TweetFollow Us on Twitter

Matrices
Volume Number:4
Issue Number:9
Column Tag:Basic School

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

Chris Stasny

Well, Chris it seems you have a little chip on your shoulder about something (Do you work for Zedcor?). In my defense, there are many “programmers” out there that have had trouble getting ZBasic event processing to do everything they want it to do. It is true that some of the problems have been fixed as the users have been debugging ZBasic ever since it was released! In my opinion, there is no Basic available today for the Macintosh which is satisfactory for doing serious software development as there are glaring holes in their capabilities. I’ve had too many phone calls from disgruntled developers trying to find a solution to their problems with no solution in sight. Call it rock throwing if you like, but face it, there isn’t any such thing as LightSpeed Basic. Thank you for your letter. We encourage others to share the technology by writing for MacTutor. We are dedicated to the distribution of useful programming information without regard to race, creed or developer status. Chris, we never claim to know everything. We could use more of your good ideas. Write to MacTutor and ask for our authors kit and share some of your “expertise”.

Dave Kelly

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,

Julian Wan

Thank you for your kind remarks. I’ve had several people ask for a separate volume of “Basic School”. Because of the cost of publishing another compiled book and since we already offer the “Best of MacTutor” Vol 1 and 2 for a reasonable price, “Basic School” will not become a volume of its own. Have heart though, I’ve used information from C, Pascal, and assembly language columns many times and so having the other language available in “Best of MacTutor” can actually help when writing BASIC programs too. I’ve also compiled a MacTutor Index HyperCard Stack which can help when you are trying to learn about a specific subject.

Dave Kelly

 

Community Search:
MacTech Search:

Software Updates via MacUpdate

Air Video Server HD 2.1.0 - Stream video...
Air Video Server HD streams videos instantly from your computer on your iPhone, iPad, iPod touch or Apple TV. No need to worry about converting or transferring files. We took everything that was... Read more
Duplicate Annihilator 5.7.5 - Find and d...
Duplicate Annihilator takes on the time-consuming task of comparing the images in your iPhoto library using effective algorithms to make sure that no duplicate escapes. Duplicate Annihilator... Read more
BusyContacts 1.0.2 - Fast, efficient con...
BusyContacts is a contact manager for OS X that makes creating, finding, and managing contacts faster and more efficient. It brings to contact management the same power, flexibility, and sharing... Read more
Capture One Pro 8.2.0.82 - RAW workflow...
Capture One Pro 8 is a professional RAW converter offering you ultimate image quality with accurate colors and incredible detail from more than 300 high-end cameras -- straight out of the box. It... Read more
Backblaze 4.0.0.872 - Online backup serv...
Backblaze is an online backup service designed from the ground-up for the Mac.With unlimited storage available for $5 per month, as well as a free 15-day trial, peace of mind is within reach with... Read more
Little Snitch 3.5.2 - Alerts you about o...
Little Snitch gives you control over your private outgoing data. Track background activity As soon as your computer connects to the Internet, applications often have permission to send any... Read more
Monolingual 1.6.4 - Remove unwanted OS X...
Monolingual is a program for removing unnecesary language resources from OS X, in order to reclaim several hundred megabytes of disk space. If you use your computer in only one (human) language, you... Read more
CleanApp 5.0 - Application deinstaller a...
CleanApp is an application deinstaller and archiver.... Your hard drive gets fuller day by day, but do you know why? CleanApp 5 provides you with insights how to reclaim disk space. There are... Read more
Fantastical 2.0 - Create calendar events...
Fantastical is the Mac calendar you'll actually enjoy using. Creating an event with Fantastical is quick, easy, and fun: Open Fantastical with a single click or keystroke Type in your event details... Read more
Cocktail 8.2 - General maintenance and o...
Cocktail is a general purpose utility for OS X that lets you clean, repair and optimize your Mac. It is a powerful digital toolset that helps hundreds of thousands of Mac users around the world get... Read more

