Fortran Language

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Fortran (FORmula TRANslation) is a high level language aimed at numerical calculations.

Fortran derived from Formula Translation is a general-purpose, compiled imperative programming language that is especially suited to numeric computation and scientific computing.

Originally developed by IBM in the 1950s for scientific and engineering applications, FORTRAN came to dominate this area of programming early on and has been in continuous use for over half a century in computationally intensive areas such as numerical weather prediction, finite element analysis, computational fluid dynamics, computational physics, crystallography and computational chemistry. It is a popular language for high-performance computing and is used for programs that benchmark and rank the world’s fastest supercomputers.

Fortran encompasses a lineage of versions, each of which evolved to add extensions to the language while usually retaining compatibility with prior versions. Successive versions have added support for structured programming and processing of character-based data (FORTRAN 77), array programming, modular programming and generic programming (Fortran 90), high performance Fortran (Fortran 95), object-oriented programming (Fortran 2003) and concurrent programming (Fortran 2008).

Fortran’s design was the basis for many other programming languages. Among the better known is BASIC, which is based on FORTRAN II with a number of syntax cleanups, notably better logical structures, and other changes to more easily work in an interactive environment. (wiki)
Recommended textbooks:Fortran 90 Explained, Metcalf and Reid, Oxford Science Publications

  • Fortran 90/95 Explained, Metcalf and Reid, Oxford University Press
  • Fortran 95/2003 Explained, Metcalf, Reid and Cohen, Oxford University Press

Internet resources are



There are a number of free compilers available

  • Online compiler
  • Fortran 77 – F77 part of the GNU GCC project
  • Fortran 95 – g95 based on the GNU GCC project
  • Fortran 95 – gfortran part of the GNU GCC 4.0 project
  • Fortran 95 – ifort Intel Fortran compilers for linux



Here is an example Fortran code:

program circle
real  :: r, area
!This program reads a real number r and prints
!the area of a circle with radius r
read (*,*) r
area = 3.14159*r*r
write (*,*)' Area = ',area
end program circle

Note that all variables are declared at the beginning of the program and before they are used.



  • Fortran77 requires that all lines except comment or continuation lines start in the 7th column (convention left over from Fortran coding sheets) and are restricted to 72 columns as follows:
    • Col 1 : Blank or c, C or * for comments
    • Col 1-5 : Statement label (optional)
    • Col 6 : Continuation of previous line (optional)
    • Col 7-72 : Statements
  • Fortran is not case specific.
  • Comments can appear anywhere in a program – use them.
  • Any symbol can be used in the 6th column for continuation but the general practice is to use +, & or numbers.
  • Implicit typing depending on first letter of variable name, i-n are integers. This easily leads to mistakes so don’t rely on it.



The standard Fortran77 layout (i.c. columns 7-72) is called fixed format but with compiler options can be extended to a line width of 132.

Some compilers accept the non-standard tab source form where tab automatically skips to the 7th column. Then line continuations can be a number in the 8th column.

Free source form is generally used for Fortran90. Continuation lines are marked by an ampersand & at the end of the line that is to be continued.



Fortran90 compilers are backward compatible so will interpret Fortran77 code. In fact, Fortan77 is a subset of Fortran90.

If you are modifying existing Fortran77 code you will have to continue to use the fixed format layout in any program that is already in fixed format.

Compilers decide whether code is fixed or free format depending on the extension of the file name e.g. prog.f or prog.f90

From now on we will present Fortran90 but mention any Fortran77 constructs that are now deprecated and replaced by Fortran90 constructs.





Start every program with the statement

    implicit none

This tells the compiler that all variables are to be declared and the compiler will report an error if any variable is not declared. This is good programming practice. Do not rely on the compiler to decide if variables are integers or real. This is different from e.g. Matlab


Variables can be of the following types:

  • REAL
  • user constructed called derived types


For example the following all declare x as a single precision real variable, that is, its representation in memory is 4 bytes long

  • real :: x
  • real(4) :: x
  • real*4 :: x


A double precision variable z which takes 8 bytes of memory can be represented in any of the following ways:

  • real(8) :: z
  • real*8 :: z
  • double precision :: z


  • COMPLEX (COMPLEX*8 or COMPLEX(4)) – single precision complex number
  • COMPLEX(8) or COMPLEX*16 – double precision complex number
  • CHARACTER(LEN=n), CHARACTER(n), or CHARACTER*n where n is an integer value represents the number of bytes in the character variable or can be *.
  • LOGICAL can be .TRUE. or .FALSE.



