Chebyshev approximation of discrete polynomials and splines - Che Seman, Fauziahanim (2004)
This work is concerned with the approximation of discrete data using polynomials and splines based on the Chebyshev approximation criterion. Five algorithms are proposed in this work to implement the Chebyshev approximation criterion. These algorithms use either cubic splines or Lagrange polynomials to construct the approximation function. The efficiency of each algorithm developed is evaluated on the basis of the number of iterations and extreme points required for convergence, and the magnitude of the errors generated. One measure of efficiency is the minimum number of knots required to capture the full behavior of the data, and the size of the error between the actual data and its approximation. These knots are the extreme points that determine how well the fitting function approximates the actual data. Since a critical step in the approximation is identifying the extreme points, in this work we have proposed a novel procedure for finding the set of extreme points for the incoming discrete data efficiently. The procedure developed in this work use polynomials and cubic splines to construct the approximation function. The output of the algorithm is a set of extreme points that can be used to construct a minimal error approximation function. A total of five algorithms based on the Lagrange polynomials and cubic splines have been developed in this work to identify the extreme points. The efficiency of each algorithm is analysed in terms of computation time and the magnitude of the errors generated. In real environment, it is hope that this theoretical work can be applied to actual data and solves problems which occur in data processing.