Lagrange Interpolation – Points to Polynomial

Given n+1 points with distinct x values, exactly one polynomial of degree ≤ n passes through all of them, and Lagrange’s formula writes it directly as a weighted sum of basis polynomials. Each basis Lᵢ equals 1 at xᵢ and 0 at every other node — for points (1,3), (2,5), (4,9): L₁(x) = (x−2)(x−4)/((1−2)(1−4)) = (x−2)(x−4)/3. Evaluating at x = 3 gives L₁ = −1/3, L₂ = 1, L₃ = 1/3, and summing yᵢLᵢ(3) = 3(−1/3) + 5(1) + 9(1/3) = 7 — correct because these points lie on the line y = 2x+1 (check: 2(3)+1 = 7). The calculator expands every basis polynomial, multiplies by yᵢ, sums, and prints the resulting polynomial. It warns about Runge’s phenomenon — high-degree interpolation oscillating at edges (as with 1/(1+25x²) on equally spaced nodes) — and recommends splines for many points.

1. 1

   Enter your points

   Distinct x values with their y values

2. 2

   Build each basis

   Lᵢ(x) products are expanded symbolically

3. 3

   Weight and sum

   P(x) = Σ yᵢLᵢ(x) is assembled

4. 4

   Evaluate anywhere

   Interpolate at your target x; extrapolation is flagged

• 1

  n+1 points determine a unique degree-n polynomial

• 2

  Basis Lᵢ is 1 at its own node, 0 at all others

• 3

  Equal spacing + high degree invites Runge oscillation

• Repeating an x value, which breaks the basis denominators

• Trusting extrapolation far outside the sampled range

• Using 20 points for a straight-line trend — degree explodes needlessly

• Sign errors assembling (x−xⱼ)/(xᵢ−xⱼ) products