TweetFollow Us on Twitter

Engineering Tips
Volume Number:4
Issue Number:2
Column Tag:The Workstation Mac

Tips for Engineering Applications

By Paul Zarchan, Cambridge, Mass.

Introduction

With the advent of the Mac 2 and many third party Macintosh speed enhancers many engineers will now consider using the Macintosh for serious computation. Many of these engineers are currently using mainframe computers where the graphics environment is primitive or nonexistant and where CPU time is expensive and has little to do with turn around time. The purpose of this article is to give a few pointers in BASIC and FORTRAN, the languages with which engineers will initially feel most comfortable, on how to make effective use of the Macintosh for both graphics and speed.

Many engineers have heard that programming on the Mac is difficult and unlike mainframe programming. This is true if the end result is a commercial product. Making a program user friendly may cause simple code to grow by more than an order of magnitude. Using the various Macintosh toolbox routines requires the learning of a new language and a new way of thinking. However ordinary engineering applications are quite easy to program on the Mac - in fact even easier to program than on a mainframe or other micro - because of the user friendly environment, superior editing tools and virtually instantaneous feedback.

This article will show how engineers can modify ordinary programs, written in FORTRAN or BASIC, which generate numerical output and enhance these programs with the addition of a few lines of code so that they can serve as input drivers to professional commercial graphics software. In this way their numerical output can easily be converted to professional quality graphs and charts with little effort. They will discover, as I did, that instantaneous graphical feedback will not only help them convey their engineering message to others quickly but will also challenge them to modify their original program to get even more useful information. In addition, techniques for generating random numbers on the Mac will be discussed so that engineers can see how easy it is to perform monte carlo and statistical analysis. Finally, for those engineers wanting to run more complex problems on the Mac in BASIC, tips will be given on how to speed up program execution tiime with minimal changes to the program.

Plotting

The powerful graphic commands in MS BASIC and the access to the Macintosh toolbox in MacFortran permit the development of plotting packages in those respective languages. However, rather than re-inventing the wheel, it is better and much easier to use a very powerful commercial graphics package for engineers that already exists - Cricket Graph. The purpose of this section is to show the few lines of code are necessary to add to existing engineering programs so that the resultant output data,whether generated in BASIC or FORTRAN, can be transferred to Cricket Graph.

Cricket Graph requires it’s numerical input to be in a tab delimited format. Unfortuneately MS BASIC writes data to a file in comma delimited format. The simple fix to the problem is to have BASIC write it’s output to the screen for viewing purposes and to also write the output to the Macintosh clipboard for eventual pasting into Cricket Graph. Listing 1 presents a sample BASIC program which computes the expressions y=i2 and z=i3 for values of i ranging between 1 and 10. The resultant output is written to both the screen and clipboard.

OPEN “CLIP:”FOR OUTPUT AS #1
FOR I=1 TO 10
Y=I*I
Z=I^3
PRINT I,Y,Z
WRITE #1,I,Y,Z
NEXT I
CLOSE #1

Listing 1 - Writing Output to Clipboard in BASIC

The code in boldface can be added to any program so that output can be written to the clipboard in tab delimited format. The next step is to quit BASIC and then open Cricket Graph and finally to use the paste command to import the data. The resultant line plot, developed with Crickett Graph and slightly enhanced in MacDraw, is shown in Fig. 1.

Figure 1 - Using Cricket Graph to Plot BASIC Output

With Cricket Graph the ranges on the abscissa and ordinate can be changed quickly by double-clicking on the axis of interest. In addition, the BASIC data can be viewed in log-log form by simply choosing the log option in Cricket Graph. Grids can be added for additional readability. Figure 2 shows the same BASIC data viewed in gridded log-log format.

Figure 2 - BASIC Output Viewed in Log-Log Format

A FORTRAN program can also be modified so that its ouput can also be placed in a tabbed delimited format. This is easier to do in FORTRAN than in BASIC because it is possible to write data to a sequential file in tabbed delimited format. In order to illustrate the technique let’s consider the Fourier Series representation of the periodic square wave shown in Fig. 3.

Figure 3 - Fourier Series Representation of Periodic Square Wave

An infinite amount of terms would yield a perfect representation of the square wave whereas a finite number of terms would approximate the square wave. This effect is known as Gibbs phenomenon. Listing 2 presents a FORTRAN program which computes the first three terms of the periodic square wave Fourier Series and also writes the output to both the screen and also to a sequential file called “SAM”.

