AN INTRODUCTION TO FORTRAN AND NUMERICAL MODELING

Transcription

1 AN INTRODUCTION TO FORTRAN AND NUMERICAL MODELING Dr. L. W. Shwartz Department of Mehanial Engineering University of Delaware September, 2000 INTRODUCTION The first part of this short doument ontains a brief desription of the Unix operating system as well as a minimal set of ommands and proedures in the FORTRAN omputer language. The purpose is to get you up any running as quikly as possible in order to do useful numerial work on the omputer. Many other ommands and proedures exist, but having many different ways to do the same thing is often onfusing. After you an use these basis, you an hange to other ommands that may be more intuitive for you. The seond part will disuss a number of engineering problems. For eah problem, a simplified mathematial model will be developed. Eah model will be translated into a Fortran omputer program. Results of running eah program will help use to understand how eah of the engineering systems works. Results will be presented graphially in both two and three dimensions using the Gnuplot program. Exerises that will help you develop your programming and modeling skills are distributed through the setions. PART I: PROGRAMMING BASICS 1.1. GETTING STARTED The Unix operating system. The University of Delaware uses Unix as the operating system for its main omputers. All other universities use Unix as do most large orporations whose prinipal businesses involves siene and engineering. Unix allows any user to use any of the omputers in the system or network from the Personal Computer (PC) or Workstation on his or her desk. Of ourse the proper permissions are neessary. The main University omputer system is known as The Composers beause eah omputer is named after a famous omposer of musi. You probably read your on opland. We will do our sientifi omputing on strauss. A more omplete name for strauss is strauss.udel.edu. Many PC s that are hard-wired into the University system, suh as the mahines in Pearson room 116, aess the omposers using an appliation alled Exeed. There are many other ways of getting onto the system. You will need a username and password. 1

2 When you get on strauss, you will be in your home diretory. There may be subdiretories there already or you an reate them yourself. Think of your home diretory as a big folder that an hold many smaller folders. Eah smaller folder an have still smaller folders. Eah folder an also ontain files. These files may be data sets, omputer programs, pitures, et. A personal mahine that uses Unix is alled a Workstation (or Unix Box by the pros!). These mahines use a graphial user interfae alled Xwindows. With Xwindows it is possible to make the desktop look very muh like MS Windows on a PC or the desktop on a Maintosh. Within Xwindows (or X for short!) you an drag files into folders, double-lik to open, et. It is possible to install Linux, a free version of Unix, on any PC. For info, visit the UD Linux Users Group website. You will then have all the advantages of a Workstation at zero ost. When Linux is installed, you an run either DOS/Windows or Linux on the same PC (but not both at the same time, although the Linux guys are working on that very issue.) Virtually all Linux programs are free and an be freely given out. Linux is now the seond most popular PC operating system. A short list of Unix ommands. Firstly, Unix is ase sensitive. Use lower ase, i.e. forget about the shift key. All ommands are followed by a arriage return (CR). Speifially CR means press the ENTER key on the keyboard. ls Lists the ontents of the urrent diretory. ls -l is like ls but provides more information about the list of files: size of file, reation date, ownership, et. man The online manual in Unix. For information on ls or any other ommand, type, for example, man ls (CR). Typing man man (CR) will give information about the man ommand. Get out of man by typing q for quit. mkdir anyname (CR) reates a subdiretory in the urrent diretory alled anyname. d anyname moves you into the subdiretory alled anyname. Just typing d without a diretory name returns you to your home diretory. d.. moves you up one level in the diretory struture. pwd means print working diretory. p filename newopy. Copies the file alled filename into the new file alled newopy. mv filename newname. Changes the name of a file ( moves it). f77 program.f -o runname. Compile the program alled program.f and all the exeutable file runname. Run the program by typing runname (CR). If -o... is omitted, the default name for the exeutable is a.out. To stop a running exeutable (if it is not doing what you want for example), type trl, meaning depress the key labelled trl and the key simultaneously. xterm reates a new terminal window (requires X). 2

