Runge Phenomenon

Abstract

In this thesis we have studied phenomenon associated with interpolation of equally spaced data known as Runge phenomenon. This phenomenon was first observed by Carl David Tolme Runge which deals with the oscillatory behavior of a higher degree interpolating polynomial near the end points of equally spaced data. In this dissertation we have discussed the conditions for the occurrence or absence of this phenomenon and illustrated our results graphically. Aasimpleaproofafor theacaseaof Runge functionaona[ð€€€a; a] has been discussed. A simple formula has been found to calculate the point, for a fixed a, beyond which Runge phenomenon makes its appearance. The role of Chebyshev polynomials in approximation theory has been briefly discussed. To further improve the convergence rate and the reduction of an error obtained through approximation of an interpolating polynomial and non-polynomial function different types of notions like Fourier series and Chebyshev series were discussed.