 REAL L
 CHARACTER*1 TAB
 TAB=Z’09'
 OPEN(1,STATUS=’NEW’,FILE=’SAM’)
 L=.5
 PI=3.14159
 C1=4./PI
 DO 10 X=0,1,.01
 F1=C1*SIN(PI*X/L)
 F3=F1+C1*SIN(3.*PI*X/L)/3.
 F5=F3+C1*SIN(5.*PI*X/L)/5.
 WRITE(9,*)X,F1,F3,F5
 WRITE(1,97)X,TAB,F1,TAB,F3,TAB,F5
 97FORMAT(F8.3,A1,F8.3,A1,F8.3,A1,F8.3)
 10CONTINUE
 CLOSE(1)
 PAUSE
 END

Listing 2 - Writing Tabbed Delimited Output to File in FORTRAN

The boldfaced statements in the above listing can be added to any FORTRAN program to write the output data, in a tabbed delimited format, to a sequential file. The sequential file can then be directly opened by Cricket Graph for plotting. For this example the resultant Cricket Graph output is displayed in line chart form in Fig. 4.

Figure 4 - FORTRAN Output Displayed in Cricket Graph

Random Numbers

Random numbers play an important role in engineering applications. They form the cornerstone of monte carlo and statistical analysis. In BASIC, uniformly distributed random numbers are available with the RND statement. Other random distributions can be constructed from the uniform distribution. For example, since the RND statement creates a uniformly distributed number between 0 and 1 with variance 1/12 and mean .5, a gaussian distributed random variable with variance 1 and zero mean can be constructed, according to the central limit theorem, by adding 12 uniformly distributed numbers together (12 times 1/12 is one) and subtracting 6. Therefore the following BASIC statement generates an approximate Gaussian distributed random variable with zero mean and unity standard deviation.

X=RND+RND+RND+RND+RND+RND+RND+RND+RND+RND+RND+RND-6

The distribution is approximately Gaussian since the maximum and minimum values of the distribution are ±6 rather than ±• . If the approximate Gaussian number generator is in a program inner loop and computer running time needs to be enhanced, fewer terms can be used in the above expression to speed up the program. For example, a six term approximate Gaussian number generator with unity variance and zero mean can be constructed from

X=1.414*(RND+RND+RND+RND+RND+RND-3)

In this case the maximum and minimum values of the approximate distribution are ±3 rather than ±• .

FORTRAN, unlike BASIC does not provide a uniform random number generator. However an efficient random number generator can be constructed in FORTRAN by making one toolbox call. TOOLBX(RANDOM) generates a uniformly distributed integer between ±32768 quickly. Listing 3 presents a FORTRAN program which, by using the above toolbox call, generates approximate gaussian random numbers and then puts the numbers into bins so that a probability density function can also be constructed.

 INTEGER BIN,RANDOM,SUM
 INTEGER*4 TOOLBX
 DIMENSION H(10000),X(10000)
 PARAMETER (RANDOM=Z’86140000')
 CHARACTER*1 TAB
 TAB=Z’09'
 OPEN(2,STATUS=’NEW’,FILE=’SAM’)
 XMAX=6.
 XMIN=-6.
 RANGE=XMAX-XMIN
 TMP=1/SQRT(6.28)
 N=1000
 BIN=50
 DO 10 I=1,N
 SUM=0
      DO 14 J=1,12
      IRAN=TOOLBX(RANDOM)
      SUM=SUM+IRAN
 14  CONTINUE
     X(I)=SUM/65536.
 10CONTINUE
 DO 20 I=1,BIN
 H(I)=0
 20CONTINUE
 DO 30 I=1,N
 K=INT(((X(I)-XMIN)/RANGE)*BIN)+.99
 IF( K<1)K=1
 IF(K>BIN)K=BIN
 H(K)=H(K)+1
 30CONTINUE
 DO 40 K=1,BIN
 PDF=(H(K)/N)*BIN/RANGE
 AB=XMIN+K*RANGE/BIN
 TH=TMP*EXP(-AB*AB/2.)
 WRITE(9,*)AB,PDF,TH
 WRITE(2,97)AB,TAB,PDF,TAB,TH
 97FORMAT(F8.3,A1,F8.3,A1,F8.3)
 40CONTINUE
 CLOSE(2)
 PAUSE
 END

Listing 3 - FORTRAN Program Illustrating Random Number Generation

The highlighted terms are those statements which are required by any FORTRAN program to generate uniformly distributed or gaussian distributed random numbers. Here X(I) is the resultant gaussian distributed number. In this particular program 1000 random numbers are calculated. The accuracy of the random number generator is illustrated by viewing the program output, shown in Fig. 5, which compares the sampled probability density function to the theoretical bell-shaped curve.