3 xterm & The ampersand returns ontrol to the alling window after opening the new terminal window. Thus both windows are usable simultaneously. textedit filename. Multi-featured file editor with pull down menus, opy-paste, et. (requires X). If filename exits, textedit opens it; otherwise it reates a new file named filename. You an use the mouse buttons to save as, save and quit telnet mahine opens onnetion to omputer with internet name or address mahine, e.g. telnet strauss, telnet strauss.udel.edu. ftp mahine opens onnetion to remote omputer mahine speifially to send or reeive files. exit loses terminal window or telnet onnetion. pine to read or send mail. pio filename. Pio is an editor that an be used when X is not available. The look is similar to pine. lpr filename. Generi Unix print ommand to send filename to a printer. You will probably need to find out what the speifi name is for the printer you want to use. gnuplot alls the plotting routine gnuplot. The gnuplot ommand set terminal dumb displays poor-quality ommand window graphis when X is not available STEPS IN WRITING A COMPUTER PROGRAM What do you want to do? Every little thing must be explained ompletely. Flowharts an be used to map out the logial flow in a omputer program. An example of a flowhart will be given below. Mathematial desription. What formulas do you want to use? The origin of the name: FORmula TRANslation = FORTRAN Write the program. Save it in a file alled myprogram.f for example. Compiling the program produes an exeutable version that the mahine understands. f77 myprogram.f auses the exeutable file alled a.out to be produed. Run the program by typing a.out (CR) De-bugging: Programs almost never run the first time... don t get disouraged. A little utility alled ftnhek an help you find mistakes. When you get numbers... do they make sense? You an hand-hek a few of them using a alulator. 3

4 1.3. PROGRAMMING IN THE FORTRAN LANGUAGE Type your program into the window reated by the Openwindows text editor. You an use the Edit ommands (right mouse button) to Copy and Paste. Almost all statements start in olumn 7. When you get a new line by hitting (CR), hit the spae bar 6 times. Only olumns 7 to 72 ontain the FORTRAN statements. Exeptions: 1. in olumn 1 for a omment (it s ignored by ompiler. It is just for you, so you know later what you are doing). 2. Entering a number in olumn 6 is a ontinuation allowing you write really long statements. Try to avoid very long statements sine they are hard to read! 3. Statement numbers an be typed into olumns 1 to 5. Thus the largest statement number is A little program alled myprogram.f is shown in the figure on the following page. It ontains omment statements saying what eah line does. You an ompile the program by typing f77 myprogram in the ommand window; then you exeute the program by typing a.out. O.k., now you do it. Just type in the non-omment lines and try some numbers. Did the program ompile orretly? If not, what did the ompiler tell you? Is it helpful? If you still have problems, try typing ftnhek myprogram.f This may give you suffiient information to solve your problem. My program ompiles and runs suessfully. The dialog in the x-term ommand window is given in Fig. 2. 4

5 the line below names the program program myprogram the following 4 lines are used to read a number that you type into the ommand window. write(6,100) 100 format( please type in the first number ) read(5,110)a 110 format(f10.0) The first line above auses the omputer to write please type in the first number into the ommand window. The third line reads the number that you type and assigns the value that you type to a variable alled a. The omputer waits for you to hit (CR) after you type the number before it will do anything else. The 2nd and 4th lines are format statements. They provide info telling the read and write statements exatly what to do. Speifially, the write statement says, write on unit=6, the information ontained in format statement 100. Note the use of the quotes. The read statement says, read from unit=5, the number a where a is written using the format f10.0. This means that the "field" is a maximum of 10 haraters long and that you will put in the deimal point. Thus numbers like are ok, but 2341 is no good (no deimal) and is no good (too long). write(6,200) 200 format( please type in the seond number ) read(5,110)b Just like before. Notie that sine the format for reading b is the same as for reading a, it is not repeated. In fat there annot be 2 lines that have the same statement number. =a+b The above obviously auses the omputer to add the numbers a and b and store the answer in the variable alled. write(6,300) 300 format( the answer is,2x,f15.5) The answer is typed into the ommand window What will atually appear is the answer is Notie that eah piee of the format statement is divided by ommas. The 2x means skip 2 plaes. f15.5 means write, using the same type of deima fration format, using 5 plaes after the deimal point. end All fortran programs should end with an end statement. Fig. 1: The program myprogram.f as it would appear in a text editor window. The lines beginning with are omments. They are ignored by FORTRAN. 5

6 > g77 myprogram.f > a.out please type in the first number 2. please type in the seond number 1.45 the answer is > Fig.2: The ommand window showing the dialog for the program myprogram.f. The symbol is the prompt whih means the omputer is waiting for you to type in a ommand VARIABLES The omputer an store a number in one of 2 ways: (i) as an integer, or (ii) as a real number. An integer is stored exatly, meaning the number 1234 is stored in the omputer memory as 1234 with no deimal point. In fat, if you type a deimal point for an integer number, you will be told that you have made a mistake. A real number, on the other hand is stored as a deimal fration. A format statement in myprogram.f has the speifiation f10.5. This means that the number is a real number. So-alled real numbers often are not stored exatly by the omputer. The way it works is that about seven signifiant figures are saved in the mahine plus an exponent, i.e. a power of 10. Thus, for example the number (1/3000) is stored as x 10. The last digit 4 is wrong, of ourse; that s beause the omputer really only saves 6.7 digits in the portion of the word alled the mantissa. The number -3 is stored in the exponent part of the word and is stored exatly. The plae in the memory of the omputer where a number is stored is alled a word. The word size used for integers is different from the size used for real numbers. For both integer words and real words, there is a largest possible number that an be stored. Finally, sine there are two types of numbers, how does the omputer know whether a given variable is to be interpreted as a real number or as an integer number? It is possible to delare the type of number eah variable is, but it is easier (and safer) to use the default settings. These are: Any variable whose first letter is i, j, k, l, m or n is taken to be an integer Any other first letter, that is a-h and o-z is taken to be real. And remember, all variables must start with a letter, not a number. Thus a2 is an aeptible variable name but 2a is not. Suppose I wanted to assign a very big number to a variable, or to write or read a very big number from the ommand window. Say the number is Using the f-format 6

