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$f(x)= \frac{1}{1+x^2}$

and when I computed the interpolating polynomial of 5 equally spaced points in [-5,5] I got

$ p(x)= 0.0053x^4 -0.1711x^2 +1$

Now I need to estimate the error in the interpolant using the error formula.

$|f(x)-p(x)|=\left|\frac{f^{(5)}(\xi)}{(5)!}\prod_{i=0}^4(x-x_i)\right|$

I'm trying to self learn this topic and I want to know if there is a way I can use f(x)-p(x) to solve this instead of using the right side of the formula.

S.Dan
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    You can find the extrema of $|f(x)-p(x)|$ by canceling the derivative. Using the change of variable $t=x^2$ will lead you to a cubic equation. –  Jan 10 '15 at 12:11
  • So how can I solve $ f^{(5)}(\xi) $ ? The latte part of the question asks to do the same for 17 equally spaced points. – S.Dan Jan 10 '15 at 12:17
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    Have a look at this paper (from page 3):http://www.math.montana.edu/~davis/Classes/MA442/Sp07/Notes/InterpError.pdf – Olivier Oloa Jan 10 '15 at 12:40
  • @S.Dan: your question is about how not to use the remainder formula. –  Jan 10 '15 at 13:50

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I think the problem asks you to do it from the right hand side. You need to find the fifth derivative of $f(x)$ then according to the property of that function, find its maximum value to estimate the largest error.

KittyL
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  • Is there a better way to find the 17th derivative of f rather than deriving it 17 times? The next part of the question asks to do the same for 17 points. – S.Dan Jan 10 '15 at 12:28
  • hmm. I guess using some computer algebra system to do it? – KittyL Jan 10 '15 at 12:30