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I've seen two different truncation formulas for the midpoint rule. A common one is $h^3 \frac{ f''}{24}$.

Another, referred to as open Newton Cotes, is $h^3 \frac{f''}{3}$.

The Newton Cotes version is from a numerical method called non self-starting heun.

Why would they differ?

Vince

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  • $\begingroup$ Could you give a little more details? The second method could possibly compare to the first by doubling the interval length, that is, the second is over two intervals of length $h$. And what has the midpoint method to do with the trapezoidal or Heun-2 method? $\endgroup$ – Lutz Lehmann Jun 14 '19 at 21:59
  • $\begingroup$ Yes, thanks, I believe doubling the length is the answer. The midpoint method is used as the predictor in the non self-starting Heun method. But instead of h+.5 as a midpoint between h and h+1, it uses h as a midpoint between h-1 and h+1. $\endgroup$ – vince Jun 15 '19 at 19:20

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