7 that we introdued, we would need to use a lot of zeros as plae holders. Instead, of ourse, we want to use sientifi notation. In fortran, this format is alled floating point form or exponential form and a proper FORTRAN assignment statement would be av=6.023e23 Other aeptable forms are av=.6023e24 av=6023.e20 av= e020 While the last form is o.k., there is no good reason for writing the number this way sine the extra zeros onvey no extra information ARITHMETIC OPERATIONS The standard arithmeti operations work just the way you expet; thus addition is +, substration is -, division is / but note multipliation uses the *. Also, raising a number to an exponent uses **. A ompliated statement using several variables and arithmeti operations must be written on a single line. We an t draw a fration, for example, with a numerator lying above a denominator. For this reason, we will use parentheses ( ) in pairs. Suppose I want to program the formula for the drag oeffiient (of a ar, say), An aeptable FORTRAN statement would be drag=dr/(.5*rho*v**2*a) There are several things to note here: 1. I have written as The parentheses are used to signify that all of left out ( ), viz drag=dr/.5*rho*v**2*a belongs in the denominator. If I had the formula would have been interpreted inorretly as 7

8 3. The exponentiation, in FORTRAN, is done first; thus v**2*a means not. If you are not sure, in a ompliated expression, that the operations are being done orretly, there is nothing wrong with inserting an extra set of ( ). Thus another perfetly fine way of writing the statement is drag=dr/(.5*rho*(v**2)*a) This also makes it a bit easier to read. All parentheses must be mathed; if you have four ( but only three ) in an assignment statement, you will get an error message and the program will not ompile. 4. The names used for the variables are arbitrary exept that, beause they are to be interpreted as real numbers, they all start with a-h or o-z. I ould have used r instead of rho, et CONVERSIONS BETWEEN REAL & INTEGER NUMBERS & VARIABLES The statement ri=i onverts an integer number on the right of the equal sign to a real number on the left. More exatly, what it does is to reate a new real variable ri and sets its value equal to the urrent value in i. Thus if i = -2 then ri will be set equal to (-2.). Note the deimal point, it is very important. It is what distinguishes a real number from an integer. The bakward onversion is more interesting. Suppose ri = 2.3, what will the following statement do? i = ri Well, sine i has to be an integer, it an t be set equal to 2.3. Instead it will trunate to the integer value i = 2 (without the deimal point of ourse). This an be very useful sometimes, but it an also ause problems. Suppose the drag formula had been oded as drag=dr/((1/2)*rho*(v**2)*a) 8

9 It will not work! The reason is that neither 1 nor 2 have deimal points. Thus they are both onsidered to be integers. Moreover the quotient 1/2 is also interpreted as an integer and is assigned the trunated value 0. Finally, beause the zero is in the denominator, drag will be assigned the value and will be printed out as Inf. Using real and integer variables or numbers in the same expression is alled mixed mode arithmeti. It is a ommon ause of mistakes and the best programming pratie is to onsistently avoid this mixing in a given FORTRAN statement. Thus all numerial onstants in formulas should be oded with a deimal point. Exponents like the 2 in v**2 an be written using an integer; however v**2. is also o.k BASIC SUPPLIED FUNCTIONS In addition to to the arithmeti operations and the exponentiation operator **, there are a number of standard funtions supplied with FORTRAN. Familar ones, with sample FORTRAN statements are sine osine tangent natural log base 10 log artangent absolute value square root yy=sin(x/2.) z=os(x/2./b) a=tan(2.5*b) q=alog(ss) q=alog10(x/2.) yy=atan(x/2.) z2=abs(z) z=sqrt(x**2 + y**2) Beause eah of these funtions alulates a real value, funtions whose names do not naturally start with a-h or o-z have an a attahed to the beginning of their names. Thus the base 10 or ommon logarithm is alled alog10 rather than log10. Also the ation of the sqrt funtion is no different from what you would get if you wrote z=(x**2 + y**2)**.5 instead. Maximums and minimums: There are several different funtions for seleting the maximum and minimum values from a set of numbers. Examples of these funtions are: real max of reals real max of integers integer max of integers integer max of reals yy=amax1(a,b,) yy=amax0(i,j,kk,mmm) kkr=max0(i,j,kk) k=max1(a,b) 9

