## Monday, May 19, 2014

### Happy Birthday Fortran!

For some of us, the name "Fortran" or "FORTRAN" evokes a computer language closely associated with massive super computers and complex mathematical models. For others, it is reminiscent of a war with C++ for the supremacy in the scientific computing world.

It has been continuously developed since its initial publication in 1957, and the latest revision came out in 2010, with another minor revision planned for 2015. Fortran is not dead, far from that, even if it has a though competition from other languages such as Haskell, Clojure or even Python.

Wikipedia has an extensive history of the language.

The first example of code in Fortran I will present is the determination of the fraction that generates a given pattern.

Let's take 0.1278, where the underlined part repeats ad infinitum. The fraction needed to obtain this value is 211/1650. For the rest of this post, I will call the part that repeats the repeated part and the part that does not repeat the prefix. The algorithm to find the fraction is well known, let's focus on the code.

It contains three parts: computing the non reduced fraction, computing the greatest common denominator (gcd) of the numerator and denominator and reducing the fraction to a numerator and a denominator that are relatively prime. Let's start with the gcd.

For this, I use Euclid's algorithm. The code in Fortran 95 to achieve this is

``` function gcd(a, b) result(c) ! Returns the GCD of a and b  integer :: a,b,c,u,l,m  if ( a > b ) then    u = a    l = b  else    u = b    l = a  end if  do while (l > 0)     m = modulo(u,l)     u=l     l=m  end do  c=u  return end function ```
It consists of a few parts: having u contains the largest value, l the smallest then looping until u modulo l  is 0. The simplification subroutine is even simpler.

```subroutine simplify(a, b, c, d) ! Returns the fraction a/b in its simplified form ! c/d where c and d are relatively prime  integer, intent(in) :: a,b  integer, intent(out) :: c,d  integer :: n,gcd  n=gcd(a,b)  c=a/n  d=b/n  return end subroutine ```

Now, the core of the problem is solved by two other functions - one that takes care of fractions with a prefix, the other one of fractions without a prefix. There are some issues in the code presented, but at this point and for a simple presentation, this is not important.

```subroutine findfractionpref(pref, rept, mpref, a, b) ! Returns the fraction a/b such as its division gives the pattern prefreptreptrept ! .... ! With the necessary multiplier  integer, intent(in) :: pref, rept, mpref  integer, intent(out) :: a, b  integer :: d1, d2, num, den  d1=1+floor(log10(real(pref)))  d2=1+floor(log10(real(rept)))  num=((pref*10**(d2)+rept)-pref)  den=10**(d1+d2)-10**(d1)  if ( mpref > 0) then   den=den*10**mpref  else   num=num*10**mpref  end if  call simplify(num, den,a , b)  return end subroutine subroutine findfractionnopref(rept, a,b) ! Returns the fraction a/b such as its division gives the pattern reptreptrept...  integer, intent(in) :: rept  integer, intent(out) :: a,b  integer :: d1, num, den  d1=1+floor(log10(real(rept)))  num=rept  den=10**d1-1  call simplify(num,den,a,b)  return end subroutine ```

Lastly, the main part of the program, used to read the various information and call the necessary subroutines.

```program fractionfinder  implicit none  integer :: pref,rept,a,b  character :: c  do   print *, 'Does your fraction include a prefix (yY/nN) or Q to quit (qQ)?'   read(*,'(A1)'), c   if ((c == 'y').or.(c == 'Y')) then    print *, 'Prefix part?'    read(*, '(I12)'), pref    print *, 'Repeated part?'    read(*,'(I12)'), rept    call findfractionpref(pref,rept,0, a,b)    print *, 'The requested fraction is ', a, '/', b   else if ((c == 'n').or.(c == 'N')) then    print *, 'Repeated part?'    read(*, '(I8)'), rept    call findfractionnopref(rept,a,b)    print *, 'The requested fraction is ', a, '/', b   else if ((c == 'q').or.(c == 'Q')) then    goto 100   end if  end do 100 print *, 'Bye bye!'  stop end program ```

Many thanks go to Rae Simpson for the help she provided with some of the terms in this post!