program CAUCHY_SEQUENCE

!  This is an example of Cauchy sequence.
!  About the sequence, I refered to the Wikipedia.
!  Concerning a program, I refered to John Burkardt's Home Page.
!  Let's enjoy Fortran90.

  integer          :: l=1, z
  real(kind(1Q0))  :: a, b, c, m, n, o, x, y

  loop_x: do
    write (*,*) 'Please enter X ,which is a natural number, and X<10^8.'
    read (*,*) x
    if (x.le.0) then
       write (*,*) 'As X is positive,',x,'is invalid.' 
       cycle loop_x
       end if
    if (x.ne.idint(x)) then 
       write (*,*) 'As X is an integer,',x,'is invalid.' 
       cycle loop_x
    end if
    if (x.ge.10**8) then
       write (*,*) 'As X<10^8,',x,'is invalid.'
       cycle loop_x
    end if
    exit loop_x
  end do loop_x

  loop_n: do
    write (*,*) 'Please enter N ,which is a rational number. 0.1^16<N<10^16.'
    read (*,*) n
    if (n.le.0.1**16) then
       write (*,*) 'As 0.1^16<N,',n,'is invalid.'
       cycle loop_n
    end if
    if (n.ge.10.0**16) then
       write (*,*) 'As N<10^16,',n,'is invalid.'
       cycle loop_n
    end if
    exit loop_n
  end do loop_n

  z = idint(x)

  write (*,*) ''
  write (*,*) '*****************************************'
  write (*,*) ' An example of Cauchy sequence'
  write (*,*) ' X =',z,',',' N =',n
  write (*,*) '*****************************************'

  write (*,'(/,a5, 2x ,a21, 2x, a21)') 'Num',  'log10(N)', '[N*sqrt{X}]/N'

  loop_c: do while (l.le.10)
    m = n / 10.0**(10-l)
    o = dlog10(m) 
    y = dble(x)
    a = m * dsqrt(y)
    b = dint(a)
    c = b / m
    write (*,'(i5, 2x, d21.16, 2x, d21.16)') l, o, c
    l = l+1
  end do loop_c

  write (*,*) ''

end program CAUCHY_SEQUENCE