10 The orresponding funtions that assign the minimum rather than maximum values are exatly the same, exept max is replaed by min. Also the pattern here is not too hard to remember. If the value you are alulating is to be an integer, the funtions that alulate the max or min start with the letter m whih is part of list i-n that is used to name integers. If the value you are alulating is real then the funtion is amax... rather than max... sine the a implies the answer is to be real. Also if the arguments are integers, the funtion name ends in 0, while if the arguments are reals, the funtion name ends in 1. As examples yy=amin1(2.3, -1., -1.e-2) assigns the value -1. to yy. Notie that I an use either the frational or the exponential way of writing a onstant. Also ibig=max1(2.3, -1., -1.e-2) sets ibig to 2 (without a period of ourse) beause it first finds the maximum among the real values inside the ( ) and then it trunates that real value into an integer value THE LOGICAL IF STATEMENT AND SOME OPERATORS Sometimes we need to perform a alulation only if a partiular statement is true. The funtion that does this is alled the logial if. The syntax is if(ondition)fortran statement The ondition is either true or false. If the statement is true, then the fortran statement is exeuted. If the ondition turns out to be false, the statement is not exeuted. Very often, the ondition involves determining whether one number is greater than (or less than) another. There are six relational operators that we need. They are equal to not equal to greater than less than greater than or equal to less than or equal to.eq..ne..gt..lt..ge..le. We also need the logial operators 10

11 .and..or. The operator.and. is used to ombine two onditions. statement1.and. statement2 is true if both statements are true. If one or both is false, then the ombination is false. The operator.or. is true if one of the statements is true. If both are true, it is also true. Notie the periods (.) on either side of ne,and, et. They are part of the operators and must be inluded. Example: The Inome Tax Table The little program in Fig. 3 alulates your United States Inome Tax. Your tax is, of ourse, determined by your inome. The United States Inome Tax is based on your so-alled adjusted gross inome (agi). Under an old Tax Rate Shedule, whih we will use, there are only three brakets : inome less than $3000/year, inome between $3000 and $24650, and inome greater than $ A different tax formula is used for eah of these brakets. Speifially, the marginal tax rates are zero, 15%, and 28% respetively. A flowhart is used to help write the omputer program. It is shown in Fig. 3FC. Tests are shown in boxes with pointed ends. The answer to eah test is either yes (Y) or no (N). Instrutions or alulations that need to be exeuted are given in retangles. The diretion of flow is from top to bottom. A FORTRAN program alled ustax.f, orresponding to the flowhart is shown in Fig. 3. Note the use of the logial if and the logial operator.and.. Exerise The inome tax table in ustax.f is the old table. Taxes were raised for high inome earners about five years ago. Two new brakets were added: inome above $100,000 is taxed at 31 % and inome above $250,000 is taxed at 36%. Modify ustax.f to aount for the new tax table. Run it at least five times to make sure that it works. Submit a listing of the new program and the results of your runs. 11

12 program ustax write(6,100) 100 format( what is your adjusted gross inome? Enter. ) read(5,110)agi 110 format(f12.0) if(agi.le.3000.)tax=0. if((agi.gt.3000.).and.(agi.le )) 1 tax=.15*(agi-3000.) if(agi.gt )tax= *(agi ) write(6,200)tax 200 format( your tax is $,f12.2) end Figure 3: The program ustax.f Read inome agi Y agi < $3000 N tax = 0 Y $3000 < agi < $24650 N tax =.15*(agi ) Y agi > $24650 N tax = *(agi ) stop Fig. 3FC: Flowhart for inome tax alulation. 12

