Piecewise Polynomial Interpolation

Numerical Analysis · Axiom Academy

LESSON Piecewise Interpolation Connecting data with multiple low-degree polynomials High-degree polynomials can oscillate wildly (Runge phenomenon). Instead of one polynomial through all points, use many low-degree polynomials , each between adjacent pairs of points. Piecewise interpolation gives local control: changing one point only affects nearby segments! Piecewise Linear Interpolation The simplest approach: connect adjacent points with straight lines. On each subinterval [xᵢ, xᵢ₊₁], the interpolant is: S(x) = yᵢ + (yᵢ₊₁ - yᵢ) · (x - xᵢ)/(xᵢ₊₁ - xᵢ) Local: each point affects only 2 segments Not smooth: corners at data points We can use higher-degree polynomials in each piece for better smoothness. Uses 4 points per piece (or conditions) We describe smoothness by how many derivatives are continuous: Function values match at joints Function and first derivative match No corners, but curvature jumps Often good enough for visualization Function, first, and second derivatives match Cubic splines are the gold standard for smooth piecewise interpolation. They achieve C² continuity while keeping each piece cubic (low degree). Next lesson!

This is the written version of the interactive lesson above. See the full Numerical Analysis course.