bandikizaui

2022-07-15

Which rule for numerical integration is more accurate, the Trapezoidal Rule or the Simpson Rule?

bap1287dg

Expert

The error formulas are:
${E}_{T}=-\frac{{h}^{2}}{12}\left(b-a\right){f}^{″}\left({\xi }_{1}\right)$
${E}_{S}=-\frac{{h}^{4}}{180}\left(b-a\right){f}^{⁗}\left({\xi }_{2}\right)$
So you can imagine that, even if ${f}^{″}$ and ${f}^{⁗}$ have similar magnitudes, if $h$ is large the trapezoidal method can perform better. Also, if $f$ is ${C}^{2}$ but not ${C}^{4}$...

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