13 1.3.6 READING NUMBERS FROM A FILE; THE OPEN STATEMENT A hiken farmer needs to find the average weight of eggs his hens are laying on a partiular day. He takes a random sample, say 10 or 20 eggs, and reords the weight of eah, in grams, in a file alled weight.dat. (He reates and types the weights into a file using the Openwindows Text Editor, putting eah weight on a separate line.) We have been hired to write a short FORTRAN program to alulate and print out the average weight. The first thing we need to do is to tell our program where to find the data. We assign a unit number to the data file weight.dat. We an hoose a small positive integer number for the unit, exept that we an t use unit=5 or unit=6 beause they are reserved for reading from, and writing onto, the ommand window. We also need to open the data file that our program is going to read from. By the way, the farmer doesn t always use the same number of sample weights on a given day and the program must be flexible enough to use a variable number of eggs. A FORTRAN program alled average.f is shown in Fig. 4. Also shown is the data file and the dialog in the ommand window. The program inludes the open statement open(unit=4,file= weight.dat,status= old ) Notie that there are three speifiations, separated by ommas. The first is unit=4 whih tells the program that the read statement should refer to that unit number, i.e. read(4,100)weight The seond tells it that unit 4 is the file weight.dat. The last speifiation says that the file weight.dat already exists, that is, it is an old file. Other status speifiations are new and unknown. New is used if you want the program to reate a file that into whih you are going to write information. Unknown is used if you wish to write repeatedly onto the same file and over-write the data that existed there before. When you examine the program average.f you will see a number of new ommands that we haven t talked about yet. An example is the rewind 4 ommand. Let s explain what the logi is. Suppose the file did not ontain any data at all. Then, if we did not aount for this possibility, the empty file would supply, in effet a zero value for the weight. But this is wrong, of ourse. So first we test to see if the first line ontains a weight by testing it using if(weight.gt.0.)goto 10 This statement inludes something else that we haven t seen before: goto 10. It tells the program that the next statement that should be exeuted is the one labeled as 10. The goto 13

14 program average open(unit=4,file= weight.dat,status= old ) n=0 sumwt=0. read(4,100)weight 100 format(f10.0) rewind 4 if(weight.gt.0.)goto 10 write(6,200) 200 format( The file is empty,...stop ) stop 10 ontinue read(4,100)weight if(weight.le.0.)goto 20 n=n+1 sumwt=sumwt+weight goto ontinue alulate average weight rn=n wtav=sumwt/rn write(6,300)n,wtav 300 format( The number of samples is,i4, Average weight =, e15.5) stop end Figure 4: The program average.f Figure 5: The data file weight.dat > g77 average.f > a.out The number of samples is 4 Average weight = E+02 > Figure 6: The ommand window for running average.f 14

15 statement is sometimes alled a transfer of ontrol statement. It ould also be written with a spae, i.e. as go to, if you prefer. So the whole statement means, if the weight is greater than zero, exeute statement 10 next. Suppose this is not true; then you don t go to 10. Instead you exeute the write statement saying that the file is empty and you stop. stop is a FORTRAN ommand meaning that the program is to terminate here and now. There an be many stop ommands in a program but only one end. So why did we rewind unit 4 after the first read ommand? First, rewind means go to the top of the file. Think of it as if a pointer exists in the file and every time a line is read, the pointer moves down one line. Rewind was neessary beause the number was read just as a test to see if the file was empty. Usually, unless the farmer messed up, the file will not be empty, in whih ase that first weight needs to be read again. Let s now onsider the logi in the program. The integer number n is being used as a ounter to see how many weights are being read before an empty line is enountered. This is done using the statement n=n+1 The variable sumwt is an aumulator. Every time we read a non-zero weight, we add it to sumwt using the statement sumwt=sumwt+weight When the program finally reads a zero weight, ontrol is transferred to line 20 by the statement if(weight.le.0.)goto 20 and then the average weight is alulated using wtav=sumwt/rn Notie that we used the ommand rn=n in order to avoid mixed-mode arithmeti. We always try to program to take into aount possible ways the program ould fail. Thus we use the relational operator.le. rather than.eq. Why? A negative weight makes no sense and we want to stop the program. How ould this happen? Well, anything is possible and.le. is safer than.eq. A more serious potential problem is in the loop starting at 10 ontinue and ending at the unonditional goto statement goto 10. Every time the program reahes goto 10, ontrol is transferred to 10 ontinue. Fortunately there is the test statement 15

16 if(weight.le.0.)goto 20 in the loop that allows the program to get out of the loop. If that statement were not there, the loop would lassify as an endless loop. The omputer would endlessly yle, reading zero weights as n inreased to ridiulously large values. Additional protetion against an endless loop ould have been provided by putting the fail-safe statement if(n.gt.1000)stop into the loop. This assumes, of ourse, that the farmer would never intentionally insert more than 1000 weights into the file weight.dat. Finally, the program average.f inludes the two statements 10 ontinue and 20 ontinue The word ontinue is a dummy ommand and is used when we wish merely to mark a position in the program by the line number. 16

17 1.3.7 DO-LOOPS Another way of preventing the possibility of an endless loop is to use a do-loop. Do-loops are a very important and powerful part of FORTRAN programming. A do-loop starts will the statement do 50 jjj=1,jmax and ends with the statement 50 ontinue In the do statement, 50 (or some other number) is the statement number, jjj is an integer variable, and jmax is an integer variable or an integer onstant representing the maximum value of jjj. jjj will be inremented by one for eah pass through the do-loop. Figure 7 shows the modified portion of the program of average.f where a do-loop has been used to replae the unonditional goto. We will use do-loops for other purposes below. 10 ontinue do 50 jj=1,100 read(4,100)weight if(weight.le.0.)goto 20 n=n+1 sumwt=sumwt+weight 50 ontinue 20 ontinue alulate average weight. Figure 7: Modifiation to program average.f illustrating use of a do-loop A more general form of the do-loop ommand is do 100 i = istart,iend,inrement where the inrement an be either a positive or negative integer. In the latter ase, iend would need to be less than istart. 17

