Euler Method

!!!PRINCIPAL
program algebra

    use algebra_lineal
    use algebra_no_lineal

    implicit none
    real(8),allocatable:: U(:)
    integer::n,m,i
    real(8)::tiniciales(2),t,deltat,z
    tiniciales(1)=0.d0
    tiniciales(2)=2.d0
    n=10
    deltat=( tiniciales(2) - tiniciales(1)) / dble(n)
    m=1
    allocate(u(m))
    u=0.d0
    u(1)=(1.d0)/(2.d0)
    open(unit=10, file='resultadosdiez.dat')
    do i=1 ,n
        call Euler ( m, U, t, deltat, FdeU)
        write(10,*)t,U,exacta(t)
        z=dble(i)
        t=z*deltat
    enddo
    deallocate(u)
    close(10)
    n=100
    allocate(u(m))
    u=0.d0
    u(1)=(1.d0)/(2.d0)
    deltat=( tiniciales(2) - tiniciales(1)) / dble(n)
    open(unit=11, file='resultadoscien.dat')
    do i=1 ,n
        call Euler ( m, U, t, deltat, FdeU)
        write(11,*)t,U,exacta(t)
        z=dble(i)
        t=z*deltat
    enddo
    deallocate(u)
    close(11)

end program

!!!FUNCIONES

  function FdeU (m,U,t)
    real(8)::FdeU(1:m)
    real(8), intent(in):: U(1:m), t
    integer, intent(in)::m
    FdeU(1)=U(1) - t**2 +1
  end function
  function exacta(t)
    real(8)::exacta,t
    exacta=(t + 1.d0)**(2.d0) - ((exp (t)) / 2.d0)
  end function

!!!EULER

      subroutine Euler ( m, U, t, dt, F)
        !*** Interface for the m--dimensional vector valued function F
        interface
        function F ( m, U, t )
        integer, intent(in) :: m
        real(8), intent(in) :: t, U(1:m)
        real(8) :: F (1:m)
        end function F
        end interface
        !*** Dummy arguments specification
        integer, intent(in) :: m
        real(8), intent(in) :: t, dt
        real(8), intent(inout) :: U(1:m)
        !*** U^{n+1} = U^n + dt * F ( U^n, t_n )
        U = U + dt * F ( m, U, t )
        end subroutine Euler