Interpolation Summary

Numerical Analysis · Axiom Academy

Key concepts from polynomial and spline interpolation Interpolation finds a function passing exactly through given data points. A unique polynomial of degree at most n exists through any n+1 distinct points. Lagrange and Newton forms represent the same polynomial differently. Divided differences provide systematic coefficient computation. The interpolation error depends on the (n+1)th derivative. Piecewise cubic polynomials joined with C^2 continuity. Natural splines have S''(a) = S''(b) = 0. Clamped splines specify endpoint derivatives. Solve a tridiagonal system for M_i values. No Runge phenomenon - stable for any number of points. With equally spaced points, high-degree polynomial interpolation can cause severe oscillations near endpoints. This gets worse as you add more points! Approximating curves - pass through first and last control points only. Defined by Bernstein basis polynomials . Convex hull property : curve stays within control polygon. de Casteljau algorithm : stable evaluation. Widely used in computer graphics and CAD. Polynomial interpolation is unique but can oscillate with many points Chebyshev nodes minimize the error term Cubic splines provide smooth, stable interpolation for any number of points Bezier curves approximate with intuitive geometric control Choose method based on: data size, smoothness needs, and derivative information

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