# Fortran: Lesson 4

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### Statement Functions

A statement function in Fortran is like a single line function definition in Basic. These are useful in defining functions that can be expressed with a single formula. A statement function should appear before any executable statement in the program, but after any type declaration statements. The format is simple - just type
f(x,y,z,…) = formula   .

You may replace f with any name you like for your function, and x, y, z, … with your own variable names. Instead of formula type the formula for your function.
Examples :
 area(r) = pi * r * r vol(r,h) = pi * r * r * h f(x,y,z) = sqrt(x / y) * cos(z)

You should declare a type for the function in a declaration statement. Here is a program using a statement function, named "area", to compute areas of circles; the program computes in double precision the area of an annulus of inner radius a and outer radius b:
									program annulus
double precision r, area, pi, a, b
parameter (pi = 3.1415926535897932D0)
area(r) = pi * r * r
print *, "Enter the inner and outer radii of the annulus: "
write (*,10) "The area of the annulus is ", area(b) - area(a)
10	format (a,f25.15)
end
In the type declaration statement just include the name of the function - do not include the parentheses or the function variables.
Observe that variables plugged into the function need not be the same variables used in defining the function.
It is possible to use a previous statement function in the definition of another. In the above program, for example, we have already defined the function area(r), so we could define further a second function "annarea", giving the area of the annulus as
								  annarea(a,b) = area(b) - area(a)

But this second function definition must appear later in the program than the first one.

### Continuation Lines

Sometimes a Fortran statement will not all fit into columns 7-72. In such a case you may continue the statement onto the next line by placing a character in column 6 of that next line. Although any character is allowed, most programmers use "+", "&", or a digit (using 2 for the first continuation line, 3 for another if necessary, and so on).
Example :
 det = a(1,1) * a(2,2) * a(3,3) + a(1,2) * a(2,3) * a(3,1) & + a(2,1) * a(3,2) * a(1,3) - a(3,1) * a(2,2) * a(1,3) & - a(2,1) * a(1,2) * a(3,3) - a(1,1) * a(3,2) * a(2,3)

### Do While Loops

A do while loop in Fortran is similar to the same loop in Basic. However, in Fortran the test must be enclosed in parentheses, and the end of the loop is identified with either end do or a labeled continue statement. As in "if … then" constructions, in loop tests one uses letter abbreviations for relations such as "≤", ">", "=", etc. Here are two loops adding the squares of the integers from 1 to 10; they differ only in the way the loops are terminated:
 N = 1 | N = 1 S = 0 | S = 0 do while (N .le. 10) | do 5 while (N .le. 10) S = S + N ** 2 | S = S + N ** 2 N = N + 1 | N = N + 1 end do | 5 continue

### Sign Function

The function sign in Fortran is called the sign transfer function. It is a function of two variables, and its definition involves two cases:
 CASE 1:   If y ≥ 0 then sign(x,y) = abs(x)   , CASE 2:   If y < 0 then sign(x,y) = - abs(x)   .

The practical effect is that sign(x,y) has the same absolute value as x, but it has the same sign as y; thus the sign of y is transferred to x. (The case y = 0 is a little special - it gives sign(x,y) always a plus sign.)
Examples :
sign(2,3) = 2   ,   sign(2, -3) = - 2   ,   sign(-2,3) = 2   ,   sign(-2, -3) = - 2   .

The variables x and y in sign(x,y) may be integers or real numbers, and either single or double precision. (And x and y may even be of different types.)
If we substitute x = 1 in the sign transfer function, we get the sign of y; that is,
 CASE 1:   If y ≥ 0 then sign(1,y) = 1   , CASE 2:   If y < 0 then sign(1,y) = - 1   .

Thus, sign(1,y) in Fortran is essentially the same as the function SGN(y) in Basic (except when y = 0, when the Fortran value is + 1 but the Basic value is 0).

# Fortran: Lesson 3

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In Fortran Lesson 1 we briefly looked at the types of variables in Fortran. To avoid mistakes in Fortran arithmetic you must pay close attention to rules regarding working with numbers of the various types. Whereas Basic is more lenient, allowing some flexibility in mixing variables and numbers of different types, Fortran is less forgiving and will make you pay for oversights. In this lesson we look more closely at some of the rules and conventions that must be observed.

