Newton's Divided Differences

Numerical Analysis · Axiom Academy

LESSON Newton's Divided Differences A recursive approach that makes adding new points easy Divided differences are a generalization of the derivative, defined recursively: The notation f[x₀, x₁, ..., xₖ] represents the k-th divided difference. Divided differences are symmetric : the value doesn't depend on the order of the points! Example: Points (1, 1), (2, 4), (4, 16), (5, 25) The highlighted values along the top diagonal are the coefficients we need! Newton's Interpolating Polynomial Using divided differences as coefficients: In nested form (efficient for evaluation): P(x) = 1 + 3(x-1) + 1(x-1)(x-2) + 0(x-1)(x-2)(x-4) = x² (which makes sense: our points come from f(x) = x²!) The Key Advantage: Adding Points With Newton's form, adding a new point is easy! Must recompute all basis polynomials Compute one new column of divided differences When points approach each other, divided differences become derivatives! This connection is fundamental to: Hermite interpolation (matching derivatives too) Taylor polynomials (all points at one location) Understanding interpolation error Summary: When to Use Newton Form When you might add more data points later When evaluating P(x) at many points (nested form) When computing forward/backward differences (equally spaced data)

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