Fortran (FORmula TRANslation) was introduced in 1957 and remains the language of choice for most scientific programming. The latest standard, Fortran 90, includes extensions that are familiar to users of C. Some of the most important features of Fortran 90 include recursive subroutines, dynamic storage allocation and pointers, user defined data structures, modules, and the ability to manipulate entire arrays.
Fortran 90 is compatible with Fortran 77 and includes syntax that is no longer considered desirable. F is a subset of Fortran 90 that includes only its modern features, is compact, and is easy to learn.
|1. Introduction||2. Do construct||3. If construct||4. Formatted output|
|5. Files||6. Arrays||7. Allocate||8. Random number sequences|
|9. Recursion||10. Character variables||12. Complex variables||14. References|
Instructions for running F on Unix workstations.
In the following, we will assume that the reader is familiar with a programming language such as True BASIC. We first give an example of an F program.
This program is similar to Program product in Chapter 2 of Gould and Tobochnik. The features of F included in the above program include:program product_example real :: m, a, force m = 2.0 ! mass in kilograms a = 4.0 ! acceleration in mks units force = m*a ! force in newtons print *, force end program product_example
The first statement must be a program statement; the last statement must have a corresponding end program statement.
Integer numerical variables and floating point numerical variables are distinguished. The names of all variables must be between 1 and 31 alphanumeric characters of which the first must be a letter and the last must not be an underscore.
The types of all variables must be declared.
Real numbers are written as 2.0 rather than 2.
The case is significant, but two names that differ only in the case of one or more letters cannot be used together. All keywords (words that are part of the language and cannot be redefined.) are written in lower case. Some names such as product are reserved and cannot be used as names.
Comments begin with a ! and can be included anywhere in the program.
Statements are written on lines which may contain up to 132 characters.
The asterisk (*) following print is the default format.
We next introduce syntax that allows us to enter the desired values of m and a from the keyboard. Note the use of the (unformatted) read statement and how character strings are printed.
2. Do constructprogram product_example2 real :: m, a, force ! SI units print *, "mass m = ?" read *, m print *, "acceleration a = ?" read *, a force = m*a print *, "force (in newtons) =", force end program product_example2
Note that n is an integer variable. In this case the do statement specifies the first and last values of n; n increases by unity (default). The block of statements inside a loop is indented for clarity.program series integer :: n real :: sum_series ! sum is a keyword sum_series = 0.0 ! add the first 100 terms of a simple series do n = 1, 100 sum_series = sum_series + 1.0/real(n)**2 print *, n,sum_series end do end program series
Because the product n*n is done using integer arithmetic, it is better to convert n to a real variable before the multiplication is done. Also note that exponentiation is done using the operator **.
3. If construct
In the next program example, the do loop is exited by satisfying a test.
The new features of F included in the above program include:program series_test ! illustrate use of do construct integer :: n ! choose large value for relative change real :: sum_series, newterm, relative_change n = 0 sum_series = 0.0 do n = n + 1 newterm = 1.0/(n*n) sum_series = sum_series + newterm relative_change = newterm/sum_series if (relative_change < 0.0001) then exit end if print *, n,relative_change,sum_series end do end program series_test
A do construct can be exited by using the exit statement.
The if construct allows the execution of a sequence of statements (a block) to depend on a condition. The if construct is a compound statement and begins with if ... then and ends with end if. The block inside the if construct is indented for clarity. Examples of more general if constructs using else and else if statements are given in Program test_factorial.
|less than or equal||<=|
|greater than or equal||>=|
The following program illustrates the use of a kind parameter and a named do construct:
program series_example ! illustrate use of kind parameter and named do loop integer, parameter :: double = 8 integer :: n real (kind = double) :: sum_series, newterm, relative_change n = 0 sum_series = 0.0 print_change: do n = n + 1 newterm = 1.0/real(n, kind = double)**2 sum_series = sum_series + newterm relative_change = newterm/sum_series if (relative_change < 0.0001) then exit print_change end if print *, n,relative_change,sum_series end do print_change end program series_example
Variables may have a particular hardware representation such as double precision by using the kind = double in parenthesis after the keyword real representing the data type. A more general use of the parameter statement is given in Program drag.
Double precision in Fortran 90 on the SGI and in F on the Power Macintosh is done by letting
double = 8However double precision in F on the SGI is done by letting
double = 2
The do and end do statements must either have the same name or both be unnamed. In general, a do construct is named to make explicit which do construct is exited in the case of nested do constructs. The use of a named do construct in the above example is unnecessary and is for illustrative purposes only.