### Integers

An integer in Fortran is a whole number; it cannot contain commas or a decimal point. Examples of numbers considered integers by Fortran are
12    ,     -1311     ,     0     ,     +43     ,     123456789     .

For positive integers the plus sign is optional, but negative integers must be preceded by a minus sign. Examples of numbers not considered integers by Fortran are
22,547     ,     3.     ,     4.0     ,     -43.57 .

Because of the decimal points, Fortran will regard 3. and 4.0 as real numbers.
An integer N in GNU Fortran must lie within the range
- 2,147,483,648 ≤ N ≤ 2,147,483,647  .

One idiosyncrasy of Fortran is that when it performs arithmetic on integers, it insists on giving an answer that is likewise an integer. If the answer is not really an integer, Fortran makes it one by discarding the decimal point and all digits thereafter. For example, Fortran will assert that
11/8 = 1    ,    15/4 = 3    ,    -4/3 = -1    ,    -50/6 = -8    ,    2/3 = 0   .

If you want Fortran to give you the correct value of 11/8, you tell it to compute 11./8., so that it interprets the numbers as real numbers and produces the correct value 1.375. Integer arithmetic in Fortran can lead to other weird surprises - for instance, the distributive law of division is invalid, as demonstrated by the example
(2 + 3)/4 = 5/4 = 1     but     (2/4) + (3/4) = 0 + 0 = 0   .

Most of the built-in functions in Fortran apply to real numbers, and attempts to apply them to integers result in compiler error messages. The compiler will protest if you ask Fortran to compute sqrt(5), but it has no problem with sqrt(5.). Likewise, if you declare N to be an integer variable and ask Fortran to compute sqrt(N) or cos(N) or log(N), your program will not compile since these functions cannot act on integers. One way around this problem is to use the intermediate function
real(x)   ,

which converts x to a real number (if it is not already one). Then, for example,
real(5) = 5.    ,    sqrt(real(5)) = sqrt(5.) = 2.23606801  .

The compiler will have no objection if N is an integer variable and you ask Fortran to compute a composition like sqrt(real(N)) or cos(real(N)).

If you declare that A is an integer and later make the assignment A = 3.45, Fortran will not complain but it will truncate 3.45 and assign A the value A = 3. Likewise, if you insert the statement A = sqrt (5.), Fortran will truncate sqrt (5.) = 2.23606801 and deduce that A = 2. But errors such as these are easily avoided if you are careful to make correct type declaration statements for all variables at the beginning of your program.

### Single Precision Real Numbers

A real number, or more precisely a single precision real number, is written with a decimal point by Fortran, even when it is a whole number. The sequence of statements

									real x
integer y
x = 3
y = 3
print *, "x = ", x, " but y = ", y, "  -  weird!"

produces the output

x = 3. but y = 3  -  weird!

GNU Fortran uses up to 9 digits, not counting the decimal point, to represent real numbers. It will report that
sqrt (3.) = 1.73205078   ,   sqrt (1100.) = 33.1662483   ,   sqrt (2.25) = 1.5   .

Fortran can use also scientific notation to represent real numbers. The sequence "En" attached to the end of a number, where n is an integer, means that the number is to be multiplied by 10n. Here are various ways of writing the number 12.345:
1.2345E1  ,  .12345E2  ,  .012345E3  ,  12.345E0  ,  12345E-3  .

In working in single precision it is futile to assign more than 9 or 10 nonzero digits to represent a number, as Fortran will change all further digits to 0. (The 10th digit can affect how Fortran does the truncation.) The assignments
									x = 123456789876543.
x = 123456789800000.
x = 1234567898E5

produce the same result if x already has been declared a single precision real number. Note that commas are not used in representing numbers; as helpful as they might be to humans, computers find them unnecessary.

### Double Precision Real Numbers

A double precision real number in GNU Fortran can be represented by up to 17 digits before truncation occurs. Double precision numbers are written in scientific notation but with D usurping the role of E. Some various ways of writing the number 12.345 as a double precision real number are

1.2345D1  ,  .12345D2  ,  .012345D3  ,  12.345D0  ,  12345D-3 .

