Chebyshev Interpolation (Polynomial Interpolation Using Chebyshev Points)

Chebyshev interpolation is a polynomial interpolation method that utilizes Chebyshev points to approximate continuous functions. The core idea involves finding a polynomial that minimizes the maximum difference between the polynomial and the original function among all possible polynomials. This is achieved through strategic selection of Chebyshev points, which are calculated as x_k = cos((2k-1)π/(2n)) for k=1,...,n in the interval [-1,1]. A key advantage of Chebyshev interpolation is its ability to provide high-quality approximations of continuous functions using a limited set of points. The method employs the Chebyshev nodes to minimize Runge's phenomenon, ensuring stable interpolation even for high-degree polynomials. In implementation, one typically constructs the interpolation polynomial using Lagrange basis functions or the barycentric interpolation formula applied to Chebyshev points. This technique finds widespread applications in computer science for function approximation algorithms and in numerical analysis for solving differential equations. The algorithm can be efficiently implemented using recurrence relations for Chebyshev polynomials T_n(x) = cos(n arccos(x)), with computational complexity O(n) for polynomial evaluation through Clenshaw's algorithm.