Figure 5 - FORTRAN Generates Gaussian Distributed Random Numbers Accurately

Speeding Up BASIC Programs

In a previous issue MacTutor issue1 it was shown how the transient response of a sixth order Butterworth filter could be simulated in FORTRAN using the second order Runge Kutta method for integrating the six differential equations. A block diagram of the Butterworth filter is repeated here for convenience in Fig. 6. Here the “1/s” terms represent integrations.

Figure 6 - Sixth Order Butterworth Filter

The resultant six first order differential equations, which can be obtained from Fig. 6, are:

A BASIC program which can integrate the above differential equations is shown in Listing 4.

DIM X(6),XOLD(6),XD(6)
START=TIMER
B1=3.86:B2=7.46:B3=9.13:B4=7.46:B5=3.86:B6=1:W0=50
XIN=1
ORDER=6
FOR I=1 TO ORDER
X(I)=0
NEXT I
T=0
H=.0005
S=0
BEGIN:
IF T>=.5 GOTO FINISH
S=S+H
FOR I=1 TO ORDER
XOLD(I)=X(I)
NEXT I
GOSUB INTEG
FOR I=1 TO ORDER
X(I)=X(I)+H*XD(I)
NEXT I
T=T+H
GOSUB INTEG
FOR I=1 TO ORDER
X(I)=.5*(XOLD(I)+X(I)+H*XD(I))
NEXT I
IF S>.004999 THEN
S=0
PRINT T,X(1)
END IF
GOTO BEGIN
FINISH:
PRINT TIMER-START
WHILE INKEY$=””:WEND
INTEG:
XD(1)=X(2)
XD(2)=X(3)
XD(3)=X(4)
XD(4)=X(5)
XD(5)=X(6)
XD(6)=(W0^6)*(XIN-B5*X(6)/(W0^5)-B4*X(5)/(W0^4)-B3*X(4)/(W0^3)-B2*X(3)/(W0^2)-B1*X(2)/W0-X(1))/B6
RETURN
END

Listing 4 - BASIC Simulation of Butterworth Filter

We can see from the listing that the integration step size, H, is .0005 sec. The running time for this program in interpretive MS BASIC is 229 sec on a Macintosh. Halving the step size or doubling the simulation time will double the running time. The program can be speeded up by using the MS BASIC Compiler and the resultant running time will decrease to 56 sec. Other methods can also be used to further enhance speed. For example the compiler and interpreter assume that all arrays are dynamic. However we can tell the compiler, but not the interpreter, that the arrays are static (as long as we aren’t going to redimension statements during the program) and significantly decrease the running times. In addition, making loop parameters integers, as is commonly done in FORTRAN, rather than default single precision real numbers will also enhance running time - especially with the compiler. Converting real numbers to integers can easily be accomplished in BASIC by adding a percent sign to the variable. Finally eliminating repeated calculations of constants from loops will also speed up the program. For example, the derivative XD(6) has many such constant computations which can easily be moved to the beginning of the program without destroying the readability of the code. The faster running BASIC program is shown in Listing 5.

DIM X(6),XOLD(6),XD(6)
START=TIMER
B1=3.86:B2=7.46:B3=9.13:B4=7.46:B5=3.86:B6=1:W0=50
XIN=1
ORDER%=6
W02=W0*W0
W03=W02*W0
W04=W03*W0
W05=W04*W0
W06=W05*W0
FOR I%=1 TO ORDER%
X(I%)=0
NEXT I%
T=0
H=.0005
S=0
BEGIN:
IF T>=.5 GOTO FINISH
S=S+H
FOR I%=1 TO ORDER%
XOLD(I%)=X(I%)
NEXT I%
GOSUB INTEG
FOR I%=1 TO ORDER%
X(I%)=X(I%)+H*XD(I%)
NEXT I%
T=T+H
GOSUB INTEG
FOR I%=1 TO ORDER%
X(I%)=.5*(XOLD(I%)+X(I%)+H*XD(I%))
NEXT I%
IF S>.004999 THEN
S=0
PRINT T,X(1)
END IF
GOTO BEGIN
FINISH:
PRINT TIMER-START
WHILE INKEY$=””:WEND
INTEG:
XD(1)=X(2)
XD(2)=X(3)
XD(3)=X(4)
XD(4)=X(5)
XD(5)=X(6)
XD(6)=W06*(XIN-B5*X(6)/W05-B4*X(5)/W04-B3*X(4)/W03-B2*X(3)/W02-B1*X(2)/W0-X(1))/B6
RETURN
END

Listing 6 - Faster BASIC Program

The terms that were changed to speed up the program are highlighted. Table 1 documents the speedup enhancements.

