Lagrange Interpolation Calculator
Compute the Lagrange interpolation polynomial for a given set of data points with our powerful calculator
Data Points
Interpolation Results
Graph Visualization
Graph will appear here when points are entered
📚 Lagrange Interpolation Method
The Lagrange interpolation polynomial is the unique polynomial of least degree that passes through a given set of points (x₀,y₀), (x₁,y₁), ..., (xₙ,yₙ).
Lagrange Basis Polynomials
The polynomial is constructed as:
where ℓᵢ(x) are the Lagrange basis polynomials:
Uniqueness
For n+1 distinct points, there exists exactly one polynomial of degree ≤n that interpolates them.
Runge's Phenomenon
High-degree polynomial interpolation can oscillate wildly between points, especially with equidistant nodes.
Computational Complexity
Evaluating the Lagrange form requires O(n²) operations, making it less efficient than Newton's form for repeated evaluations.
Applications
Used in numerical analysis, computer graphics, CAD systems, and anywhere smooth approximation of discrete data is needed.
📋 Example Problems
Linear Interpolation (2 points)
Points: (1,1), (3,2)
Solution: L(x) = ½(x+1)
Quadratic Interpolation (3 points)
Points: (1,1), (2,4), (3,9)
Solution: L(x) = x²
Cubic Interpolation (4 points)
Points: (0,1), (1,0), (2,1), (3,0)
Solution: L(x) = (x³ - 4x² + 3x + 2)/2
Note: This calculator provides exact mathematical results for the given points. For high-degree polynomials, numerical stability may be affected by the specific point distribution.