Lagrange Interpolation Calculator
Construct the Lagrange interpolating polynomial through given data points and evaluate at x.
Inputs
At least 2 distinct x values.
Corresponding y = f(x) values.
P(x)
3.25000000
Polynomial degree
3
Step by step
Values used
x values (comma-separated) = 0, 1, 2, 3; y values (comma-separated) = 1, 2, 5, 10; Evaluate at x = 1.50
Lagrange formula
P(x) = Σᵢ yᵢ · Πⱼ≠ᵢ (x - xⱼ)/(xᵢ - xⱼ)
P(x)
= 3.25000000
Polynomial degree
= 3
How it works
Lagrange interpolation constructs the unique polynomial of degree ≤ n-1 passing through n data points. It uses the formula P(x) = Σ yᵢ·Lᵢ(x) where each basis polynomial Lᵢ(x) = Π_{j≠i} (x-xⱼ)/(xᵢ-xⱼ) equals 1 at xᵢ and 0 at all other data points. Enter x and y values as comma-separated lists.
Formula
Lagrange formula
P(x) = Σᵢ yᵢ · Πⱼ≠ᵢ (x - xⱼ)/(xᵢ - xⱼ)
- P(x)
- Interpolating polynomial at x
- (xᵢ,yᵢ)
- Data points
- Lᵢ(x)
- i-th Lagrange basis polynomial
Frequently Asked Questions
What is Runge's phenomenon?
With many equally-spaced points, high-degree polynomials can oscillate wildly between points. Use Chebyshev nodes or splines for many data points.
How many points do I need?
n points determine a unique polynomial of degree n-1. More points give higher-degree polynomials but risk oscillation.
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