Node-Dependent Conditioning, Lebesgue Growth, and Noise Amplification in Polynomial and Spline Interpolation

This paper presents a reproducible computational study of conditioning and stability in polynomial and spline interpolation. The central question is how node geometry and approximation structure affect interpolation error and the amplification of small perturbations. Newton and barycentric polynomial interpolation are compared on uniform and Chebyshev nodes, and natural cubic splines are used as a local piecewise alternative. The study estimates Lebesgue constants, measures interpolation error on dense grids, and performs Monte Carlo perturbation tests using the Runge function as a difficult example and the exponential function as a favorable smooth example. The results show that uniform-node polynomial interpolation becomes increasingly ill-conditioned as the degree grows, with rapidly increasing Lebesgue constants and severe amplification of small data noise. Chebyshev nodes keep the Lebesgue constants small and produce stable polynomial approximations, while natural cubic splines remain robust because of their local structure. Barycentric interpolation improves the evaluation form of the interpolating polynomial but does not remove the underlying node-dependent conditioning of the interpolation operator. Secondary composite quadrature benchmarks are included to connect the interpolation study with broader numerical approximation practice. The results emphasize that reliable approximation depends not only on exact interpolation conditions or formal order, but also on operator conditioning, node placement, and whether the approximation is global or local.