Matrices in True Basic

By Dave Kelly, MacTutor Editorial Board

Using Cramer’s Rule and True Basic to Solve Simultaneous Equations

It is only a matter of time until each Engineering or Math student comes across the need to solve simultaneous equations. Fortunately, we live in a time when someone has already figured out the solutions and we can readily use the methods which have been discovered. In this column I hope to show how True Basic’s MAT statements can be used with Cramer’s Rule to solve simultaneous equations.

In most areas of engineering, the time comes from time to time to solve equations. Many of these solutions can be found using a pencil and paper and a little bit of logical thinking. We learn from algebra that to solve a set of equations, there needs to be one equation generated for each unknown variable. It is the responsibility of the engineer to use his expertise in his field of study to determine what the equations are. The solution of the equations are sometimes trivial, but may take time to manipulate the equations with simple algebra. Eventually the answer is found.

Computers are supposed to make life easier. Right? Well, of course! So let’s see how we can solve some equations using True Basic. First, we need a problem to solve: We want to find the current in each of three paths of a circuit. The circuit is shown in figure 1.

To determine the equations which we will use to solve the problem we will use Ohm’s law and Kirchhoff’s voltage law. For the non-engineer, Ohm’s law is a mathematical way to express the relationship between voltage, current, and resistance:

E = I * R

Figure 1.

where R is resistance, E is the voltage across the resistance, I is the current through the resistance. This simple law is basic to a beginning electronics class. Ohm’s law is named after a German physicist, George Simon Ohm, who published a pamphlet in 1827 which contained the results of his efforts to describe and relate currents and voltages mathematically. It has since been shown, however, that this result was discovered 46 years earlier in England by Henry Cavendish.

Kirchhoff’s voltage law is named after Gustav Robert Kirchhoff, a German university professor who was born about the time Ohm was doing his experimental work. Kirchhoff’s law states that the algebraic sum of the voltages around any closed path in a circuit is zero. I won’t attempt to prove this as there are many electronics books which explain how Kirchhoff’s voltage law works.

Figure 2.

Our first equation is derived by taking the loop shown in Figure 2. The path of the loop does is not important as long as the path is closed. The equation is:

-7 + 1(i1 - i2) + 3 (i3 - i2) + 1i3 = 0

which equates to:

i1 - 4i2 + 4i3 = 7.

Our second equation is derived by taking the loop where current i2 is as shown in Figure 1. The equation is:

1(i2 - i1 ) + 2 i2  + 3(i2 - i3) = 0

which equates to:

-i1 - 6i2 - 3i3 = 0.

The last equation comes from the relationship of the current source and the currents i1 and i2:

i1 - i3  =  7.

Now we have three equations and three unknown values to solve for. An easy method to use is given by Cramer’s Rule. Gabriel Cramer (1704-1752), was a Swiss mathematician, also known by his book on the theory of curves, which appeared in 1750 in Geneva. Cramer’s Rule states that if we have a system of linear equations

a11x1 + a12x2 + + a1nxn = b1,
a21x1 + a22x2 + + a2nxn = b2,
            ,
an1x1 + an2x2 + + annxn = bn

with n equations and n unknowns., the n X n coefficient matrix of the system is denoted by A. The determinant A 0, the components of the solution of the system are given by the formula,

where the matrix Bk is the same as A except that the elements aik, 1 ¾ i ¾ n, in the kth column of A have been replace by the terms bi, 1 ¾ i ¾ n, respectively.

Cramer’s Rule says that n simultaneous equations with n unknowns can be determined by taking determinants of the matrix. Each of the unknowns is calculated as follows:

and

and

and

where

The great thing about this is that with True Basic the solution is easy to calculate because of the MAT statements included in True Basic. MAT statements include MAT INPUT (for inputing an array), MAT LINE INPUT, MAT PLOT (plotting to a True Basic chart), MAT PRINT (to print a matrix, MAT READ (read an array from data statements), MAT REDIM (to reset dimensions of a matrix), and MAT WRITE (to write an array to a file). The MAT statement is an enhancement to Basic that is very welcome in the scientific/engineering community. It has been included for many years in Hewlett Packard Basic (now version 5.0).

Using Cramer’s Rule by hand the evaluation of the determinant for i3 is:

evaluation of the other determinants gives i1= 9 Amps and i2= 2.5 Amps. The evaluation of these determinants are fairly easy by hand, but with more equations and more unknowns, the it gets to be a lot of number crunching. The program at the end will do the number crunching for an n X n matrix, limited by the amount of memory you have. The number of equations is calculated in a function which uses the factorial function included in the True Basic Libraries to figure how much data is available in the DATA statements. The coefficients are entered into the DATA statements with equation coefficients in the first DATA statement and the solution coefficients in the second DATA statement. For solving for a large number of equations/unknowns you may break up the DATA into multiple lines.

Once the number of equations is determined, the program calls the sim_solve subroutine to solve the equation. Since the MAT statement does all the work, the subroutine is short and after a couple of MAT assignment statements, the matrix is ready to be crunched. True Basic also supplies us with the det function which will automatically calculate the determinant of an n X n matrix. It sure beats keeping track of a bunch of numbers by hand like I did when I went to school. Calculations are now done and our circuit has been analyzed! Now if they would only put a Mac on my desk I could use it!