When assigning a value to a double precision variable you should use this D-scientific notation, as otherwise the value will be read only in single precision. For example, if A is double precision and you want to assign A the value 3.2, you should write

									A = 3.2D0

instead of just A = 3.2. (See Base 2 Conversion Errors below for more explanation.)

When a number is input from the keyboard in response to a "read *" command, the user need not worry about types or input format. Suppose for example that x is single or double precision, and the user is to enter a value for x in response to the command "read *, x". If the user enters simply "3" (integer format), GNU Fortran will change 3 to the proper format (to 3. if x is single precision and to 3D0 if x is double precision) before assigning it to x. Likewise, if x is double precision and the user enters 3.1 (single precision format), Fortran converts 3.1 to 3.1D0 before assigning it to x. (However, with an ordinary assignment statement "x = 3.1" from within the program, the number is not changed to double precision format before being assigned to x.)
A number x can be converted to double precision by the function
dble(x)   .

### Base 2 Conversion Errors

Whereas humans, having 10 fingers, do arithmetic in base 10, computers have no fingers but do arithmetic with on-off switches and therefore use base 2. As we know, some numbers have infinite decimal representations in base 10, such as
1/3 = .33333 …       ,       2/7 = .285714285714 …   .

There is no way to represent such numbers in base 10 with a finite number of digits without making a round-off error. Computers have the same problem working in base 2. In general, the only numbers representable with a finite number of digits in base 2 can be written in the form m/n, where m and n are integers and n is an integral power of 2. Examples are
6 (= 6/20)   ,   5/2   ,   3/8   ,   29/16   ,   537/256   ,   -3/1024 .

When we ask computers to do arithmetic for us, there is an inevitable source of error. We give the computer the numbers in base 10, and the computer must change them all over to base 2. For most numbers there is a round-off error, as the computer can work with only a finite number of digits at a time, and most numbers do not have a finite representation in base 2. If the computer is working in single precision Fortran, it works in about 9 digits (base 10), and so the round-off error will occur in about the 8th or 9th base 10 digit. In double precision this error appears much later, in about the 16th or 17th base 10 digit. If the arithmetic the computer performs is very complicated, these round-off errors can accumulate on top of each other until the total error in the end result is much larger. After the computer has done its job in base 2, it converts all numbers back to base 10 and reports its results.
Even if the computer does no arithmetic at all, but just prints out the numbers, the base 2 conversion error still appears. Here is a program illustrating the phenomenon:
									program demo
real x
double precision y, z
x = 1.1
y = 1.1
z = 1.1D0
print *, "x =", x, " , y =", y, " , z =", z
end
The somewhat surprising output when this program is run in GNU Fortran is
x = 1.10000002   ,   y = 1.10000002   ,   z = 1.1   .

The variable x is single precision, and base 2 conversion round-off error shows up in the 9th digit. Although y is double precision, it has the same round-off error as x because the value 1.1 is assigned to y only in single precision mode. (What happens is Fortran converts 1.1 to base 2 before changing it to double precision and assigning it to y.) Since z is double precision, and it is assigned the value 1.1 in double precision mode, round-off error occurs much later, far beyond the nine digits in which the results are printed. Thus the value of z prints exactly as it is received. Using write and format statements (see below), it is possible to print z using 17 digits; if you do so, you will find that Fortran reports z = 1.1000000000000001, where the final erroneous 1 appears as the 17th digit.
Base 2 round-off error occurs in the preceding example because 1.1 = 11/10, and 10 is not a power of 2. If you modify the program by replacing 1.1 with 1.125 = 9/8, there will be no round-off error because 8 = 23 is a power of 2 - so the values of x, y, and z will print exactly as assigned. (Try it!!)