Pie In The Sky: A Pizza Odyssey (Games)
Pie In The Sky: A Pizza Odyssey 1.0 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0 (iTunes) Description: A game about delivering pizza. In space. | Read more »
Android's Popular OfficeSuite Now A...
Once only available for Android devices, OfficeSuite has finally landed on the app store. The Mobile Systems app lets you view, edit, create, and share Word, Excel, and PowerPoint documents as well as convert them to/from PDFs. It's touted as being... | Read more »
Fast & Furious: Legacy's Creati...
| Read more »
N-Fusion and 505's Ember is Totally...
| Read more »
These are All the Apple Watch Apps and G...
The Apple Watch is less than a month from hitting store shelves, and once you get your hands on it you're probably going to want some apps and games to install. Fear not! We've compiled a list of all the Apple Watch apps and games we've been able to... | Read more »
Appy to Have Known You - Lee Hamlet Look...
Being at 148Apps these past 2 years has been an awesome experience that has taught me a great deal, and working with such a great team has been a privilege. Thank you to Rob Rich, and to both Rob LeFebvre and Jeff Scott before him, for helping me... | Read more »
Hands-On With Allstar Heroes - A Promisi...
Let’s get this out of the way quickly. Allstar Heroes looks a lot like a certain other recent action RPG release, but it turns out that while it’s not yet available here, Allstar Heroes has been around for much longer than that other title. Now that... | Read more »
Macho Man and Steve Austin Join the Rank...
WWE Immortals, by Warner Bros. Interactive Entertainment and WWE, has gotten a superstar update. You'll now have access to Macho Man Randy Savage and Steve Austin. Both characters have two different versions: Macho Man Randy Savage Renegade or Macho... | Read more »
Fearless Fantasy is Fantastic for the iF...
I actually had my first look at Fearless Fantasy last year at E3, but it was on a PC so there wasn't much for me to talk about. But now that I've been able to play with a pre-release version of the iOS build, there's quite a bit for me to talk... | Read more »
MLB Manager 2015 (Games)
MLB Manager 2015 5.0.14 Device: iOS Universal Category: Games Price: $4.99, Version: 5.0.14 (iTunes) Description: Guide your favorite MLB franchise to glory! MLB Manager 2015, officially licensed by MLB.com and based on the award-... | Read more »

Price Scanner via MacPrices.net

Apple offering refurbished 27-inch 5K iMacs f...
The Apple Store is offering Apple Certified Refurbished 27″ 3.5GHz 5K iMacs for $2119 including free shipping. Their price is $380 off the price of new models, and it’s the lowest price available for... Read more
16GB iPad mini on sale for $199, save $50
Walmart has 16GB iPad minis (1st generation) available for $199.99 on their online store, including free shipping. Their price is $50 off MSRP. Online orders only. Read more
New 128GB MacBook Airs on sale for $50 off MS...
 B&H Photo has 128GB 11″ and 13″ 2015 MacBook Airs on sale today for $50 off MSRP including free shipping plus NY sales tax only: - 11″ 1.6GHz/128GB MacBook Air (Model #MJVM2LL/A): $849 $50 off... Read more
13-inch 2.6GHz Retina MacBook Pro (refurbishe...
The Apple Store has Apple Certified Refurbished 13″ 2.6GHz/128GB Retina MacBook Pros available for $979 including free shipping. Original MSRP for this model was $1299. Read more
Save up to $600 with Apple refurbished Mac Pr...
The Apple Store is offering Apple Certified Refurbished Mac Pros for up to $600 off the cost of new models. An Apple one-year warranty is included with each Mac Pro, and shipping is free. The... Read more
Samsung Galaxy S 6 and Galaxy S 6 edge U.S. P...
Samsung Electronics America, Inc. has announced the Galaxy S 6 and Galaxy S 6 edge will be available in the U.S. beginning April 10, with pre-orders being accepted now. “We have completely reimagined... Read more
13-inch 2.5GHz MacBook Pro (refurbished) avai...
The Apple Store has Apple Certified Refurbished 13″ 2.5GHz MacBook Pros available for $829, or $270 off the cost of new models. Apple’s one-year warranty is standard, and shipping is free: - 13″ 2.... Read more
Save up to $80 on iPad Air 2s, NY tax only, f...
 B&H Photo has iPad Air 2s on sale for $80 off MSRP including free shipping plus NY sales tax only: - 16GB iPad Air 2 WiFi: $469.99 $30 off - 64GB iPad Air 2 WiFi: $549.99 $50 off - 128GB iPad... Read more
iMacs on sale for up to $205 off MSRP
B&H Photo has 21″ and 27″ iMacs on sale for up to $205 off MSRP including free shipping plus NY sales tax only: - 21″ 1.4GHz iMac: $1019 $80 off - 21″ 2.7GHz iMac: $1189 $110 off - 21″ 2.9GHz... Read more
Färbe Technik Offers iPhone Battery Charge LI...
Färbe Technik, which manufactures and markets of mobile accessories for Apple, Blackberry and Samsung mobile devices, is offering tips on how to keep your iPhone charged while in the field: •... Read more

Jobs Board

DevOps Software Engineer - *Apple* Pay, iOS...
**Job Summary** Imagine what you could do here. At Apple , great ideas have a way of becoming great products, services, and customer experiences very quickly. Bring Read more
*Apple* Retail - Multiple Positions (US) - A...
Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, you're also the Read more
Sr. Technical Services Consultant, *Apple*...
**Job Summary** Apple Professional Services (APS) has an opening for a senior technical position that contributes to Apple 's efforts for strategic and transactional Read more
Lead *Apple* Solutions Consultant - Retail...
**Job Summary** Job Summary The Lead ASC is an Apple employee who serves as the Apple business manager and influencer in a hyper-business critical Reseller's store Read more
*Apple* Pay - Site Reliability Engineer - Ap...
**Job Summary** Imagine what you could do here. At Apple , great ideas have a way of becoming great products, services, and customer experiences very quickly. Bring Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.