Here are some examples of array declarations, there are several ways to do this.

  • a is a real one-dimensional array of length 4, indexed 1 to 4.
    real :: a(4)
  • a is a real one-dimensional array of length 20
    integer, parameter :: n=20
    real  :: a(n)
  • b is an integer one-dimensional array of length 20, indexed 0 to 19
    integer :: b(0:19)
  • c is double precision 2-dimensional array
    double precision, dimension(10,10) :: c

In these the array a has elements a(1) to a(20), b has elements b(0) to b(19) and c has elements c(1,1) to c(10,10)



Compile and execute the following code variables.f90 by doing

    >f90 variables.f90 -o variables
program variables

implicit none

integer    :: i
integer(2) :: j
integer(4) :: k
integer(8) :: l

real    :: a
real(4) :: b
real(8) :: c

write (*,*) ' Huge:   ',huge(i),huge(j),huge(k),huge(l)
write (*,*) ' Digits: ',digits(i),digits(j),digits(k),digits(l)

write (*,*) ''

write (*,*) 'Huge:   ',huge(a),huge(b),huge(c)
write (*,'(a,i,i,i)') 'Digits: ',digits(a),digits(b),digits(c)
write (*,'(a8,e,e,e)') 'Epsilon: ',epsilon(a),epsilon(b),epsilon(c)
write (*,'(a9,en,en,en)') 'Epsilon: ',epsilon(a),epsilon(b),epsilon(c)
write (*,'(a10,3es)') 'Epsilon: ',epsilon(a),epsilon(b),epsilon(c)
write (*,'(a11,3e20.10e4)') 'Epsilon: ',epsilon(a),epsilon(b),epsilon(c)

end program variables



The output from variables.f90 is

 Huge:     2147483647  32767  2147483647   9223372036854775807
 Digits:           31          15          31          63
 Huge:     3.4028235E+38  3.4028235E+38  1.797693134862316E+308
Digits:           24          24          53
Epsilon:  0.1192093E-06  0.1192093E-06   0.2220446049250313E-15
Epsilon: 119.2092896E-09119.2092896E-09 222.0446049250313081E-18
 Epsilon:   1.1920929E-07  1.1920929E-07   2.2204460492503131E-16
  Epsilon:   0.1192092896E-0006  0.1192092896E-0006  0.2220446049E-0015

Note how the output is presented and how this relates to the write statements.

  • * is a special character meaning standard or default
  • a represents ASCII characters
  • i represents integers
  • e represents exponential notation
  • es represents scientific notation (similar to e)
  • en represents engineering notation


huge and digits represent intrinsic functions in the Fortran language and are defined in the standard. huge returns the largest number representable by a number of the same type and kind as the argument. digits returns the number of significant digits for numbers of the same type and kind as the argument.



In Fortran90 the KIND parameter was introduced to allow more flexibility for the user in declaring variable precision. The following code shows how KIND can be used.

program kindtype

implicit none

integer, parameter   :: i4=SELECTED_INT_KIND(4)
integer, parameter   :: i8=SELECTED_INT_KIND(8)
integer, parameter   :: r4=SELECTED_REAL_KIND(6,37)
integer, parameter   :: r8=SELECTED_REAL_KIND(15,307)
integer(KIND=i4) :: ia
integer(KIND=i8) :: ib
real(KIND=r4) :: ra
real(KIND=r8) :: rb
print *,' Integer 4 ',huge(ia),kind(ia)
print *,' Integer 8 ',huge(ib),kind(ib)
print *,' Real 4 ',huge(ra),kind(ra)
print *,' Real 8 ',huge(rb),kind(rb)
end program kindtype



Compile and run this example kindtype.f90 and compare the output with the results from variables.f90.

