What is Interpolation?

Numerical Analysis · Axiom Academy

Estimating values between known data points Click on the graph to add points. Watch the interpolating polynomial update! Given n+1 data points (x₀, y₀), (x₁, y₁), ..., (xₙ, yₙ) with distinct xᵢ values, find a polynomial P(x) of degree at most n such that: Key theorem: This polynomial exists and is unique! Readings at noon each day. What was the temperature at 3 PM? Population every 10 years. Estimate mid-decade values. Key frames at certain times. Smooth motion between them. Easy to evaluate: Just addition and multiplication Easy to differentiate and integrate: Known closed forms Dense in continuous functions: Can approximate any continuous function arbitrarily well (Weierstrass) Lagrange Interpolation: Elegant formula, easy to understand Newton's Divided Differences: Efficient for adding new points Splines: Piecewise polynomials for smoother results

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