Enter Data Points
Add (x, y) pairs — x values must be distinct
| Point | x value | y = f(x) |
|---|
Max 8 points | Min 2 points
Find L(x) at x =
Enter x to interpolate/extrapolate
Examples:
Lagrange Interpolation — Formula & Method
Lagrange interpolation finds the unique polynomial of degree ≤ n−1 that passes through n given data points. It is one of the most fundamental methods in numerical analysis.
L(x) = Σᵢ yᵢ · ℓᵢ(x)
ℓᵢ(x) = Πj≠i (x − xⱼ) / (xᵢ − xⱼ)
where ℓᵢ(xᵢ) = 1 and ℓᵢ(xⱼ) = 0 for j ≠ i
ℓᵢ(x) = Πj≠i (x − xⱼ) / (xᵢ − xⱼ)
where ℓᵢ(xᵢ) = 1 and ℓᵢ(xⱼ) = 0 for j ≠ i
How to Use This Calculator
- Enter at least 2 data points (x, y) in the table
- All x values must be distinct (no duplicate x)
- Enter the target x value to find L(x) at that point
- Click Calculate to get interpolated value, basis polynomials and steps
- Maximum 8 data points supported
Applications of Lagrange Interpolation
- Numerical analysis: Approximating unknown functions from sample data
- Computer graphics: Smooth curve generation and animation paths
- Engineering: Data fitting, sensor calibration, signal processing
- Finance: Yield curve interpolation in bond pricing
- Physics: Estimating measurements between experimental data points
Lagrange Interpolation Examples
2 points (1,2),(3,4) → L(2) = 3.0 (linear)
3 points y=x²: (1,1),(2,4),(3,9) → L(2.5) = 6.25
4 points y=x³: (1,1),(2,8),(3,27),(4,64) → L(2.5) = 15.625
At data point: L(xᵢ) = yᵢ exactly (always passes through)
3 points y=x²: (1,1),(2,4),(3,9) → L(2.5) = 6.25
4 points y=x³: (1,1),(2,8),(3,27),(4,64) → L(2.5) = 15.625
At data point: L(xᵢ) = yᵢ exactly (always passes through)
Lagrange vs Newton Interpolation
- Lagrange: Simple formula, easy to understand. Best for small datasets. Requires full recomputation when adding a new point.
- Newton divided differences: More efficient when adding points incrementally. Same polynomial, different representation.
- Spline interpolation: Avoids Runge's phenomenon. Better for large datasets. Piecewise polynomial — not one big polynomial.
Key Properties of Lagrange Basis Polynomials
- ℓᵢ(xᵢ) = 1 — equals 1 at its own data point
- ℓᵢ(xⱼ) = 0 for j≠i — equals 0 at all other data points
- Σ ℓᵢ(x) = 1 — basis polynomials always sum to 1
- Degree of each ℓᵢ(x) = n−1 for n data points
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