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I have a question. I am looking at the proof of the error of the Simpson's rule. Let $p_{3} \in P_{3}$ be the polynomial that interpolates $f$ at the points $x_{0},x_{1},x_{2}$. Then, from the interpolation error, we have $f(x)-p_{3}(x)=\frac{1}{24}f^{(4)}(w(x))(x-a)(x-\frac{a+b}{2})(x-b)$, or not? But, why is $w$ a function of $x$? And then, at the end of the proof, it is proved that $w$ is independent from $x$. Can't I say from the beginning that $w$ is not a function of $x$?

Pedro
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evinda
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  • Isn't it $\frac{1}{(n+1)!}=\frac{1}{4!}=\frac{1}{24}$ ? – evinda Jan 24 '14 at 16:37
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    See equation (not section) 4.1 here: http://www.math.psu.edu/shen_w/451/NoteWeb/NumComp2012.pdf for the rror estimate (maybe I am missing something - not yet awake). – Amzoti Jan 24 '14 at 16:45
  • The equation 4.1 is for the composite Simpson's rule, but what's with the simple one? – evinda Jan 24 '14 at 17:58

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