Given $f(x)=\exp (i 2\pi x)$. What is the minimum number of data points $(x,f(x))$ so that $\int_0^1 f(x)\,dx$ is exact when calculated with Simpson's $-1/3$ rule of numerical integration? The answer given is $9$. I have no idea how to prove it.
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Is $f(x)$ a complex function? – leonbloy Jul 08 '17 at 00:41
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If it is exact at $9$, you can just compute it at all lower numbers of points and observe that it is not exact. – Ross Millikan Jul 08 '17 at 00:45
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Take the two end point terms together, the terms with a factor 4 and the terms with a factor 2. The terms with a factor 2 and the terms with a factor 4 are both geometric series, so you can write down the formulas for them, so this yields and exact formula for then n point Simpson approximation. – Count Iblis Jul 08 '17 at 01:45