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I'm running a numerical integral over a function which I can only know on specific points as the problem is defined on a lattice.

I've been using a trapezoidal method which has for known error:

$$\Delta = \frac{1}{12} \text{mesh}^3 \;*\text{length} * \text{max}(f'')$$

mesh will be equal to 1 in my case but I'm not sure how to evaluate the max of the second derivative, which I don't know naively. Is the bound still correct if I use a discrete second derivative instead? There is no additional information I can access.

Learning is a mess
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1 Answers1

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No. The behavior of $f$ between two interpolation nodes could be wild. Consider for instance this example. If $f(x)=\sin(100\pi\,x)$, then $f(k/100)=0$ for $0\le k\le100$. If you interpolate with nodes $k/100$, $0\le k\le100$ on the interval $[0,1]$, the interpolation function will be identically $0$, and the sup norm of the error $1$.

Julián Aguirre
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