{1}
! Simultaneous Equation Solver
! Dave Kelly
! ©1988 MacTutor

LIBRARY “Fnmlib”
DECLARE DEF Factrl

DEF Get_Equation_Count            !Read the number of equations
    WHEN ERROR IN
         LET Count=0
         DO
            READ value
            LET Count=Count+1
         LOOP
    USE
         RESTORE
    END WHEN
    LET n=0
    DO
       LET n=n+1
    LOOP UNTIL Factrl(n)>Count
    LET Count=n-1
    LET Get_Equation_Count=Count
END DEF

! Start Main Program
LET number_of_equations=Get_Equation_Count
DIM a(2,2),b(2,2),c(2,1),x(2)
! Set up all the equations
MAT READ a(number_of_equations,number_of_equations)
MAT READ c(number_of_equations,1)
MAT REDIM x(number_of_equations)
CALL sim_solve(a,b,c,x,number_of_equations)
MAT PRINT x                       ! Print the solutions
DATA 1,-4,4,-1,6,-3,1,0,-1        ! equation coefficients
DATA 7,0,7                        ! equation data
END

SUB sim_solve(a(,),b(,),c(,),x(),number_of_equations)
    IF det(a)=0 THEN EXIT SUB
    FOR j=1 to number_of_equations
        MAT b=a
        FOR i= 1 to number_of_equations
            LET b(i,j)=c(i,1)
        NEXT i
        LET x(j)=det(b)/det(a)
    NEXT j
END SUB

Someone told Dave Kelly that he was a programmer. They were kind, but not very accurate. His article “Hierarchical Menus & Colors Notes” [VOL. 4 NO. 6] contained more errors than accuracies. He claims that ZBasic source code must be compiled and the resources attached to make use of them. Why not use the function: “RefNum% = FN OPENRESFILE(FileName$)”?

This makes resources available during the programming phase and requires the deletion of only one command before creating the application.

The next glaring error was his slap-dash program that should have shown how to use submenus from ZBasic.[Excuse me, but I showed A way to to submenus from ZBasic, not THE only way. Actually, I like your way better now that you have shown it to us. Thanks -DK] As the following program demonstrates, submenus do not require special event handling loops, nor is it necessary to append special resources.

{2}
‘**********************************************
‘                    SubMenus from ZBasic
‘**********************************************
WINDOW OFF
COORDINATE WINDOW
WINDOW 1
TEXT 0,12,0,0
PRINT@(1,1)”MENU#”
PRINT@(1,3)”Item#”
‘----------------------Set The Menu------------------
MENU 1,0,1,”File”:MENU 1,1,1,”Quit/Q”
‘
SubMenu=150 ‘This is the number we’ll use for our SubMenu
MENU SubMenu,0,1,”I’m outa here”  ‘We’ll delete this from the menu bar 
as soon as we’ve attached it to a menu item.
MENU SubMenu,1,1,”Plain/P;Bold/B<B;Italic/I<I;Outline/O<O;Shadow/S<S”
SubMenuHndl&=FN GETMHANDLE(SubMenu)  ‘Handle to the SubMenu
‘
‘now delete the name from the menu bar
‘CALL DELETEMENU(SubMenu)  ‘(the menu itself still remains available)
‘
MENU 2,0,1,”Format”‘We’ll add the submenu here
‘
‘Create a menu item with the command key equivalent of
‘CHR$(27) and mark the item with the number of the
‘SubMenu instead of a “2” for a check mark
‘
MenuName$=”Style/”+CHR$(27)
MENU 2,1,SubMenu,MenuName$
‘
‘Use a negative insertion number(-1) and the handle to
‘our SubMenu - the “CALL INSERTMENU(SubMenuHandle,-1)”
‘should take place immediately after the root menu is created
‘
CALL INSERTMENU(SubMenuHndl&,-1)
MENU 2,2,1,”Itz Eazy With Z”  ‘Add more items and menus if you like
CALL DRAWMENUBAR ‘Redraw the bar to exclude the hidden menu
‘------------------------Events--------------------------------
ON MENU GOSUB “Handle Menu”
MENU ON ‘We’ll just track menu events
“Loop”  ‘Loop and wait
GOTO “Loop” ‘Getting dizzy?
MENU OFF
“Handle Menu”’----------------Menu Handling------------------
MenuID=MENU(0)
ItemID=MENU(1)
MENU  ‘Get results of the menu action
IF MenuID=1 AND ItemID=1 THEN END  ‘End if user selected “Quit”
PRINT@(9,1)MenuID”       “
PRINT@(9,3)ItemID”       “ ‘Else-Show menu info
RETURN  ‘Back to the “Loop”

Wake up, Dave. And try to spend a little time developing your programming skills and a little less time throwing rocks.

Regards

I’m a new subscriber to MacTutor and have particularly enjoyed your column “Basic School”. I’m a BASIC programmer and frankly don’t have the time to really learn the other high level languages such as C or Pascal. It is, therefore, heartening to find a source of instruction for BASIC on the MAC.

Do you think it would be feasible to compile your articles into a separate volume? I believe such a book would sell quite well. There are many recreational and small application programmers who both love BASIC and the MAC. Keep up the great work.

Sincerely,