Condition Interpreter Compiler

Nominal 229 sec 56 sec

Make Arrays Static 229 sec 42 sec

Use Integers in Loops 211 sec 26 sec

Move Constant Computations 120 sec 22 sec

Out of Loops

Table 1 - Improving Speed of BASIC

We can see from Table 1 that there is a big payoff in using static arrays and integers when using the compiler. On the other hand, moving constant computations out of loops is more effective, in this case, when using the interpreter. For this example we can see that the net speed-up in being careful and using the compiler, rather than the interpreter, is more than a factor of 10 with only minor modifications to the original code. The best policy is to first get the programmed debugged and running using the interpreter because errors can be found and corrections can be made virtually instaneously. The tips for speed enhancement can then be applied when the program is checked out and is ready to be compiled.

References

1. Zarchan, P. “New Mac Workstation Potential”,

MacTutor, March 1987, pp. 15-21.

 

Community Search:
MacTech Search:

Software Updates via MacUpdate

BetterTouchTool 1.84 - Customize Multi-T...
BetterTouchTool adds many new, fully customizable gestures to the Magic Mouse, Multi-Touch MacBook trackpad, and Magic Trackpad. These gestures are customizable: Magic Mouse: Pinch in / out (zoom... Read more
Dropbox 8.4.21 - Cloud backup and synchr...
Dropbox is an application that creates a special Finder folder that automatically syncs online and between your computers. It allows you to both backup files and keep them up-to-date between systems... Read more
OmniGraffle Pro 6.6.1 - Create diagrams,...
OmniGraffle Pro helps you draw beautiful diagrams, family trees, flow charts, org charts, layouts, and (mathematically speaking) any other directed or non-directed graphs. We've had people use... Read more
OmniGraffle 6.6.1 - Create diagrams, flo...
OmniGraffle helps you draw beautiful diagrams, family trees, flow charts, org charts, layouts, and (mathematically speaking) any other directed or non-directed graphs. We've had people use Graffle to... Read more
f.lux 37.7 - Adjusts the color of your d...
f.lux makes the color of your computer's display adapt to the time of day, warm at night and like sunlight during the day. Ever notice how people texting at night have that eerie blue glow? Or wake... Read more
BBEdit 11.6.1 - Powerful text and HTML e...
BBEdit is the leading professional HTML and text editor for the Mac. Specifically crafted in response to the needs of Web authors and software developers, this award-winning product provides a... Read more
ScreenFlow 6.1 - Create screen recording...
ScreenFlow is powerful, easy-to-use screencasting software for the Mac. With ScreenFlow you can record the contents of your entire monitor while also capturing your video camera, microphone and your... Read more
Microsoft Office 2016 15.25 - Popular pr...
Microsoft Office 2016 - Unmistakably Office, designed for Mac. The new versions of Word, Excel, PowerPoint, Outlook and OneNote provide the best of both worlds for Mac users - the familiar Office... Read more
FileZilla 3.21.0 - Fast and reliable FTP...
FileZilla (ported from Windows) is a fast and reliable FTP client and server with lots of useful features and an intuitive interface. Version 3.21.0: Fixed Vulnerabilities Fixed a string format... Read more
Fantastical 2.2.5 - Create calendar even...
Fantastical 2 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... Read more

Pokemon GO update: Take me to your leade...
The Team Leaders in Pokemon GO have had it pretty easy up until now. They show up when players reach level 5, make their cases for joining their respective teams, and that's pretty much it. Light work, as Floyd Mayweather might say. [Read more] | Read more »
Ruismaker FM (Music)
Ruismaker FM 1.0 Device: iOS Universal Category: Music Price: $4.99, Version: 1.0 (iTunes) Description: Following up on the success of Ruismaker, here's her crazy twin-sister, designed for people who want to design their own... | Read more »
Space Marshals 2 (Games)
Space Marshals 2 1.0.15 Device: iOS iPhone Category: Games Price: $5.99, Version: 1.0.15 (iTunes) Description: The sci-fi wild west adventure in outer space continues with Space Marshals 2. This tactical top-down shooter puts you in... | Read more »
Dungeon Warfare (Games)
Dungeon Warfare 1.0 Device: iOS Universal Category: Games Price: $3.99, Version: 1.0 (iTunes) Description: Dungeon Warfare is a challenging tower defense game where you become a dungeon lord to defend your dungeon against greedy... | Read more »
Solitairica (Games)
Solitairica 1.0.7 Device: iOS Universal Category: Games Price: $3.99, Version: 1.0.7 (iTunes) Description: Solitairica takes RPG combat and challenging rogue-like progression to a fresh new place—the world of solitaire! | Read more »
Bowmasters tips, tricks and hints
At least for this writer, archery was one of the more pleasant surprises of the 2016 Rio Olympics. As opposed to target shooting with guns, which was dreadfully boring, watching people shoot arrows at targets was pretty darn cool. [Read more] | Read more »
Best apps for watching live TV
The Olympics have come and gone, leaving nearly everyone in a temporary state of "What the heck am I going to watch on TV right now?" Besides old reruns of Golden Girls, but that goes without saying. [Read more] | Read more »
What is Flip Diving, and why has it take...
Move over Pokemon GO. There's a new king in town, and it's "the world's #1 cliff diving game." [Read more] | Read more »
5 places where Pokemon GO is still numbe...
In the U.S., the bloom is off the Pokemon Go rose ever so slightly. It's still doing great, sitting atop the top grossing chart as it has for some time, but it's no longer among the top 10 free apps in downloads, possibly because darn near... | Read more »
Madden NFL Mobile: How defense has chang...
Saying that defense is not a priority in Madden NFL Mobile is a bit of an understatement. In asynchronous head-to-head play, you don't take control of your defenders at all, as the AI manages them while your opponent plays offense. When it's your... | Read more »

