph you requested.

     >plot d;3*d;0,10

     should work. Some errors have mathematical causes:

     >store y = acos -4
     Error 03: The argument to acos is out of its domain.

     the argument to the arccosine function must be between -1 and +1.

     >x = 0
        = 0
     >z = 5 + 6/x
     Error 06: Division by zero.

     Division by zero yields an infinite quantity, which Al-Jabr cannot handle.

     >h = 3.45e -3
     Error 07: Syntax error in expression.

     No spaces are permitted whithin a number. Try

     >h = 3.45e-3

     Some errors are more subtle:

     >plot x;3 - 2*x^2;3,-5
     Error 11: Invalid horizontal axis range.

     the minimum horizontal axis value (3) cannot be greater than the maximum (-
     5)!

     >plot x;3 - 2*x^2;-5,3

     should work.

          Well by now you should have obtained a working knowledge of the
     functioning of Al-Jabr. You might want to experiment some on your own. If
     not you can enter

     >quit

     at the command prompt to exit to dos. For a more detailed guide to using
     Al-Jabr, I'd advise you to read the manual. If you encounter a problem and
     need to contact me, read the readme.doc file. I hope you enjoy using Al-
     Jabr. I welcome any constructive suggestions and criticisms of the program,
     manual, or this tutorial.