### Mixed Type Arithmetic

In general, arithmetic in Fortran that mixes numbers of different types should be avoided, as the rules are quickly forgotten and mistakes are easily made. If Fortran is asked in some arithmetic operation to combine an integer number with a real one, usually it will wait until it is forced to combine the two and then convert the integer to real mode. Here are some calculations illustrating the process followed by Fortran, and showing why you should stay away from this nonsense:
 5. * (3 / 4) = 5. * 0 = 5. * 0. = 0. (5. * 3) / 4 = (5. * 3.) / 4 = 15. / 4 = 15. / 4. = 3.75 5. + 3 / 4 = 5. + 0 = 5. + 0. = 5. 5 + 3. / 4 = 5 + 3. / 4. = 5 + .75 = 5. + .75 = 5.75

If x and y are declared as double precision variables, and you want to multiply x by a number, say 2.1 for example, to get y, you should write
								y = 2.1D0 * x
Writing just y = 2.1 * x will retain single precision when 2.1 is converted to base 2, thereby introducing a larger base 2 round-off error and defeating your efforts at double precision. Similar remarks apply to other arithmetic operations. Errors of this nature are easily made when working in double precision. The best way to avoid them is to follow religiously this general rule:
Do not mix numbers of different types in Fortran arithmetic!!

### Exponentials and Roots

Already we point out an exception to the above rule - it is OK to use integers as exponents of real numbers. That is because, when serving as an exponent, an integer acts more as a "counter of multiplications" rather than as an active participant in the arithmetic. For instance, when Fortran does the calculation 1.25, it performs the multiplications
1.2 * 1.2 * 1.2 *1.2 * 1.2   ,

and the integer 5 never enters into the calculations! Thus, although it may appear so at first glance, the computation of 1.25 does not really mix an integer with a real number in any arithmetic operation. The same can be said of negative integers as exponents. The calculation of 1.2-5 involves multiplying five factors of 1.2, and then taking the reciprocal of the result - so the number -5 is not involved in the actual arithmetic.
Rational exponents must be handled carefully. A common mistake of novice Fortran programmers is to write something like 5 ** (2/3) and expect Fortran to compute the value of 52/3. But Fortran will view 2 and 3 as integers and compute 2/3 = 0, and conclude that 5 ** (2/3) = 5 ** 0 = 1. The correct expression for computing 52/3 is
5. ** (2./3.)   ,

wherein all numbers are viewed as real numbers.
Roots of numbers are computed in the same manner. To compute the seventh root of 3 you would use the expression
3. ** (1./7.)   .

If N is an integer variable and you wish to compute the N-th root of the real variable x, do not write x ** (1/N), as Fortran will interpret 1/N as 0 when N > 1. Instead write x ** (1./real (N)), so that 1 and N are first converted to real variables.

### Write and Format Statements

Just as in Basic we use TAB and PRINT USING commands to more precisely control program output, in Fortran we can use write commands with format statements. While these can get complicated, the most commonly used options are pretty easy to use. A typical write statement is
									write (*,20) x, y, z
The "*" in the parentheses instructs Fortran to write to the screen, while "20" refers to the label of the format statement for this write command. The x, y, and z are the variables to be printed. A format statement for this write command might be
20      format (3f10.4)    .

Inside the parentheses, the "3" indicates that 3 entities will be printed, the "f" denotes that these will be floating point real numbers (not exponential notation), the "10" stipulates that 10 places will be used for printing (counting the sign, decimal point, and the digits), and ".4" mandates 4 digits after the decimal point. Some printouts formatted this way are
12345.6789    ,    -1234.5678    ,    10002.3400   .

The letter "f" in this context is a format code letter; here are some of the more commonly used format code letters, with their implications:
 f real number, floating point format e single precision real number, exponential notation d double precision real number, exponential notation i integer a text string (character) x space / vertical space (line feed) t tab indicator

Strings (in quotes) may be placed in format statements, separated by commas. Here are examples of write statements with corresponding format statements; at the right of each is a description of the corresponding output:
 write (*,10) n, x, y 10   format (i4,4x,f10.4,2x,f10.4) integer n printed using 4 places, then 4 spaces, then real numbers x and y printed with 2 spaces between, each using 10 places and 4 decimal places write (*,20) area 20   format ("The area is ",f8.5) string in quotes is printed, then the real number area is printed, using 8 places with 5 decimal places write (*,30) "The area is ", area 30   format (a,f8.5) same output as immediately above write (*,40) x, y, z 40   format (3d20.14) 3 double precision numbers x, y, z printed, each reserving 20 spaces, with 14 decimal places write (*,50) student, score 50   format (a20,4x,i3) student, a text string up to 20 characters, is printed, then 4 spaces, then score, an integer using a maximum of 3 places write (*,60) r, A 60   format (t10,f4.2,/,t10,f6.2) tabs to column 10, prints real number r, goes to next line, tabs to column 10, prints real number A

