*       symptotico polyonymo gia <<isapehonta shmeia>>?????????
        
        real*8 x(105),f(105),df(105,105),p,y,h,pp
        integer i,j,k,c(12),s
        
        ! Small program for define every c! for 0<c<12.
        do i=1,12
        c(i)=1
         do j=1,i
         c(i)=c(i)*j
         end do
        end do
        ! end of the small program.
                
        print* , 'Give number <K> of points (x,y) you want '
        print* ,' to make a symptotic of K-1 order: '
        read* ,k
        print*,'+++++++++++++++++++++++++++++++++++++++'
        
        do i=1,k
          print* , 'Give a value for x:   ',i
          read* , x(i)
          print* , 'Give a value for f(x):',i
          read* , f(i)
          print*,'-------------------------------------'
        end do
        
        do i=1,k-1
          df(1,i)=f(i+1)-f(i)
        end do
        
        kk=k-2
        do i=2,k-1
        
         do j=1,kk
          df(i,j)=df(i-1,j+1)-df(i-1,j)
         end do
        
        kk=kk-1
        end do 
                
        print* , '--------------------------------------------- '
        print* , '--------------------------------------------- '
        
        print* , 'Give x for which you want to predict the f(x): '
        read* , y
                
        h=abs(x(2)-x(1))
        s=abs(int((y-x(1))/h))  ! int(x) converts x to the nearest integer.        
        ! if y makes s bigger than 12, then the result will be wrong.
        
        print*,'SSSSSSSSSSS= ',s
        pp=0.0d0
        do i=1,k-1
        pp=pp+((c(s))/(c(i)*c(s-i)))*df(i,1)
        end do
        
        p=pp+f(1)
        
        print* , 'The value of f(x) for x=' , y , ' is: ' , p
                
        pause
        end