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I am trying to implement a program the numerical method to solve ODE called Block BDF as explained in this article: https://waset.org/journals/waset/v38/v38-49.pdf

As it is variable step-size, I need to compute the LTE (local truncation), that is $LTE=y_{n+2}^{(5)}-y_{n+2}^{(4)}$. There is a formula in the article for $y_{n+2}^{(5)}$ but I cannot find one for $y_{n+2}^{(4)}$; where $y_{(n+2)}^{(k)}$ means the $k-$order approximation to $y_{(n+2)}$.

Can someone please help me finding the formula for fourth order approximation? (As I'm only trying to implement, information only about the formula can be helpful, but explanation is encouraged).

Thank you very much!

PS. The more general method is BDF: http://en.wikipedia.org/wiki/Backward_differentiation_formula but this method works in block of 2 (works for $y(n+1)$ and $y(n+2)$ at the same step).

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