18 1.3.8 SUBSCRIPTED VARIABLES AND THE DIMENSION STATEMENT There are many reasons why we might want to onsider a group of numbers as members of the same lass and assign a ommon name to them. Suppose, for example, that a person who sells fenes needs a program to alulate the total length of the perimeter of an irregularly-shaped piee of land. We are given the artesian oordinates for the loation of the (previously-ereted) fene posts on the boundary of the plot of land. The FORTRAN program peri.f, the input data file, and the ommand window are shown in the figures program peri Calulates the total perimeter length of a polygon, given the (x,y) oordinates of the verties. dimension x(100),y(100) open(unit=4,file= xydata.dat,status= old ) read(4,*)nvert do 20 n=1,nvert 20 read(4,*)x(n),y(n) sumlen=0. do 40 i=1,nvert-1 dis2=(x(i+1)-x(i))**2+(y(i+1)-y(i))**2 sumlen=sumlen+sqrt(dis2) 40 ontinue dis2=(x(nvert)-x(1))**2+(y(nvert)-y(1))**2 sumlen=sumlen+sqrt(dis2) perim=sumlen print 110, nvert,perim 110 format(2x, Number of verties =,i5, Total perimeter =,e14.6, 1 feet ) stop end Figure 8: The program peri.f Figure 9: The data file xydata.dat that is read by peri.f 18

19 > g77 peri.f > a.out Number of verties = 4 Total perimeter = E+02 feet > Figure 10: The ommand window after running peri.f FUNCTION SUBPROGRAMS AND SUBROUTINES Eah of the FORTRAN programs onsidered so far onsists of a single main program. When writing large ompliated programs, it is good pratie to divide up the program into a main program and one or more subprograms. There are several reasons for doing this: 1. It an make the program easier to understand beause the logial flow is more lear. 2. It an make it easier to find mistakes sine the ompiler will tell the programmer in whih subprogram the diffiulty probably is. Consider the so-alled funtion subprogram. It works like the basi supplied funtions sin(x), abs(y), et. exept that the programmer supplies his own way of alulating the partiular funtion. As an example, onsider again an inome tax alulation. In the previous example, we did not onsider one aspet of the tax ode, the dedutions allowed for dependents. This is an amount that is subtrated from the adjusted gross inome before the tax table is applied to the rest. Suppose the dedution for eah dependent is $3000 unless the adjusted gross inome is greater than $100,000 in whih ase the dedution, per dependent is only $2000. Suppose also that there are only two tax rates; 15 per ent on inome up to $30,000 and 28 per ent on the inome above this. [We will use this formula even though there is a little problem with it; the dedution should be gradually redued, rather than as a step jump.] A FORTRAN program that uses two different funtion subprograms to solve this inome tax problem is shown in Fig

20 program taxdep program to alulate inome tax with two different dedutions for dependents. Illustrates use of funtion subprograms. ommon/inome/agi write(6,100) 100 format( what is your adjusted gross inome? Enter. ) read(5,110)agi 110 format(f12.0) write(6,120) 120 format( Enter number of dependents as an integer ) read(5,130)idep 130 format(i5) if(agi.le )tax=tax1(idep) if(agi.gt )tax=tax2(idep) tax=amax1(tax,0.) write(6,200)tax 200 format( your tax is $,f12.2) end funtion tax1(i) ommon/inome/agi ri=i ti=agi-3000.*ri if(ti.gt.30000)goto 10 tax1=.15*ti return 10 ontinue tax1=.15* tax1=tax1+.28*(ti ) return end funtion tax2(i) ommon/inome/agi ri=i ti=agi-2000.*ri if(ti.gt.30000)goto 10 tax2=.15*ti return 10 ontinue tax2=.15* tax2=tax2+.28*(ti ) return end Figure 11: The program taxdep.f 20

21 In the program, the user supplies the adjusted gross inome (agi) and the number of dependents. Then, depending on agi, either one or the other of the funtions tax1 or tax2 is alled. The argument of eah funtion is idep, the number of dependents. The two funtion subprograms are loated in the file after the mainprogram taxdep. They are the same exept for the dedution-per-dependent in the fourth line of eah subprogram and. Beause the Openwindows Text Editor has ut-paste apability, it is easy to generate tax2 by starting with a opy of tax1 and making the appropriate hanges. Notie some other things as well: 1. The seond line in eah subprogram, as well as in the main program is a named ommon delaration ommon/inome/agi Obviously eah funtion tax1 or tax2 needs to know what the agi is. This is aomplished by using the ommon delaration whih tells eah subprogram to store or find the number in the same loation in the omputer memory, i.e. in a ommon loation. The name of the partiular ommon region in memory was alled inome. If there is only one ommon blok, as here, the name ould be omitted. A number of different variables or dimensioned arrays an be put in a single ommon blok. If more than one item is mentioned in a ommon statement, the items are separated by ommas. 2. The two piees of information that tax1 and tax2 need are the agi and the number of dependents idep. idep is transferred as the argument of eah funtion while agi is transferred via ommon. Atually both number ould have been transferred by the funtion all. We ould have used instead tax=tax1(idep,agi) In this ase the ommon statements would not have been neessary (and should not be used). Of ourse the number and type (real or integer) of funtion arguments must be the same in the title of the funtion as they are when they are alled. Thus if the above two arguments are used, we must also hange the funtion title to funtion tax1(i,agi) Notie that the name of the variable does not neessarily have to be the same in the funtion all as it is in the funtion name. Here we used idep in the all and just i in the name. 3. In eah funtion the ommand return appears twie. Return transfers ontrol bak to the main program after the tax value has been alulated. A funtion that uses logi, like the ones used here, will typially have more than one return. 21

22 Subroutines A subroutine works muh like a funtion subprogram exept it has more powerful apability. As we have seem, a funtion subprogram is used to return one alulated value to the main program. A subroutine, on the other hand, an do just about any task that might otherwise be done in the main program. Often very long omputer programs have a very short main program. The main program basially ontains a sequene of alls to a number of subroutines. The subroutines do most of the atual work; the main program ats like a table of ontents. It tells what the program is about without going into any of the details. For example, the program peri.f an be re-written to use a subroutine to do the perimeter alulation and another subroutine to do the printing. Figure 8-1 shows perisub.f, the modified version of peri.f. program perisub Calulates the total perimeter length of a polygon, given the (x,y) oordinates of the verties. ommon nvert,x(100),y(100) open(unit=4,file= xydata.dat,status= old ) read(4,*)nvert do 20 n=1,nvert 20 read(4,*)x(n),y(n) all al(perim) all pprint(nvert,perim) stop end subroutine al(perim) ommon nvert,x(100),y(100) sumlen=0. do 40 i=1,nvert-1 dis2=(x(i+1)-x(i))**2+(y(i+1)-y(i))**2 sumlen=sumlen+sqrt(dis2) 40 ontinue dis2=(x(nvert)-x(1))**2+(y(nvert)-y(1))**2 sumlen=sumlen+sqrt(dis2) perim=sumlen return end subroutine pprint(nvert,perim) print 110, nvert,perim 110 format(2x, Number of verties =,i5, Total perimeter =,e14.6, 1 feet ) return end Figure 8-1: Modifiations to program peri.f illustrating use of subroutines. Eah subroutine is invoked using the word all and eah subroutine has a return statement 22

23 to transfer ontrol bak to the main program. Subroutine al requires information from the main program. This information an be provided either by putting the numbers in ommon or by passing then through as arguments of the subroutine all. The ommon blok is used to transfer the variable nvert and the two dimensioned arrays x(100) and y(100). The all is used to transfer the alulated variable perim bak to the main program. When dimensioned arrays are mentioned in ommon, that is the plae where they are dimensioned and a separate dimension statement should not be given. The subroutine pprint needs the numbers nvert and perim. They are both passed through the subroutine all. We ould have also used a separate subroutine to read the data instead of reading it in the main program. See if you an make that work. If you do that, the main program would then only onsist of alls to the subprograms. Note that the main program and the subroutines are all inluded in the same file. They are all ompiled together by the single ommand f77 perisub.f DRAWING PLOTS OF YOUR OUTPUT; THE GNUPLOT UTILITY Most UNIX systems have a publi-domain easy-to-use plotting program alled gnuplot that works in X-windows. Lets identify the problem in the tax rate shedule used by the program taxdep.f. The problem with the tax algorithm is that inreasing your inome from just under $100,000 to just over $100,000 auses a step-jump inrease in your tax; you an atually have less money after taxes if you make a little more. To see how bad this problem is, let s use taxdep.f to alulate the tax for inomes from $1000 to $200,000 in inrements of $1000. Rather than run the program 200 times, we will put a do-loop in the program (See Fig. 12). We just show, in this figure, the modified main program, alled taxdep1. We do not need to make any hanges in the two funtions tax1 and tax2. 23

