Considering the Trapezoidal Rule and the Simpson Rule: Which one is more accurate?
For me, it seems to be the Simpson Rule (because it uses quadratic functions compared to linear functions with the trapezoidal rule), but I recently found statements that it isn't true for all cases (but for the most).
Could anyone explain to my why that is? Maybe there exists a trivial example to show this? How can I then state that simpson rule is "in most cases" more accurate? Just by having made this experience?
Thank you very much!
The error formulas are: $$ E_T = -\frac{h^2}{12} (b-a)f''(\xi_1) $$
$$ E_S = -\frac{h^4}{180}(b-a) f''''(\xi_2) $$
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 $C^4$...