Interpolation: a Taylor polynomial approach

Abstract

Interpolation is a technique that calculates the unknown values from known given values within a certain range. Whereas the process of calculating unknown values beyond a certain given range is called extrapolation. However, the term interpolation includes extrapolation. Many operators require weak interpolation theory to accurately describe their cartographic properties. For example, Anthony P.Austin (2016) discussed several topics related to interpolation and how it is used in numerical analysis. Biswajit Das, Dhritikesh Chakrabarty (2016) had developed an interpolation formula derived from Lagrange’s interpolation formula. The formula obtained had been applied to represent the numerical data, on the total population of India by a suitable polynomial. Slawomir Sujecki (2013) proposed an extension of the concept of the Taylor series to arbitrary functions that are physically meaningful. The prime objective of our work is to construct a model for interpolation to get a better approximation compared to some existing methods. In this research, we aimed to approximate the Taylor polynomial of unknown function by known data set. We obtained a system of equations by substituting known data set to the Taylor polynomial and found the derivatives needed for the interpolation model using the system of equations. Then we intended to compare the proposed method with some existing methods taking single polynomial, trigonometric and exponential functions as test functions. Also, we compared our model with a well-known existing method, polynomial interpolation. The proposed model overlaps on the polynomial function and exponential function when 5 points are taken with an even comparatively larger step size. The error of our model is less than the error of polynomial interpolation throughout the range in both cases. However, the proposed model deviates from the test function when using a small number of feed data for the trigonometric function. When increased the feed data, our model and the trigonometric function fall on the same curve whereas the usual polynomial interpolation deviates much from the trigonometric function. Consequently, we can conclude that the proposed model performs better than polynomial interpolation for polynomial, trigonometric and exponential functions. And in the case of the increased amount of feed data, we record improved accuracy at interpolating procedure.