The solution to du/dt = Au = [ 0 -1; 1 0] u (eigenvalues i and -i) goes around in a circle: u = (cos t, sin t). Suppose we approximate du/dt by forward, backward, and centered differences F, B, and C:
(F) U(n+1) - U(n) = AU(n)
(B) U(n+1) - U(n) = AU(n+1)
(C) U(n+1) - U(n) = .5A(U(n+1)+U(n))
Find the eigenvalues of I + A, (I-A)^-1, and (I-.5A)^-1(I+.5A). For which difference equation does the solution U(n) stay on a circle?