You can use loops with format statements to print arrays; here are examples:
 do i = 1, 10           write (*,70) a(i)        end do 70   format (f5.2) an array a of real numbers, indexed from 1 to 10, is printed; each entry occupies 5 places with 2 decimal places, and is printed on a separate line write (*,80) (a(i), i = 1, 10) 80   format (f5.2) same output as immediately above write (*,90) (a(i), i = 1, 10) 90   format (10f5.2) same output as above, except that all entries are printed on the same line do i = 1, 5            write (*,7) (m(i,j), j = 1, 6) 7         format (6i3)        end do prints a 5 x 6 two-dimensional array m of integers, with each integer entry m(i,j) occupying 3 places. Each row of the matrix appears on its own line.

Here are other useful things to know about formatting:

1. If you do not specify a format, GNU Fortran will print real numbers using about 9 digits, even if you do calculations in double precision. If you want to print in double precision you must use write and format statements. When double precision is used the maximum number of digits possible is 17. A format specifier something like format (fm.n), where m is at least 20, is required to take full advantage of double precision.
2. If a value is too large to be printed in the specified format, Fortran will just print a string of asterisks (eg: ********** ). If you get such an output, you have to fix your format statement.
3. Real numbers are rounded off (not truncated) to fit the specified formatting.
4. If your formatting specifies more positions than the number requires, blanks are inserted to the left of the number.
5. Format statements may appear anywhere in a program after the variable declarations and before the end statement.
6. Unless your format statement is very simple, the chances are that your output won't look like you want on the first try - just fiddle with the formatting until you get it right.
Following are examples of stored values, formatting specifications for printing the values, and resulting output. (The "^" symbol indicates a blank).
 Stored Value Format Specifier Output 1.234567 f8.2 ^^^^1.23 0.00001 f5.3 0.000 -12345 i5 ***** -12345 i6 -12345 12345 i6 ^12345 0.00001234 e10.3 ^0.123E-04 0.0001234 e12.4 ^^0.1234E-03 1234567.89 e9.2 ^0.12E+07 aloha a8 ^^^aloha 1.23456789123D0 d17.10 ^0.1234567891E+01

# Fortran: Lesson 2

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We look at more of the commonly used features and commands of Fortran.

### Logical Expressions

A logical expression is a relation between variables or expressions that can have a value of TRUE or FALSE. Such expressions are used in "if Ö then" constructions and in loops, when testing whether to execute certain steps of a program. Relations and connectives appearing in logical expressions are listed in the following table; you will see how some of these are used in later examples.

 Relation/Connective Meaning .lt. less than .gt. greater than .le. less than or equal to .ge. greater than or equal to .eq. equals .ne. not equal to .and. and .or. or .not. not .xor. "exclusive" or (i.e., only one is true) .eqv. equivalent (i.e., same truth values) .neqv. not equivalent

### If ... Then ... Else Constructions

"If … Then … Else" constructions in Fortran are pretty much like those in Basic, with but a few minor modifications. First, instead of using in the tests symbols like "=", "<", ">=", etc., as in Basic, you must use the abbreviations in the preceding table. Also, tests must be enclosed in parentheses, and "else if" may be two words. Here are several examples:

1. if (x .gt. 0) print *, "x is positive"
2. if (x .ge. y .and. x .ge. z) go to 40
3. 											if (x .ge. 0) then
y = sqrt(x)
print *, y, " squared = ", x
end if
4. 											if (x .ge. 0) then
y = sqrt(x)
print *, y, " squared = ", x
else
print *, "x has no square root"
end if
5. 											if (x .ge. 0) then
print *, "x is positive"
y = sqrt(x)
else if (x .lt. 0) then
print *, "x is negative"
go to 60
else if (x .eq. 0) then
print *, "x is zero"
y = 0
end if

Observe that, as in examples 1) and 2), the one-line "if" statement does not use "then". Moreover, "else" appears on a line by itself, while "else if" shares the line with the test condition and "then".