Note the use of the PARAMETER statement used to define a constant that may be used several times throughout the program.

Try adding another variable or two with different values for the selection of the kind and see what results.





+     Addition
-     Subtraction
*     Multiplication
/     Division
**    Exponentiation

There is an order of precedence with ** the highest followed by * and /. Next are the unary operators + and – and last the binary operators + and -. For example,

A**B*C is evaluated as (A**B)*C

A/B*C is evaluated as (A/B)*C

Use parentheses to force a particular order of evaluation.



If every operand in a numeric expression is of the same type, the result is also of that type.

If operands of different data types are combined in an expression, the data type of the result is the same as the highest ranking operand. For example,

double precision :: x, y
y = x*2

The integer constant 2 will be promoted to double precision 2.0D0 before doing the multiplication.

Best not to rely on this but be sure that all expressions contain variables or constants of the same type.



Care must be taken when using all integer variables. The code segment

real   :: a
a = 3/2

will produce the result a=1.0 not a=1.5

Compile and run the code exercise_int.f90 and see how the value of a can be calculated.



Loops are used for repeated of execution of similar instructions. For example,

program looping
implicit none
integer  :: i, n
real*8   :: sum
n = 10
sum = 0.0D0
do i=1,n
  sum = sum + dble(i)
  write (*,*) ' Value of sum at step ',i,' is ',sum
end program looping


Things to note:

  • Loop is bounded by do and enddo
  • This can be replaced by a statement number such as
          do 5 i=1,n
             sum = sum + dble(i)
    5     continue
  • Loop index is an integer.
  • There can be loops within loops using a different index.
  • Note 0.0D0 meaning zero in double precision.
  • Note the matching of types in the summation step.



Write Fortran code to calculate and print out the value of the variable a where a=5+1/i for i=1,..,n and n=10.

Then modify the code so that a is a one-dimensional array and the iteration of the do loop for i calculates a(i).



Logical expressions can only have the values .TRUE. or .FALSE.
There are several relational operators, Fortran77 uses the form on the left and Fortran90 can use either form.

.LT.  meaning  <   less than
.LE.           <=  less than or equal
.GT.           >   greater than
.GE.           >=  greater than or equal
.EQ.           ==   equal
.NE.           /=  not equal

There are also logical operators, .AND., .OR., .NOT.

Here is an example

logical  :: x,y,z
x = .TRUE.
y = x .AND. 2>1
z = x .OR. 2<1

In this example all three logical variables will be defined as .TRUE.



Can be written on one line e.g.

if (resid < 5.0D-10) stop

or, in the general form e.g.

if (resid < 5.0D-10) then
   write (*,*)' Residual is less than 5.0D-10'
    write (*,*)' Continue execution'

There can even be more levels

if (resid < 5.0D-10) then
   write (*,*)' Residual is less than 5.0D-10'
elseif (num_iters > 100) then
   write (*,*)' Number of iterations exceeded'
   go to 25
   write (*,*)' Continue execution'



When the IF statement is used within a DO loop and you want to exit the loop but continue the code use EXIT as in the following

loop: do i=1,10
        if (resid < 5.0D-10) exit loop
      enddo loop

Note the use of optional naming of the DO loop.



The DO WHILE statement executes the range of a DO construct while a specified condition remains true. For example,

do while (resid >= 5.0D-10)
   resid = abs(x(i))
   write (*,*) ' Continue execution'
   i = i+1
end do

abs stands for absolute value.



Extend the code you wrote for Exercise 1 so that it exits the loop when a(i) is within .01 of the asymptotic result 5.

Do this first using an IF statement then using the DO WHILE construct. For the DO WHILE case use the very first example code you wrote where a is not an array. You will need to initialise both the variable a and a variable for the iteration.

Print out the value of the iteration count at which the code completes. Do these two methods give the same answer?