Subprograms are called from the main program or other subprograms. As an example, the following program adds and multiplies two numbers that are inputed from the keyboard. Note that the variables x and y are public and are available to the main program.
module common public :: initial,add,multiply integer, parameter, public :: double = 8 real (kind = double), public :: x,y contains subroutine initial() print *, "x = ?" read *,x print *, "y = ?" read *,y end subroutine initial subroutine add(sum2) real (kind = double), intent (in out) :: sum2 sum2 = x + y end subroutine add subroutine multiply(product2) real (kind = double), intent (in out) :: product2 product2 = x*y end subroutine multiply end module common program tasks ! illustrate use of module and subroutines ! note how variables are passed use common real (kind = double) :: sum2, product2 call initial() ! initialize variables call add(sum2) ! add two variables call multiply(product2) print *, "sum =", sum2, "product =", product2 end program tasks
Subprograms (subroutines and functions) are contained in modules. The form of a module, subroutine, and a function is similar to that of a main program.
A module is accessed in the main program by the use statement.
Subroutines are invoked in the main program by using the call statement.
A subprogram always has access to other entities in the module.
The subprograms in a module are preceded by a contains statement.
Variables and subprograms may be declared public in a module and be available to the main program (and other modules).
Information can also be passed as an argument to each subprogram as are the variables sum2 and product2. A open parenthesis () is needed even if there are no arguments. The intent of each dummy argument of a program must be indicated.
intent in means that the dummy argument cannot be changed within the subprogram.
4. Formatted output<br> The structure of Program cool is similar to Program tasks. Note the use of the modulo function and the use of format specifications. In place of the asterisk * denoting the default format, we have used a format specification which is a list of edit descriptors. An example from Program cool is
The t (tab) edit descriptor is used to skip to a specified position on an output line. The edit descriptor a (alphanumeric) is for character strings. An example of the f (floating point) descriptor is given byprint "(t7,a,t16,a,t28,a)", "time","T_coffee","T_coffee - T_room"
The edit descriptor f13.4 means that a total of thirteen positions are reserved for printing a real value rounded to 4 places after the decimal point. (The decimal point and a minus sign occupy two out of the thirteen positions.) The edit descriptor 2f13.4 means that the edit descriptor f13.4 is done twice. The other common edit descriptor is i (integer).print "(f10.2,2f13.4)",t,T_coffee,T_coffee - T_room
Comment on Program drag
The only new syntax in Program drag is the use of the parameter statement:
A parameter is a named constant. The value of a parameter is fixed by its declaration and cannot be changed during the execution of a program.real (kind = double), public, parameter :: g = 9.8
Program save_data illustrates how to open a new file, write data in a file, close a file, and read data from an existing file.
Input/output statements refer to a particular file by specifying its unit. The read and write statements do not refer to a file directly, but refer to a file number which must be connected to a file. There are many variations on the open statement, but the above example is typical. The values of the action specifier are read, write, and readwrite (default). Values for status are old, new, replace, or scratch.program save_data ! illustrate writing and reading file integer :: i,j,x character(len = 32) :: file_name print *, "name of file?" read *, file_name open (unit=5,file=file_name,action="write",status="new") do i = 1,4 x = i*i write (unit=5,fmt=*) i,x end do close(unit=5) ! open(unit=1,file=file_name,action="read",status="old") open(unit=1,file=file_name,position="rewind",action="read",status="old") do i = 1,4 read (unit=1,fmt = *) j,x print *, j,x end do close(unit=1) end program save_data
If we plan to reuse data on the same system with the same compiler, we can use unformatted input/output to save the overhead, extra space, and the roundoff error associated with the conversion of the internal representation of a value to its external representation. Of course, the latter is machine and compiler dependent. Unformatted access is very useful when data is generated by one program and then analyzed by a separate program on the same computer. To generate unformatted files, omit the format specification. Examples of programs which use direct access and records are available.
The definition and use of arrays is illustrated in Program vector.
The main features of arrays include:module common public :: initial,cross contains subroutine initial(a,b) real, dimension (:), intent(out) :: a,b a(1:3) = (/ 2.0, -3.0, -4.0 /) b(1:3) = (/ 6.0, 5.0, 1.0 /) end subroutine initial subroutine cross(r,s) real, dimension (:), intent(in) :: r,s real, dimension (3) :: cross_product ! note use of dummy variables integer :: component,i,j do component = 1,3 i = modulo(component,3) + 1 j = modulo(i,3) + 1 cross_product(component) = r(i)*s(j) - s(i)*r(j) end do print *, "" print *, "three components of the vector product:" print "(a,t10,a,t16,a)", "x","y","z" print *, cross_product end subroutine cross end module common program vector ! illustrate use of arrays use common real, dimension (3) :: a,b real :: dot call initial(a,b) dot = dot_product(a,b) print *, "dot product = ", dot call cross(a,b) end program vector
An array is declared in the declaration section of a program, module, or procedure using the dimension attribute. Examples include
real, dimension (10) :: x,y
real, dimension (1:10) :: x,y
integer, dimension (-10:10) :: prob
integer, dimension (10,10) :: spin
The default value of the lower bound of an array is 1. For this reason the first two statements are equivalent to the first.
