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 reinventing 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 doubleclicking on the axis of interest. In addition, the BASIC data can be viewed in loglog 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 loglog format.
Figure 2  BASIC Output Viewed in LogLog 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+RND6
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+RND3)
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=XMAXXMIN
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 bellshaped 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 TIMERSTART
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)*(XINB5*X(6)/(W0^5)B4*X(5)/(W0^4)B3*X(4)/(W0^3)B2*X(3)/(W0^2)B1*X(2)/W0X(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 TIMERSTART
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*(XINB5*X(6)/W05B4*X(5)/W04B3*X(4)/W03B2*X(3)/W02B1*X(2)/W0X(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 speedup 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. 1521.