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

iFFmpeg 6.4.0 - Convert multimedia files...
iFFmpeg is a comprehensive media tool to convert movie, audio and media files between formats. The FFmpeg command line instructions can be very hard to master/understand, so iFFmpeg does all the hard... Read more
Thunderbird 52.2.0 - Email client from M...
As of July 2012, Thunderbird has transitioned to a new governance model, with new features being developed by the broader free software and open source community, and security fixes and improvements... Read more
Airfoil 5.6.1 - Send audio from any app...
Airfoil allows you to send any audio to AirPort Express units, Apple TVs, and even other Macs and PCs, all in sync! It's your audio - everywhere. With Airfoil you can take audio from any... Read more
Apple Remote Desktop Client 3.9.3 - Clie...
Apple Remote Desktop Client is the best way to manage the Mac computers on your network. Distribute software, provide real-time online help to end users, create detailed software and hardware reports... Read more
jAlbum Pro 14.0 - Organize your digital...
jAlbum Pro has all the features you love in jAlbum, but comes with a commercial license. You can create gorgeous custom photo galleries for the Web without writing a line of code! Beginner-friendly... Read more
jAlbum 14.0 - Create custom photo galler...
With jAlbum, you can create gorgeous custom photo galleries for the Web without writing a line of code! Beginner-friendly, with pro results - Simply drag and drop photos into groups, choose a design... Read more
Vitamin-R 2.48 - Personal productivity t...
Vitamin-R creates the optimal conditions for your brain to work at its best by structuring your work into short bursts of distraction-free, highly focused activity alternating with opportunities for... Read more
Beamer 3.3 - Stream any movie file from...
Beamer streams to your Apple TV.... Plays any movie file - Just like the popular desktop movie players, Beamer accepts all common formats, codecs and resolutions. AVI, MKV, MOV, MP4, WMV, FLV. To... Read more
Shredo 1.2.2 - $9.99
Shredo is a beautiful, functional file-shredding and privacy scan utility. It permanently shreds files, folders, and external volumes' contents to keep information secure and impossible for anyone to... Read more
Final Draft 10.0.3 - Industry-leading sc...
Final Draft allows you to use your creative energy to focus on the content; let Final Draft take care of the style. Final Draft is the number-one selling application specifically designed for writing... Read more

Latest Forum Discussions

See All

The House of da Vinci (Games)
The House of da Vinci 1.0.0 Device: iOS Universal Category: Games Price: $4.99, Version: 1.0.0 (iTunes) Description: Enter The House of Da Vinci, a new must-try 3D puzzle adventure game. Solve mechanical puzzles, discover hidden... | Read more »
Solve the disappearance of history’s gre...
Blue Brain Games invites you to indulge in an immersive hands-on 3D puzzle adventure in similar vein to The Room series, with its debut release The House of Da Vinci. Set during the historic period of the Italian Renaissance (when Leonardo himself... | Read more »
Age of Rivals (Games)
Age of Rivals 3.3 Device: iOS Universal Category: Games Price: $.99, Version: 3.3 (iTunes) Description: Deep civilization-building strategy in a fast-paced card game! | Read more »
Panthera Frontier (Games)
Panthera Frontier 1.0 Device: iOS Universal Category: Games Price: $4.99, Version: 1.0 (iTunes) Description: | Read more »
Angry Birds Evolution beginner's gu...
Angry Birds changes things up a fair bit in its latest iteration, Angry Birds Evolution. The familiar sling-shot physics mechanics are still there, but the game now features team-based gameplay, RPG elements, and a new top-down view. With all of... | Read more »
Sega Forever is for the retro game fans
Sega is launching a new retro games service titled Sega Forever, in a move that's sure to delight games enthusiasts with a bit of nostalgia. Sega's releasing five classic games for free. The titles include Sonic the Hedgehog, Phantasy Star II,... | Read more »
The Little Acre (Games)
The Little Acre 1.0 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0 (iTunes) Description: | Read more »
Lovely adventure game 'The Little A...
The Little Acre makes its way from PC to iOS today and it's a real charmer. It's a lovely little adventure game set in 1950's Ireland, gorgeously animated in a style slightly reminiscent of old Disney films. The game won a lot of praise for its... | Read more »
Futurama: Worlds of Tomorrow officially...
Futurama: Worlds of Tomorrow arrives next Thursday, and it features . . . Stephen Hawking? Turns out a few of your favorite science super stars, including Bill Nye, Neil DeGrasse Tyson, and George Takei will be making an appearance alongside the... | Read more »
MU Origin’s new update offers exciting c...
MU Origin is kicking off the summer with style. Its latest update is a real doozy, offering up plenty of fresh content. New challenges, some extra help for newcomers, and new costumes are just a few of the things you will find in the new patch. | Read more »

