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We propose a non grid-based interpolation scheme based on the informationfrom the data collected from the vicinity of the query point. As anon-grid-based interpolation, the data points can be distributed randomly in asmall region, and the interpolation is constructed so that it naturally makesuse of the information not only on the function value but also on its higherorder derivatives. The main advantage of the present approach is that theprecision of the interpolation can be adjusted in accordance to the quantity ofthe data, in other words, a balance between the precision and the computationalcost can be achieved by properly choosing the size of the neighborhood wherethe data points are collected. The method is applicable to univariate as wellas multivariate functions. We show that the proposed scheme is efficient andprecise. The present approach is then employed to study the eigenvalue problem.

A non grid-based interpolation scheme for the eigenvalue problem