### Stop

A stop statement stops the execution of a program. For instance, the sequence of statements below terminates the program whenever n is less than zero:

									if (n .lt. 0) then
print *, "Error - your age cannot be negative!"
stop
end if

Do not confuse stop and end. Use end only as the very last statement in the program, and use stop only to terminate the program before this last statement. Violating these rules will fatally confuse the compiler - it regards an end statement as the program's physical end.

### Labels and Go To

Labels and the "go to" statement work as in Basic, except that a label must be a number, and it must be typed in columns 2-5. Here is an example of a go to command directing the action to a labeled statement:

									if (x .lt. 0) go to 10
print *, "The square root of x is ", sqrt(x)
stop
10	print *, "x is negative and has no square root"

### Character Variables

A character variable is analogous to a string variable in Basic. A character variable must be declared at the beginning of the program, and attached to it in the declaration must be a number following an asterisk "*"; this number indicates the maximum number of symbols in the string. For example, the declaration statement
character name*20, ans*1

indicates that "name" is a character variable holding no more than 20 symbols, while "ans" is a character variable holding only one symbol.

A string in Fortran may be enclosed in either double quotes, as in "hello", or in single quotes, as in 'goodbye'.

### Do Loops

"For … Next" loops in Basic become "Do Loops" in Fortran. Such a loop begins with a do statement, and ends with either end do, or a labeled continue statement. Here are two loops that add the squares of the integers from 1 to 10:

										sum = 0
do i = 1, 10
sum = sum + i ** 2
end do
print *, "The sum is", sum
									sum = 0
do 5 i = 1, 10
sum = sum + i ** 2
5	continue
print *, "The sum is", sum

The end do and continue statements serve only to identify the end of the loop. The limits of the loop may be variables as well as numbers (e.g.: do i = m, n). As in Basic you may indicate a step size, which can be positive or negative. For example, the statement

do i = 1, 9, 2
specifies that the loop variable i run over the odd numbers 1, 3, 5, 7, 9.
Loops can be nested, and nested loops can end on the same continue statement (but not on the same end do statement). Here are two instances of nested loops assigning the entries of a 10 x 10 matrix:
										do i = 1, 10
do j = 1, 10
a(i,j) = i + j
end do
end do
									do 5 i = 1, 10
do 5 j = 1, 10
a(i,j) = i + j
5	continue

# Fortran: Lesson 1

# 1
# 1

### Introduction

Fortran is one of the oldest programming languages devised, but it is also still one of the most popular, especially among engineers and applied scientists. It was developed in the 1950's at IBM. Part of the reason for Fortran's durability is that it is particularly well-suited for mathematical programming; moreover, there are millions of useful programs written in Fortran, created at considerable time and expense, and understandably people are reluctant to trash these old programs and switch to a new programming language.

The name Fortran originally referred to "Formula Translation", but it has long since taken on its own meaning. There are several versions of Fortran around, among them Fortran 77, Fortran 90, and Fortran 95. (The number denotes the year of introduction.) Fortran 77 is probably still the most used, and it is the version installed on UHUNIX and in the UH math lab. Even though this semester we have thus far studied Basic, at the same time we have studied Fortran, because commands and procedures are very similar in the two languages. Moving from QuickBasic to Fortran is more a matter of change of terminology than anything else.

### Editing Fortran

Unlike in Basic, a Fortran program is not typed in a "Fortran window". Instead, a program is typed and saved with an editor (i.e., a word processor), and the program is then turned into an executable file by a Fortran compiler. To begin the process of creating a Fortran program in the math lab, you must open an editor. It is preferable to use a simple editor - such as Notepad or the DOS editor - because fancy word processors might add extraneous formatting notation that will hang up Fortran.

A most peculiar feature of Fortran 77 is its line structure, which is a carryover from the old days when programs were typed on punch cards. A punch card had 80 columns, and so does a line of Fortran code. A "c" in column 1 indicates a comment (similar to REM in Basic). Columns 2-5 (usually left blank) are reserved for line numbers. Column 6 is used only to indicate a continuation of a line too long to fit on the card. Columns 7-72 contain the instructions of the program. Columns 73-80 were originally used for numbering the punch cards, but are rarely used nowadays - leave them blank and the compiler will ignore them.