In the previous exercise we introduced ABS for the absolute value of a double precision variable. These are Fortran intrinsic functions. Other examples are

sin(a)   sin of the angle a
cos(b)   cosine of the angle b
sqrt(x)  square root of x
exp(z)   exponential of z
log(x)   natural logarithm of x
max(a,b) maximum of a and b
mod(a,b) remainder when a is divided by b

See the Fortran Reference Manual for a complete list: Fortran 95.pdf   Fortran 77.pdf



There are two types of subprograms in Fortran, functions and subroutines. The difference is that a function subprogram is invoked in an expression and returns a single value. A subroutine is invoked by a CALL statement and does not return a particular value.

For example,

real*8 :: x,y
x = func(y)

shows a function being used, whilst

real*8 :: x,y
call subr(x,y)

demonstrates a subroutine call where x returns the result and y contains the input value.

Subprograms are used to simplify coding by keeping the main program as uncluttered as possible and calculations which are used several times coded as subprograms.



To show how to use subprograms we will use this simple integration code.

program integration1

implicit none

integer  :: i, j
real(8)  :: x, y, integ

integ = 0.0D0

do j=1,10
  y = dble(j)*1.0D0
  do i=1,10
     x = dble(i) * 2.5D-1
     integ = integ + sin(x+y)

write (*,'(a,e20.10e3)')' Integration value = ',integ

end program integration1

Compile and run this code, integration1.f90.



We can use a function call to calculate sin(x+y) as follows:

program integration2
implicit none

integer  :: i, j
real(8)  :: x, y, func, integ
real(8)  :: val

integ = 0.0D0

do j=1,10
  y = dble(j)*1.0D0
  do i=1,10
     x = dble(i) * 2.5D-1
     val = func(x,y)
     integ = integ + val

write (*,'(a,e20.10e3)')' Integration value = ',integ

end program integration2


function func(x,y) result(result)
  implicit none
  real(8) :: x,y,result
  result = sin(x+y)
end function func

Note the use of the optional Fortran90 keyword RESULT.

The function subprogram can be in a separate file from the main program and compiled separately but then linked in to create the executable.



The calculation of sin(x+y) can be written as a subroutine as follows:

program integration3
implicit none

integer  :: i, j
real(8)  :: x, y, integ
real(8)  :: val

integ = 0.0D0

do j=1,10
  y = dble(j)*1.0D0
  do i=1,10
     x = dble(i) * 2.5D-1
     call subr(x,y,val)
     integ = integ + val

write (*,'(a,e20.10e3)')' Integration value = ',integ

end program integration3

subroutine subr(a,b,c)
  implicit none
  real(8) :: a,b,c
  c = sin(a+b)
end subroutine subr



Start with the following code fragment

program quadratic
implicit none
real(8) :: a,b,c
complex(8) :: r1,r2
a = 2.0D0
b = 9.0D0
c = 4.0D0

then continue the program to calculate the roots of the quadratic ax**2+b*x+c=0 using a subroutine call.

If the roots are complex print a warning to that effect but not the solutions. Print the solutions if they are real.

Try with different values of a, b and c.

There are two possible solutions in quadratic.f90 and quadratic2.f90.



In most scientific code subroutines will require using arrays for both input and output. In Fortran77 this will generally require passing the size of the arrays as a dummy argument as well as the arrays. We will look at other ways to do this in the Fortran90 course.


The integration code can be written using arrays in the subroutines.

program integrationarray
implicit none

integer  :: i, j
integer, parameter :: n=10
real(8), dimension(n)  :: x, y
real(8), dimension(n,n) :: val
real(8)  :: integ

integ = 0.0D0

do j=1,10
  y(j) = dble(j)*1.0D0
  x(j) = dble(j) * 2.5D-1
call subr(x,y,val,n)
do j=1,n
  do i=1,n
     integ = integ + val(i,j)

write (*,'(a,e20.10e3)')' Integration value = ',integ

end program integrationarray


subroutine subr(a,b,c,n)
  implicit none
  integer :: n, i, j
  real(8) :: a(n),b(n),c(n,n)
  do j=1,n
     do i=1,n
     c(i,j) = sin(a(i)+b(j))
end subroutine subr





There are many ways of writing out data.

