Given n+1 distinct points (x_i, y_i), Lagrange interpolation constructs the unique polynomial of degree ≤n passing through all points using L_i(x) = ∏_{j≠i} (x - x_j)/(x_i - x_j). The Lagrange form P(x) = Σ y_i L_i(x) is elegant and explicit but becomes numerically unstable when adding new points since all basis functions must be recomputed.
The interpolation problem is fundamental: given n+1 data points (x₀, y₀), (x₁, y₁), ..., (xₙ, yₙ) with distinct x-values, find a polynomial P(x) of degree at most n that passes through all of them. A uniqueness theorem guarantees that exactly one such polynomial exists — if two polynomials of degree ≤ n agree at n+1 points, their difference has n+1 roots but degree at most n, which forces it to be identically zero. The Lagrange construction gives an explicit formula for this unique polynomial.
The idea is to build basis polynomials L₀(x), L₁(x), ..., Lₙ(x), each of degree n, with a remarkable property: L_i(x_j) = 1 when j = i and L_i(x_j) = 0 when j ≠ i. Each basis polynomial "selects" exactly one data point and ignores the others. The formula is L_i(x) = ∏_{j≠i} (x − x_j) / (x_i − x_j). The numerator product ensures L_i vanishes at every node except x_i (each factor x − x_j is zero when x = x_j). The denominator normalizes the result so that L_i(x_i) = 1. Once you have these basis polynomials, the interpolant is simply P(x) = Σᵢ yᵢ L_i(x) — at any node x_k, every term vanishes except y_k · L_k(x_k) = y_k, confirming the polynomial passes through every data point.
The Lagrange form is mathematically elegant and invaluable for theoretical analysis — for instance, it makes the interpolation error formula transparent. However, it has a practical limitation: if you add a new data point (xₙ₊₁, yₙ₊₁), every basis polynomial must be recomputed from scratch because each L_i depends on all the nodes. The new factor (x − xₙ₊₁) must be inserted into every numerator, and every denominator must be updated. This makes incremental updates expensive, costing O(n²) work.
Newton's divided-difference form addresses this limitation by writing the same unique polynomial in a different basis — one where adding a new point requires computing just one new coefficient and appending one new term. The two forms represent the same polynomial (since it is unique) but are suited to different tasks: Lagrange for theory and closed-form expressions, Newton for computation and incremental updates. Understanding both forms, and why they must agree, deepens your grasp of polynomial interpolation as a whole and prepares you for the error analysis that governs when interpolation can be trusted and when it breaks down (as in Runge's phenomenon).