24 program taxdep1 dimension gross(500),tx(500),rate(500),ati(500) ommon/inome/agi idep=4 do 20 ii=1,200 ri=ii agi=1000.*ri gross(ii)=agi if(agi.le )tax=tax1(idep) if(agi.gt )tax=tax2(idep) tax=amax1(tax,0.) tx(ii)=tax rate(ii)=tx(ii)/gross(ii) ati(ii)=gross(ii)-tx(ii) 20 ontinue open(unit=3,file= txresult,status= unknown ) do 40 i=1,200 write(3,200)gross(i),tx(i),rate(i),ati(i) 200 format(4e15.5) 40 ontinue lose(unit=3) end Figure 12: Modifiations to main program taxdep taxdep1 We use the dimension statement to reate four arrays, eah of length 500. The dimension statement should be the first statement after the title (ignoring any omments, of ourse). The four arrays, in order, will ontain the gross inome, the alulated tax (tx), the tax rate (rate), and the aftertax-inome (ati). The number of dependents (idep) is set to 4 in the program. There are atually two do-loops in the program. The first (do 20...) alulates the tax, as before, as well as the rate and the ati. As these values are alulated, they are saved in the dimensioned arrays. The seond do-loop (do 40...) writes the arrays into a file alled txresult. Note the use of the open and lose statements for unit=3. The lose statement is new to us. Using it is good programming pratie. Modern omputer proessors use pipelining. This means, in effet, that they do more than one thing at a time. Expliitly losing the file prevents problems that an sometimes happen beause of pipelining. Notie that we use e rather than f format to write the numbers into the file txresult. The output numbers an be either quite large or quite small and the e-format an be ounted on to work while the f-format might overflow. 24

25 Notie that the maximum dimension, i.e. 500, is greater than the maximum value of ii or i used in the program. This is fine. What would be no good is if the program tried to alulate an array element whose index is larger than 500. Figure 13 shows the after-tax-inomes alulated for gross inomes between $97,000 and $105,000. This plot is reated by gnuplot whih is invoked simply by typing gnuplot at the prompt in the ommand window. The dialog is shown in Fig After-Tax Inome ($) txresult gross inome $ Figure 13: After tax inome plotted using gnuplot. > gnuplot G N U P L O T Linux version 3.7 pathlevel 0.2 last modified Wed Aug 18 14:06:10 CEST 1999 Type help to aess the on-line referene manual The gnuplot FAQ is available from < ig25/gnuplot-faq/> Send omments and requests for help to <info-gnuplot@dartmouth.edu> Send bugs, suggestions and mods to <bug-gnuplot@dartmouth.edu> Terminal type set to x11 gnuplot> set grid gnuplot> plot[97000:105000] txresult u 1:4 with linesp gnuplot> set xlabel gross inome ($) gnuplot> set title After-Tax Inome ($) gnuplot> plot[97000:105000] txresult u 1:4 with linesp gnuplot> Figure 14: The dialog used to reate the gnuplot shown in Fig

26 Gnuplot is alled by typing gnuplot in the ommand window as shown. The next few lines are written by gnuplot giving you information about the authors, et. The gnuplot prompt is gnuplot> as shown. The datafile txresult ontains 4 olumns of numbers reated by running the program taxdep1.f. The first olumn ontains the gross inome numbers and the fourth olumn the orresponding values of ati, the after tax inome. The plot ommand plot[97000:105000] txresult u 1:4 with linesp means plot, with horizontal axis values from to , the ontents of file txresult using the data from olumn 1 as the x data and the data from olumn 4 as the y data. Make the plot using linespoints, that is put a big dot at eah data point and onnet the data points with lines. The simplest gnuplot ommand plot txresult would plot using the default settings that the x olumn is olumn 1 and the y olumn is olumn 2. All pairs (x,y) are plotted and the points are marked with dots. They are not onneted with lines. Gnuplot has many other apabilities inluding three-dimensional plotting. A full manual an be found on the Web. It also has a good on-line help faility invoked by typing help at the gnuplot> prompt. 26

27 Some exerises The equation is alled transendental. There is no diret proedure for solving transendental equations like there is for solving quadrati equations, for example. One method is alled iteration. In this method, we find a sequene of numbers using We ontinue to use the formula until the onverge, thst is until the values don t hange any more. (i) Write a Fortran program to solve. Find orret to 6 signifiant figures. Chek your answer with a hand alulator. (ii) Any iteration requires a starting value, say, and I haven t told you what value to use. Does that mean that all starting values will give the same final result for? Try a number of values of and say, in words, what happens. (iii) Iteration methods do not always work. Consider where is a positive onstant. Try iteration for =.2 and =5. What happens? Find a ondition on that guarantees onvergene. (iv) Gnuplot an onveniently be used to solve transendental equations beause it an plot many funtions automatially. Also, one an set the and ranges of the plot. This an be used to zoom in on the answer. How many values of satisfy Find eah of these values orret to 5 signifiant figures. 27