Lagrange interpolation: P(x) = Σ yᵢ Lᵢ(x) where Lᵢ(x) = Π(x−xⱼ)/(xᵢ−xⱼ) for j≠i. Given n points, produces unique polynomial of degree ≤ n−1. No system of equations needed — direct formula. Basis polynomials Lᵢ satisfy Lᵢ(xⱼ) = 1 if i=j, 0 otherwise.
How to Use
Enter data points:
- Points: (x,y) pairs
- Evaluate: P(x) at any x
- Output: Polynomial + basis
How It Works
Each Lᵢ(x) is 1 at xᵢ and 0 at all other xⱼ. The interpolating polynomial is P(x) = y₁L₁(x) + y₂L₂(x) + ... + yₙLₙ(x).
Limitations
High-degree polynomials can oscillate wildly (Runge's phenomenon). For many points, use piecewise interpolation (splines) instead.
Step-by-Step Instructions
- 1Enter x-y data points.
- 2View basis polynomials.
- 3See interpolating polynomial.
- 4Evaluate at any x.
- 5Add or remove points.