Polynomial interpolation approximations of arbitrary continuous functions

The polynomial interpolations,owing to its simplicity in computation, have wide applications in numerical approximation and engineering.However,when the number of interpolating nodes is large,the Lagrange interpolation is numerically unstable.The barycentric lagrange interpolation,using the second kind of Chebyshev points as interpolating nodes,has excellent numerical stability.The paper aims at giving a function f(x) in the interval and finding a polynomial function so that the error ‖f(x)-pn(x)‖∞ approaches to accuracy of machine.In this paper,the interpolating polynomial is numerical computation in the second kind of Chebyshev points by using barycentric Lagrange interpolation.The order of polynomial is obtained by determining the number of interpolating nodes according to the require accuracy.The computer program in MATLAB and some numerical examples are given to demonstrate the effectiveness of the proposed method.