I am aware of the formula of this method. However, is it true that the method produces more accurate polynomial when the $x$ points are closer to each other? if so or not, why?
Moreover, If I am given a set of $n$ points i.e. $n_1,n_2,n_3,\cdots, n_n$ and I want to approximate the value of the function of a point laying between $n_1,n_2$ using second order Lagrange polynomial. Does the best interpolating polynomial is the one which brackets this point? i.e the one which takes as input points $n_1,n_2$? if so or not, Why?
Thanks in advance.