  This paper adapts look-ahead and backward finite difference formulas to
compute future eigenvectors and eigenvalues of piecewise smooth time-varying
symmetric matrix flows $A(t)$. It is based on the Zhang Neural Network (ZNN)
model for time-varying problems and uses the associated error function $E(t) =
A(t)V(t) - V(t) D(t)$ or $e_i(t) = A(t)v_i(t) -\la_i(t)v_i(t)$ with the Zhang
design stipulation that $\dot E(t) = - \eta E(t)$ or $\dot e_i(t) = - \eta
e_i(t)$ with $\eta > 0$ so that $E(t)$ and $e(t)$ decrease exponentially over
time. This leads to a discrete-time differential equation of the form $P(t_k)
\dot z(t_k) = q(t_k)$ for the eigendata vector $z(t_k)$ of $A(t_k)$. Convergent
look-ahead finite difference formulas of varying error orders then allow us to
express $z(t_{k+1})$ in terms of earlier $A$ and $z$ data. Numerical tests,
comparisons and open questions complete the paper.
