This chapter proposes effective methods and algorithms for the interpolation and extrapolation of large numerical data sets using root-polynomial and root-fractional-rational functions. The corresponding analytical expressions for calculating the coefficients of root-fractional-rational functions are given. Also given are corresponding examples of using these functions to solve complex interpolation and extrapolation problems, including problems in electron optics, probability theory, and mathematical statistics, as well as descriptions of membership functions in fuzzy-logic problems. The provided studies have shown that modified analytical expressions for root-polynomial and root-fractional-rational functions, into which a positive deviation is introduced, can effectively be used to reduce the relative error of interpolation and extrapolation. The convergence of proposed interpolation and extrapolation methods in the case of using deviation is also guaranteed. The use of the proposed methods for interpolation and extrapolation for stiff function data sets is also described in this chapter. Therefore, further implementation of proposed methods in the developed computer software will significantly reduce the time for solving complex mathematical simulation problems both on local computers and in cloud and fog computing in local and global computer networks. In any case, the use of proposed methods and their development will provide significant impetus for further development and improvement of both local computer software and network software for cloud and fog computing.