  We propose a non grid-based interpolation scheme based on the information
from the data collected from the vicinity of the query point. As a
non-grid-based interpolation, the data points can be distributed randomly in a
small region, and the interpolation is constructed so that it naturally makes
use of the information not only on the function value but also on its higher
order derivatives. The main advantage of the present approach is that the
precision of the interpolation can be adjusted in accordance to the quantity of
the data, in other words, a balance between the precision and the computational
cost can be achieved by properly choosing the size of the neighborhood where
the data points are collected. The method is applicable to univariate as well
as multivariate functions. We show that the proposed scheme is efficient and
precise. The present approach is then employed to study the eigenvalue problem.
