Multistep Method: Gear's Formula Interpolation

Question

Please explain how to do this. How can we use Lagrange Interpolation to derive this formula? Thanks in advance.

Answer 1

\begin{eqnarray} q(t) &=& y_{n-1} {(t-t_{n})(t-t_{n+1}) \over (t_{n-1}-t_{n})(t_{n-1}-t_{n+1}) } + y_{n} {(t-t_{n-1})(t-t_{n+1}) \over (t_{n}-t_{n-1})(t_{n}-t_{n+1}) } + y_{n+1} {(t-t_{n-1})(t-t_{n}) \over (t_{n+1}-t_{n-1})(t_{n+1}-t_{n}) }\\ &=&{1 \over 2 h^2} (y_{n-1} (t-t_{n})(t-t_{n+1}) - 2y_{n} (t-t_{n-1})(t-t_{n+1}) + y_{n+1} (t-t_{n-1})(t-t_{n}) ) \end{eqnarray}

Grinding through the details gives $q'(t_{n+1}) = {1 \over 2h} (y_{n-1}- 4y_n + 3 y_{n+1})$.

Equating $q'(t_{n+1}) = f(t_{n+1},y_{n+1})$ gives the desired formula.