Fortran is case insensitive - that is, it does not distinguish between capital and small letters. Thus x and X refer to the same variable. Many programmers for simplicity use all small letters, but you may do as you like. Also, after column six Fortran does not recognize spaces (except for spaces inside quotations as in print statements). In general, spaces are mostly for the purpose of making code more readable by humans. When you type a Fortran program with an editor, make certain the editor indents more than six spaces; then if you begin every line with an indent you do not have to worry about counting six spaces at the beginnings of lines.

Let us go through the steps of editing, compiling, and running a short program. First open Notepad under Windows, or type "edit" (and return) under a DOS prompt to open the DOS editor. (When you double-click the Fortran icon on a math lab computer, you get a DOS prompt.) Beginning each line with an indent (except for the fourth line, where the "c" must be placed in the first column), type the program exhibited below; the program computes the area of a circle of radius r, as input by the user. The resulting file that you save is called the source file for the program.

									program circlearea
real r, area, pi
parameter (pi = 3.14159)
c	This program computes the area of a circle.
print *, "What is the radius?"
area = pi * r ** 2
print *, "The area is", area
print *, "Bye!"
end

The first statement above gives the program name, the second declares that "r", "area", and "pi" will be single precision real quantities, and the third announces that pi has the value 3.14159. The fourth statement, beginning with "c" in column 1, is a comment describing what the program does; such comments are for the benefit of the programmer and are ignored by Fortran. The fifth statement prompts the user for the radius of the circle, and the sixth accepts this input. The seventh statement computes the area and the eighth informs the user of this area. Finally, the last two statements bid goodbye and terminate the program.

The name for a source file in Fortran must end with the extension ".f" before the compiler recognizes it. After you have typed the above program, save the file as area.f. (If you type the file in Notepad, include the whole name in quotes when you save it, as otherwise the extension .txt will be added to the name.) The file will be saved to your h directory in the math lab. Under a DOS prompt you can view the files in this directory by typing dir and enter; under Windows you can double-click "My Computer" and then the icon for the h drive.

### Compiling

After you have created and saved a source file, you next must compile this file. Open a Fortran window and enter

g77 name.f

where in place of name you insert the name of your source file. (If the source file resides in a directory different from that of the Fortran program, you will have to include also the directory path of the file.) To compile the file of our example above, in the math computer lab you just enter g77 area.f.

If your program has mistakes (which usually happens on the first attempt at compiling), instead of a compiled file you will get Fortran error messages pointing out problems. Some of these messages can be hard to decipher, but after reading hundreds of them you will get better at it. If your program has no mistakes Fortran will simply return a DOS prompt - that is good news because it means Fortran has successfully created a compiled file. By default this new file is given the name a.exe. (You can give the compiled file a name of your own choosing by typing g77 area.f -o name.exe to compile the program - but usually there is no reason not to accept the default name.) Your compiled file, also located in the h directory, is now executable - that means the program is ready to run.

### Running a Program

If your compiled file has the default name a.exe, you simply type a and return to run it (or name and return if you gave the file another name). After you run the program and see how it works, you can return to your editor and revise it as you wish. It is perhaps better to keep two windows open - both the Fortran window and the editing window - so that you can quickly switch from one to the other with a mouse-click. After revising a program, you must save and compile it again before changes take effect.

If you do enough Fortran programming, sooner or later you will err and create and run a program that never stops. In such a situation, type "Control-C" to interrupt the execution of the program.

Now that we have discussed the basic nuts and bolts of creating and running a Fortran program, we discuss some terminology and commands. You will probably find that most of these remind you of similar things in Basic.

### Program

Every Fortran program must begin with a program line, giving the name of the program. Here are examples:

									program quadratic
program mortgage
program primes

### Variables, Declarations, Types

After the program name come the declaration statements, stating the types of the variables used in the program. A variable name consists of characters chosen from the letters a-z and the digits 0-9; the first character of the name must be a letter. You are not allowed to use your program name as a variable, nor are you allowed to use words reserved for the Fortran language, such as "program", "real", "end", etc.
The variable types in Fortran are
1)  integer (in the range from about - 2 billion to + 2 billion)
2)  real (single precision real variable)
3)  double precision (double precision real variable)
4)  character (string variable)
5)  complex (complex variable)
6)  logical (logical variable)
As illustration, the declaration statements
										real r, area
integer M, N
double precision a, b

declare that r and area are single precision real variables, that M and N are integers, and that a and b are double precision real variables.

