Note that \(\ X(\tau_{k})\ \simeq\ {\overset{\overline{}}{x}}_{k}\) ,\(\text{Y\ }\left(\tau_{k}\right)\simeq\ {\overset{\overline{}}{y}}_{k}\),\(\mathcal{M}\ \left(\tau_{k}\right)\simeq\ {\overset{\overline{}}{M}}_{k}\)and \(V_{m}\ (\tau_{k})\ \simeq\ {\overset{\overline{}}{v_{m}}}_{k}\)for \(k\ =\ 0,\ 1,\ \ldots,\ N\). Also, by differentiating the first
approximations (28) and evaluating at the collocation or interpolation
points \(\left\{\tau_{k}\right\}_{k=1}^{N}\) we can achieve: