The Fibonacci Sequence

The Fibonacci sequence is defined by

One characteristic of the sequence is that the ratio tends to the Golden Ration

We want to write a program that computes the first 50 terms in the Fibonacci sequence; at each step we check whether the above ratio is sufficiently close to the Golden Ratio; and if it is, we print a message and end the computations.

The implementation follows.

PROGRAM fibonacci
!
  implicit none
  real :: x0=1.0, x1=1.0, x2
  real :: ratio, golden, tol=1.e-6
  integer :: i, n_max=50
!
  golden = (1.0+sqrt(5.0))/2.0
!  
  do i=2,n_max
    x2 = x0+x1
    print *,"x",i," = ",x2
    ratio = x2/x1 
    if ( abs(ratio-golden) .lt. tol ) then
      print *,"ratio did converge at i = ",i
      exit
    end if
    x0=x1; x1=x2
  end do
!
  if ( i .gt. n_max ) then
      print *,"ratio did not converge, i = ",i
  end if
!
end program fibonacci

During each sweep through the DO loop we compute and print the new element in the sequence, x2. Then, we calculate the x2/x1 ratio, and compare it to the golden ration; if they are suffieciently close we exit the loop. If the loop is terminated normally, the counter is n_max+1, and in this case the ratio did not converge. At the end of each sweep we prepare for the next iteration by assigning x1, x2 to x0,x1).