print *,result


write (*,*) result


write (6,*) result


write (6,'(f10.4)') result


write (*,'(10F10.4)') (result(i),i=1,10)
    write (*,200) result 
200 format(f10.4)


To open files and read or write data from or to them.

open (unit=2,file='ascii_data',form='formatted',status='old')
read (2,'(10f20.4)') (input(i),i=1,10)
open (unit=3,file='binary_data',form='unformatted',status='unknown',iostat=ierr)
if (ierr = 0) write(3,*) data

Unit numbers 5 and 6 are conventionally reserved for STDIN and STDOUT



Copy the code integration4.f90 and the file input.dat. Compile integration4.f90 and run it. The output all appears in the file integration4.dat.

Try editing input.dat and rerunning the code.

Have a good look at the Fortran code and make sure you understand how it works.



program integration4
implicit none
real(8)  :: factor = 1.0D0
integer, parameter :: filelen=20
real(8) :: x, y, integ, init
real(8) :: userfunc
real(8), dimension(2) :: step
real(8), dimension(4) :: boundary

call loadinput('input.dat', filelen, init, boundary, step)

integ = init

y = boundary(3)

do while (y <= boundary(4))
   x = boundary(1)
   do while (x <=boundary(2))
      integ = integ + userfunc(x,y,factor)
      x = x + step(1)
   y = y + step(2)

call writeoutput('integration4.dat', filelen, integ, boundary, x, y)

end program integration4


function userfunc(x,y,factor) result(func)
  implicit none
  real(8) :: x, y, func, factor
  func = factor * sin(x + y)
end function userfunc

subroutine loadinput(inputfile, filelen, init, boundary, step)
  implicit none
  integer :: filelen
  real(8) :: init
  real(8), dimension(2) :: step
  real(8), dimension(4) :: boundary
  character(filelen) :: inputfile

  open(unit=1, file=inputfile, action='read')

  read(1, *) init
  read(1, *) boundary
  read(1, *) step
end subroutine loadinput


subroutine writeoutput(inputfile, filelen, integ, boundary, x, y)
  integer :: filelen
  real(8) :: integ, x, y
  real(8), dimension(4) :: boundary
  character(filelen) :: inputfile

  open(unit=1, file=inputfile, action='write')

  write(1, *) 'Integration value = ',integ
  write(1, *) 'x-range:   ',boundary(1),'<= x <= ',boundary(2)
  write(1, *) 'y-range:   ',boundary(3),'<= y <= ',boundary(4)
  write(1, *) 'Final x and y:   ',x,y
end subroutine writeoutput





There is a vast literature of highly optimised Fortran code for doing many things, much of it is open source.

For example, the LAPACK library provides a large suite of dense linear algebra routines. The source for LAPACK is available from NETLIB at so you can install it anywhere.

To get more information on, enter

man lapack

To show how it is used we will call the routine ZGEEV to find the eigendecomposition of a 2-dimensional complex matrix.


man zgeev

to see the full documentation for this routine.


program eigen
implicit none
integer, parameter :: n=10
integer, parameter :: lwork=3*n
complex(8), dimension(n,n) :: array, leftvectors, rightvectors
complex(8), dimension(n) :: eigenvalues
complex(8), dimension(lwork) :: work1
real(8), dimension(2*n) :: work2
real(8), dimension(2) :: rand
integer :: i, j, info
do j=1,n
   do i=1,n
      call random_number(rand)
      array(i,j) = cmplx(rand(1),rand(2))
call zgeev('V','V',n,array,n,eigenvalues,leftvectors,n,rightvectors,  &
      n, work1,lwork,work2,info)
if (info==0) then
   write (*,*)' Eigenvalues '
   do i=1,n
      write (*,*) eigenvalues(i)
   write (*,*)' ZGEEV failed with info = ',info
end program eigen


The code eigen.f90 shows this routine in action. Have a good look at it, then compile and run it as follows to link to the LAPACK library.

f90 -o eiegen eigen.f90 -lcxml

Notice that there are two work arrays declared which must be passed to the subroutine. Also an error parameter is returned to flag the success or otherwise of the routine. Always check these error parameters.

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