The lower bound of an array can be negative.
The last statement is an example of two-dimensional array.
Rather than assigning each array element explicitly, we can use an array constructor to give an array a set of values. An array constructor is a one-dimensional a list of values, separated by commas, and delimited by "(/" and "/)". An example is
is equivalent to the separate assignmentsa(1:3) = (/ 2.0, -3.0, -4.0 /)
a(1) = 2.0 a(2) = -3.0 a(3) = -4.0
Note that the array cross_product can be referenced by one statement:
print *, cross_product
F has many vector and matrix multiplication functions. For example, the function dot_function operates on two vectors and returns their scalar product. Some useful array reduction functions are maxval, minval, product, and sum.
7. Allocate statement
One of the better features of Fortran 90 is dynamic storage allocation. That is, the size of an array can be changed during the execution of the program. The use of the allocate and deallocate statements are illustrated in the following. Note the use of the implied do loop.
An example of passing arrays:program dynamic_array ! simple example of dynamic arrays real, dimension (:), allocatable :: x integer :: i,N N = 2 allocate(x(N:2*N)) ! implied do loop x(N:2*N) = (/ (i*i, i = N, 2*N) /) print *, x deallocate(x) allocate(x(N:3*N)) x = (/ (i*i, i = N, 3*N) /) print *, x end program dynamic_array
module param integer, public, parameter :: double = 8 end module param module common use param private public :: initial integer, public :: N contains subroutine initial(x) real (kind = double), intent(inout), dimension(:) :: x N = 100 x(1) = 1.0 end subroutine initial end module common program test use param use common real (kind = double), allocatable,dimension (:) :: x N = 10 allocate(x(N)) call initial(x) end program test
8. Random number sequences
Fortran 90 includes several useful built-in procedures. One of the most useful ones is subroutine random_number. Although it is a good idea to write your own random number generator using an algorithm that you have tested on the particular problem of interest, it is convenient to use subroutine random_number when you are debugging your program or if accuracy is not important. The following program illustrates several uses of subroutine random_number and random_seed. Note that the argument rnd of random_number must be real, has intent out, and can be either a scalar or an array.
Note how subroutine random_seed is used to specify the seed. This specification is useful when the same random number sequence is used to test a program.program random_example real :: rnd real, dimension (:), allocatable :: x integer, dimension(2) :: seed, seed_old integer :: L,i,n_min,n_max,ran_int,sizer ! generate random integers between n_min and n_max ! dimension of seed is one in F and two in Fortran 90. call random_seed(sizer) print *, sizer ! illustrate use of put and get seed(1) = 1239 seed(2) = 9863 ! need for Fortran 90 call random_seed(put=seed) call random_seed(get=seed_old) ! confirm value of seed print *, "seed = ", seed_old L = 100 ! length of sequence n_min = 1 n_max = 10 do i = 1,L call random_number(rnd) ran_int = (n_max - n_min + 1)*rnd + n_min print *,ran_int end do ! assign random numbers to array x as another example allocate(x(L)) call random_number(x) print "(4f13.6)", x call random_seed(get=seed_old) print *, "seed = ", seed_old end program random_example
A simple example of a recursive definition is the factorial function:
A recursive definition of the factorial is
factorial(n) = n! = n(n-1)(n-2) ... 1
A program that closely parallels the above definition follows. Note how the word recursive is used.
factorial(1) = 1 factorial(n) = n factorial(n-1)
module fact public :: f contains recursive function f(n) result (factorial_result) integer, intent (in) :: n integer :: factorial_result if (n <= 0) then factorial_result = 1 else factorial_result = n*f(n-1) end if end function f end module fact program test_factorial use fact integer :: n print *, "integer n?" read *, n print "(i4, a, i10)", n, "! = ", f(n) end program test_factorial
A less simple example (taken from pp. 98-99 in The Fun of Computing,John G. Kemeny, True BASIC (1990)) is given two integers, n and m, what is their greatest common divisor, that is, the largest integer that divides both? For example, if n = 1000 and m = 32, than the greatest common divisor (gcd) is gcd = 8.
