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I used polynomial interpolation of degree 5 to approximate one 0 of the Bessel function $J_0$. I used the following data:

enter image description here

With the data above, I computed a polynomial using the images of the Bessel funcion as nodes and the $x$ values as the images. After that, I evaluated the polynomial at 0 to obtain the approximation of the 0.

I noticed that if I only use the 6 $x$ values with positive image I get a better approximation than if I use the ones with negative image, as well as a better approximation than if I use 3 with positive image and 3 with negative image (the closest ones to the 0 of the Bessel function). I cannot understand why this happens, any help? I hope I have explained myself clearly... I would guess this has something to do with the fact that the function is continuous and decreasingly monotone but I'm not sure... thank you in advance.

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Repeating your steps, what I obtained (as you did, for sure) is

  1. the solution is $\color{red}{2.404825}295$
  2. the solution is $\color{red}{2.404}216735$
  3. the solution is $\color{red}{2.404825}765$

while the first zero of $J_0(x)\approx \color{red}{2.404825558}$.

Looking, for each case, at the second derivative, what I noticed is that it cancels respectively at $0.31179$ , $0.01091$ and $0.208339$.

This could explain that.

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