Price Scanner via MacPrices.net

Apple refurbished 13-inch MacBook Airs availa...
Apple has Certified Refurbished 2016 and 2015 13″ MacBook Airs now available starting at $849. An Apple one-year warranty is included with each MacBook, and shipping is free: - 2016 13″ 1.6GHz/8GB/... Read more
Apple refurbished iPad mini 2s available for...
Apple is offering Certified Refurbished iPad mini 2s for up to $80 off the cost of new minis. An Apple one-year warranty is included with each model, and shipping is free: - 16GB iPad mini 2 WiFi: $... Read more
Save up to $600 with Apple refurbished Mac Pr...
Apple has Certified Refurbished Mac Pros available 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 following... Read more
Mac Pros on sale for $200 off MSRP
B&H Photo has Mac Pros on sale for $200 off MSRP. Shipping is free, and B&H charges sales tax in NY only: - 3.7GHz 4-core Mac Pro: $2799, $200 off MSRP - 3.5GHz 6-core Mac Pro: $3799, $200... Read more
Will We See A 10.5″ iPad Pro in 2017? – The ‘...
A MacRumors report, cites a research note from KGI Securities analyst Ming-Chi Kuo, saying a new size iPad model is in the works. According to the highly respected Cho, who has a strong track record... Read more
IOGEAR USB-C Docking Station Transforms Lapto...
IOGEAR has announced the launch of its innovative USB-C Docking Station with Power Delivery which turns USB-C enabled laptops into desktop workstations. The new IOGEAR USB-C Docking Station features... Read more
12-inch Retina MacBooks on sale for up to $10...
Amazon has 2016 12″ Apple Retina MacBooks on sale for $100 off MSRP. Shipping is free: - 12″ 1.1GHz Space Gray Retina MacBook: $1199 $100 off MSRP - 12″ 1.1GHz Silver Retina MacBook: $1224.99 $75 off... Read more
13-inch 2.5GHz MacBook Pro (Apple refurbished...
Apple has 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″ 2.5GHz MacBook Pros... Read more
21-inch iMacs on sale for up to $120 off MSRP
B&H Photo has 21″ iMacs on sale for up to $120 off MSRP including free shipping plus NY sales tax only: - 21″ 3.1GHz iMac 4K: $1379 $120 off MSRP - 21″ 2.8GHz iMac: $1199.99 $100 off MSRP - 21″ 1... Read more
Typinator 6.10 comes with 50 improvements – G...
Ergonis Software today announced release of Typinator 6.10, a new version of their text expander utility for macOS. Typinator 6.10 comes with 50 improvements, including new features, compatibility... Read more

Jobs Board

*Apple* Mobile Master - Best Buy (United Sta...
What does a Best Buy Apple Mobile Master do? At Best Buy, our mission is to leverage the unique talents and passions of our employees to inspire, delight, and enrich Read more
*Apple* Retail - Multiple Positions Akron, O...
Job Description: Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
Simply Mac *Apple* Specialist- Repair Techn...
…The Technician is a master at working with our customers to diagnose and repair Apple devices in a manner that exceeds the expectations set forth by Apple Read more
*Apple* Retail - Multiple Positions Germanto...
Job Description: Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
*Apple* Professional Learning Specialist - A...
# Apple Professional Learning Specialist Job Number: 51234379 Portland, Maine, Maine, United States Posted: Aug. 18, 2016 Weekly Hours: 40.00 **Job Summary** The Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.