One method for finding the gcd is to integer divide n by m. We write n = q m + r, where q is the quotient and r is the remainder. If r = 0, then m divides n and m is the gcd. Otherwise, any divisor of m and r also divides n, and hence gcd(n,m) = gcd(m,r). Because r < m, we have made progress. As an example, take n = 1024 and m = 24. Then q = 42 and r = 16. So we want gcd(24,16). Now q = 1 and r = 8 and we calculate gcd(16,8). Finally q = 2, and r = 0 so gcd = 8. The following program implements this idea.
module gcd_def public :: gcd contains recursive function gcd(n,m) result (gcd_result) integer, intent (in) :: n,m integer :: gcd_result integer :: remainder remainder = modulo(n,m) if (remainder == 0) then gcd_result = m else gcd_result = gcd(m,remainder) end if end function gcd end module gcd_def program greatest use gcd_def integer :: n,m print *, "enter two integers n, m" read *, n,m print "(a,i6,a,i6,a ,i6)", "gcd of",n," and",m,"=",gcd(n,m) end program greatest
The example of recursion given in almost all introductory textbooks is the towers of Hanoi. To save space, a discussion is given elsewhere together with the program. (not finished)
The volume of a d-dimensional hypersphere of unit radius can be related to the area of a (d - 1)-dimensional hypersphere. The following program uses a recursive subroutine to integrate numerically a d-dimensional hypersphere:
module common public :: initialize,integrate integer, parameter, public :: double = 8 real (kind = double), parameter, public :: zero = 0.0 real (kind = double), public :: h, volume integer, public :: d contains subroutine initialize() print *, "dimension d?" read *, d ! spatial dimension print *, "integration interval h?" read *, h volume = 0.0 end subroutine initialize recursive subroutine integrate(lower_r2, remaining_d) ! lower_r2 is contribution to r^2 from lower dimensions real(kind = double),intent (in) :: lower_r2 integer, intent (in) :: remaining_d ! # dimensions to integrate real (kind = double) :: x x = 0.5*h ! mid-point approximation if (remaining_d > 1) then lower_d: do call integrate(lower_r2 + x**2, remaining_d - 1) x = x + h if (x > 1) then exit lower_d end if end do lower_d else last_d: do if (x**2 + lower_r2 <= 1) then volume = volume + h**(d - 1)*(1 - lower_r2 - x**2)**0.5 end if x = x + h if (x > 1) then exit last_d end if end do last_d end if end subroutine integrate end module common program hypersphere ! original program by Jon Goldstein use common call initialize() call integrate(zero, d - 1) volume = (2**d)*volume ! only considered positive octant print *, volume end program hypersphere
10. Character variables
The only instrinsic operator for character expressions is the concatenation operator //. For example, the concatenation of the character constants string and beans is written as
"string"//"beans"The result, stringbeans, may be assigned to a character variable.
A useful example of concatenation is given in the following program.
Note the use of the write statement to build a character string for numeric and character components.program write_files ! test program to open n files and write data integer :: i,n character(len = 15) :: file_name n = 11 do i = 1,n ! assign number.dat to file_name using write statement write(unit=file_name,fmt="(i2.2,a)") i,".dat" ! // is concatenation operator file_name = "config"//file_name open (unit=1,file=file_name,action="write",status="replace") write (unit=1, fmt=*) i*i,file_name close(unit=1) end do end program write_files
11. Complex variables
Fortran 90 is uniquely suited to handle complex variables. The following program illustrates the way complex variables are defined and used.
program complex_example integer, parameter :: double = 2 real (kind = double), parameter :: pi = 3.141592654 complex (kind = double) :: b,bstar,f,arg real (kind = double) :: c complex :: a integer :: d ! A complex constant is written as two real numbers, separated by ! a comma and enclosed in parentheses. a = (2,-3) ! If one of part has a kind, the other part must have same kind b = (0.5_double,0.8_double) print *, "a =", a ! note that a has less precision than b print *, "a*a =", a*a print *, "b =", b print *, "a*b =", a*b c = real(b) ! real part of b print *, "real part of b =", c c = aimag(b) ! imaginary part of b print *, "imaginary part of b =", c d = int(a) print *, "real part of a (converted to integer) =", d arg = cmplx(0.0,pi) b = exp(arg) ! done in two lines for ease of reading only bstar = conjg(b) ! complex conjugate of b f = abs(b) ! absolute value of b print *, "properties of b =", b,bstar,b*bstar,f end program complex_example
Walter S. Brainerd, Charles H. Goldberg, and Jeanne C. Adams, Programmer's Guide to F, Unicomp (1996).
Michael Metcalf and John Reid, The F Programming Language, Oxford University Press (1996).
Program save_data was modified by Ty Faechner, 18 June 2003.
Please send comments and corrections to Harvey Gould, firstname.lastname@example.org.Updated 6 January 2006.