Price Scanner via MacPrices.net

Clearance 2016 MacBook Pros available for up...
B&H Photo has clearance 2016 13″ and 15″ MacBook Pros in stock today and on sale for up to $400 off original MSRP. Shipping is free, and B&H charges NY & NJ sales tax only: - 15″ 2.7GHz... Read more
Apple Ranks 9th In comScore Top 50 U.S. Digit...
comScore, Inc. has released its monthly ranking of U.S. online activity at the top digital media properties for May 2017 based on data from comScore Media Metrix Multi-Platform. * Entity has... Read more
10.5-inch iPad Pros available for up to $20 o...
B&H Photo has the new 2017 10.5″ iPad Pros available for up to $20 off MSRP including free shipping plus NY & NJ sales tax only: - 64GB iPad Pro WiFi: $649 - 256GB iPad Pro WiFi: $749 - 512GB... Read more
Three Off-The-Beaten-Track iOS Apps That Dese...
One of the great things about using iPads and iPhones is the vast selection of apps available for most anything you want or need to do. The three outlined in this article have been in my core app... Read more
Apple No. 1 Spot In Gartner Top 100 Vendors i...
Gartner, Inc. has unveiled the top global 100 vendors in IT in 2016 based on their revenue across IT (excluding communication services) and component market segments. In the Gartner Global Top 100:... Read more
Clearance iMacs available for up to $300 off...
B&H Photo has clearance 21″ and 27″ Apple iMacs available starting at $949, each including free shipping plus NY & NJ sales tax only: - 27″ 3.3GHz iMac 5K: $1999 $300 off original MSRP - 27″... Read more
1.4GHz Mac mini, refurbished, available for $...
The Apple Store has Apple Certified Refurbished 1.4GHz Mac minis available for $419. Apple’s one-year warranty is included, and shipping is free. Their price is $80 off MSRP, and it’s the lowest... Read more
Free Mixoo iOS Photo Editing App Gets Major U...
Button Software Technology has announced Mixoo 3.0, an important feature update to the company’s photo editor for iOS devices. Mixoo offers amazing ways to edit as well as decorate personal photo... Read more
FNable 2.0 Function Key Enabling Software Get...
Briksoftware has released FNable 2.0 With Touch Bar Support, a major new version of the function key enabling software now compatible with Touch Bar keyboards. FNable’s simplicity that allows users... Read more
Clearance 2016 12-inch Retina MacBooks, Apple...
Apple recently dropped prices on Certified Refurbished 2016 12″ Retina MacBooks, with models now available starting at $1019. Apple will include a standard one-year warranty with each MacBook, and... Read more

Jobs Board

*Apple* Solutions Consultant (Asc) - PT - Cu...
…important role that the ASC serves is that of providing an excellent Apple Customer Experience. Responsibilities include:* Promoting Apple products and solutions Read more
Product Metrics Manager - *Apple* Media Pro...
- Identify blind spots in existing product metrics- Presenting on how Apple Media Products are used and perceived by our customers- Work with data science and Read more
*Apple* Professional Learning Specialist - A...
…Professional Learning Plans for educators in schools supported by Apple Professional Learning;Coordinate both leadership and teacher professional learning to Read more
Senior Payments Architect - *Apple* Pay - A...
…selected candidate will challenge assumptions and design solutions first and foremost with Apple 's customers in mind despite the complexity that is often behind the Read more
Product Manager, *Apple* News - Apple Inc....
…on new product areas, many of which have great potential impact to Apple .Drives the definition of broad, complex, and high impact products for worldwide markets, Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.