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I have the following function:

$$\frac{1024\,x^4}{e^{(4x)^{4}}}$$

And I applied the composed Simpson rule to obtain the value of the integral of that function from $0$ to $0,62715.$ Then, I applied the Romberg Method to obtain a better approximation for the same number of sub-intervals. But when I have done that, I discovered that the Romberg method, for the same number of subdivisions made my approximation worse.

The formula that I've used for each extrapolation of the Romberg Method was:

$$R(k,m) = \frac{4^m\cdot R(k+1,m-1)-R(k,m-1)}{4^m-1} $$

Is it even possible that the Romberg method makes the aproximation worse?

Romberg Table

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  • Yes, why wouldn't they? From finitely many values of a smooth function you can't deduce 'anything' (concrete as this sequence of errors being decreasing). The only information that you have is what happens as the number of points tends to infinity. – conditionalMethod Nov 27 '19 at 11:38
  • Smooth functions are very flexible. What you have in those terms is information about finitely many points of its graph. Having those points fixed, you can prescribe the values at other points to make the actual value of the integral anything you want. You can make it such that the sequence of error terms spell out your birthday, your mother's maiden name, all the books of Stephen King before it finally tends to zero. – conditionalMethod Nov 27 '19 at 11:50
  • I've added a image of my table to the answer. How did you calculated the first column? It is different from mine. I've checked the first column with online simpson rule calculators like this one: https://www.emathhelp.net/calculators/calculus-2/simpsons-rule-calculator/?f=%281024x%5E4%29%2F%28e%5E%284x%29%5E4%29&a=0&b=0.62715&n=2&steps=on – Ru44Be4N7 Nov 27 '19 at 13:47

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