If you do not declare the type of a variable, Fortran will by default make it an integer if it starts with one of the letters i through n, and will make it a single precision real variable otherwise. However, it is normal (and good) programming practice to declare the type of every variable, as otherwise mistakes are easily made.

The implicit quantifier before a type declaration makes all variables starting with the listed letters of the specified type. For example, the declarations

									implicit integer (i-m)
implicit real (r-t)

make variables starting with i, j, k, l, m integers, and those starting with r, s, t real. However, the implicit quantifier is probably best avoided, as programmers with short memories will make mistakes.

A declaration statement is nonexecutable - that is, it provides information but does not instruct Fortran to carry out any action. Declarations must appear before any executable statement (a statement that does tell Fortran to take some action).

### Assignment

The equals sign "=" assigns the variable on the left side the value of the number or expression on the right side (exactly as in Basic).

### Parameter

The parameter statement works like CONST in Basic - it specifies a value for a constant. The syntax is
parameter (name = constant expression)

where name is replaced by the name of the constant, and constant expression by an expression involving only constants. Thus
parameter (pi = 3.14159)

specifies a value for the constant pi, while the succeeding statement
									parameter (pi = 3.14159)
parameter (a = 2* pi, b = pi/2)

fixes values of new constants a and b in terms of the old constant pi. Remember that once a constant is defined you are not allowed to change its value later.

All parameter statements must appear before the first executable statement.

A comment is similar to an REM statement in Basic. You can indicate a comment by placing a "c" in column 1 and then the comment in columns 7-72. Alternatively, you can use an exclamation point "!" to indicate a comment; it may occur anywhere in the line (except columns 2-6). Everything on a line after an exclamation point becomes a comment.

### Print *

The command "print *" is analogous to PRINT in Basic; it instructs Fortran to print items to the screen. Examples are

									print *, x
print *, "The solution is ", x
print *, 'The radius is', r, 'and the area is', area

Note that a comma follows "print *", and that commas (instead of semicolons as in Basic) appear between successive items to be printed. Observe also that either double or single quotes may enclose strings. The command "print *" on a line by itself (without a comma) serves as a line feed.

The command "read *" is analogous to INPUT in Basic. Examples are

								read *, radius
read *, A, B, C

In the first example the program pauses to allow the user to enter the radius. In the second example the user types the values of A, B, and C, separated by returns; alternatively, the user can type A, B, and C separated only by commas, and then one final return.

### End

The end statement marks the end of the main Fortran program or of a subprogram. (It cannot be used in the middle of the program, as in Basic.)

### Operations of Arithmetic

Here are the common arithmetical operations in Fortran:

 Addition x + y Subtraction x - y Multiplication x * y Division x / y Exponentiation x ** y

Fortran performs exponentiations first, then multiplications and divisions, and lastly additions and subtractions. (When in doubt, use parentheses!)

Be careful with division. If m and n are integers, then m/n is truncated to its integer part. Thus 3/4 is evaluated as 0, and 25/6 as 4. When working with constants rather than variables you can avoid this problem by using periods after integers. For example 3./4. is evaluated in the standard way as .75, as Fortran treats 3. and 4. as real variables rather than as integers.

### Intrinsic Functions

Many standard mathematical functions are built into Fortran - these are called intrinsic functions. Below is a table of some of the functions most commonly used in mathematical programming. All trig functions work in radians. (Note that arguments of functions must be enclosed in parentheses.)

									abs(x)
c	absolute value of x

acos(x)
c	arccosine of x

asin(x)
c	arcsine of x

atan(x)
c	arctangent of x

cos(x)
c	cosine of x

cosh(x)
c	hyperbolic cosine of x

dble(x)
c	converts x to double precision type

exp(x)
c	exponential function of x (base e)

log(x)
c	natural logarithm of x (base e)

mod(n,m)
c	remainder when n is divided by m

real(x)
c	converts x to real (single precision) type

sign(x,y)
c	changes the sign of x to that of y

sin(x)
c	sine of x

sinh(x)
c	hyperbolic sine of x

sqrt(x)
c	square root of x

tan(x)
c	tangent of x

tanh(x)
c	hyperbolic tangent of x