```      subroutine gradient_p(x, g)
double precision x(dims), g(dims), g_x(dims, dims), y
integer k, l
do k = 1, dims
do l = 1, dims
g_x(k, l) = 0d0
enddo
g_x(k, k) = 1d0
enddo
call g_p(x, g_x, y, g)
end

subroutine naive_euler(w, r)
double precision w(controls), r
double precision x(dims), xdot(dims), delta_t, g(dims)
double precision xddot(dims), t(dims), x_new(dims), s(dims)
double precision delta_t_f, x_t_f(dims), charges(ncharges, dims)
double precision g_charges(dims, ncharges, dims)
common /closure/ charges
common /g_closure/ g_charges
integer j
delta_t = 1e-1
charges(1, 1) = 10d0
charges(1, 2) = 10d0-w(1)
charges(2, 1) = 10d0
charges(2, 2) = 0d0
do j = 1, dims
g_charges(dims, 1, 1) = 0d0
g_charges(dims, 1, 2) = 0d0
g_charges(dims, 2, 1) = 0d0
g_charges(dims, 2, 2) = 0d0
enddo
x(1) = 0d0
x(2) = 8d0
xdot(1) = 0.75d0
xdot(2) = 0d0
call ktimesv(dims, -1d0, g, xddot)
call ktimesv(dims, delta_t, xdot, t)
call vplus(dims, x, t, x_new)
if (x_new(2).gt.0d0) then
do j = 1, dims
x(j) = x_new(j)
enddo
call ktimesv(dims, delta_t, xddot, t)
call vplus(dims, xdot, t, s)
do j = 1, dims
xdot(j) = s(j)
enddo
goto 1
endif
delta_t_f = -x(2)/xdot(2)
call ktimesv(dims, delta_t_f, xdot, t)
call vplus(dims, x, t, x_t_f)
r = x_t_f(1)*x_t_f(1)
end
```

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