This is more of a theoretical question regarding my choice of x's for my interpolation. I'm wondering if someone can explain to me why when I choose different x's for approximating a value at a point, I get completely wildly different answers. Take the following:
$$ \begin{array}{c|lcr} n & \text{x} & \text{f(x)} & \text{f'(x)} \\ \hline 0 & 0.30 & 0.29552 & 0.95534 \\ 1 & 0.32 & 0.31457 & 0.94924 \\ 2 & 0.35 & 0.34290 & 0.93937 \end{array} $$
Now I want to approximate, using Hermite Interpolation, the value at $H_4(0.34)$ (but this kind of logic applies to everything but spline interpolation I think, correct me if I'm wrong)
My initial guess was to just assume the x's in order, that is let
$$ x_0 = 0.30 ;x_1 = 0.32 ;x_2 = 0.35 $$
This turned out to be wrong. The answer for this polynomial was wildly out of the acceptable range. It ended up that $H_4(0.34)=0.602832$ using this method.
Now, if you go back and you choose
$$ x_0 = 0.35 ;x_1 = 0.32 ;x_2 = 0.30 $$
Then this gives a good approximation.
Why does this work? Can I choose any arbitrary x, or can I only use central/forward/backward difference methods?
If anyone could clear this up for me I would greatly appreciate it.
Thank you!