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I have the composite Simpson's rule as $$ \frac{h}{6}(f(a)+f(b)) + \frac{h}{3}(f(a+h)+f(a+2h)+...+f(a+(n-1)h) + \frac{2h}{3}(f(a+\frac{h}{2})+(f(a+3\frac{h}{2})+...+f(a+(2n-1)\frac{h}{2}) $$

My professor has done the proofs for the midpoint and trapezoidal rule error formula, but left it to us for Simpson.

Show that if f(t) has a continuous fourth derivative then: $$ \int_{a}^b f(x)dx$ = Q_5(f) - \frac{(b-a)^{5}}{2880n^{4}} f^{4}(\theta) $$

My proof is already over 3 pages long and I feel like there must be a simpler way. I started by using a=-1,b=1 and using $\varphi$ so $$ \int_{-1}^1 \varphi(t)dt = \frac{2}{3}(\frac{1}{2}\varphi(-1)+2\varphi(0)+\frac{1}{2}\varphi(1))-\frac{1}{90}\varphi^{4}(\theta ) $$

I start by integrating by parts 4 times. Then I compute the integral from -1 to 1 twice (for t-1,t+1) and the integral from -1 to 0 and 0 to 1. Then I add them all. and divide by 3, then use the mean value theorem. Am I on the right track? It seems so tedious, and I am not getting correct answers - but again, this could be because of the tedious nature. Thanks!

bob
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  • The error bound has been studied here – Mittens Oct 24 '22 at 21:14
  • I am discussing the Composite Simpsons rule, not the Simpson's rule – bob Oct 25 '22 at 22:08
  • One you have the bound for the single Simpson rule the composite bound is easily obtained. – Mittens Oct 25 '22 at 22:09
  • That is pretty much my question haha - I'm not sure how to get there. My professor has given us the Simpson's error – bob Oct 26 '22 at 17:22
  • You can read Süli, Endre; Mayers, David (2003). An Introduction to Numerical Analysis. Cambridge University Press which is referred to by wikipedia https://en.wikipedia.org/wiki/Simpson%27s_rule#CITEREFS%C3%BCliMayers2003. In p211 (7.18) it just changes $h=(x_2-x_0)/2$ ($(x_2-x_0)/2$ is explicitly shown in that QA but it is implied in QA question formula) in the above QA link by Mittens to $\frac{(b-a)}{2n}$ in your notation. Then when summing up all $n$ terms it needs to multiply $n$. Then we get the formula. – An5Drama Jun 24 '24 at 05:45

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