You have built a degree-5 Newton interpolating polynomial through 6 data points. A 7th data point now arrives. What is the correct procedure?
ARecompute the entire Lagrange basis from scratch with all 7 points
BRebuild the full divided difference table using all 7 points
CCompute one new column in the divided difference table and append one new term to the polynomial
DSolve a new 7×7 linear system to find updated polynomial coefficients
This is Newton's divided differences' key advantage over Lagrange: adding a new point requires only computing the new diagonal entry in the divided difference table (one new column of differences), then appending a single new term f[x₀,…,x₆](x−x₀)(x−x₁)…(x−x₅) to the polynomial. All previous terms — and all previous divided differences — remain unchanged. Lagrange interpolation (option A) requires completely rebuilding all basis polynomials Lᵢ(x). The Vandermonde approach (option D) requires solving a new linear system. Newton's form makes incremental data addition O(n) rather than O(n²).
Question 2 Multiple Choice
As two interpolation nodes x₀ and x₁ approach the same value (x₁ → x₀), what does the first-order divided difference f[x₀, x₁] = (f(x₁)−f(x₀))/(x₁−x₀) approach?
AZero, since the numerator and denominator both approach zero
BInfinity, since dividing by a vanishing quantity is undefined
The first-order divided difference is exactly the difference quotient from calculus: (f(x₁)−f(x₀))/(x₁−x₀). As x₁ → x₀, this is precisely the definition of the derivative f′(x₀). More generally, the kth-order divided difference f[x₀,…,xₖ] approaches f^(k)(x₀)/k! as all nodes collapse to x₀. This connection explains why divided differences are well-defined even for coincident nodes (via derivatives) and why Newton's form is more numerically stable than Lagrange near-coincident nodes — it is computing genuine approximations to derivatives.
Question 3 True / False
Newton's divided difference formula and Lagrange interpolation produce different polynomials when applied to the same set of data points.
TTrue
FFalse
Answer: False
Both methods produce exactly the same polynomial — the unique polynomial of degree ≤ n passing through n+1 given points. The uniqueness theorem for polynomial interpolation guarantees there is only one such polynomial. Newton's divided difference form and Lagrange's basis-polynomial form are two different representations of the same mathematical object, analogous to how the same number can be written as a fraction or a decimal. The difference is computational: Newton's form is easier to update incrementally and numerically more stable for near-coincident nodes.
Question 4 True / False
When a new data point is added to a Newton interpolating polynomial, the coefficients of all previously computed terms remain unchanged.
TTrue
FFalse
Answer: True
This is the defining feature of Newton's form. The polynomial P(x) = f[x₀] + f[x₀,x₁](x−x₀) + f[x₀,x₁,x₂](x−x₀)(x−x₁) + ⋯ is built incrementally: each term uses divided differences computed from x₀ through xₖ, and those differences do not change when a new point xₙ₊₁ is added. Only a new divided difference (one new column in the table) and one new term are computed. This stands in contrast to Lagrange, where every basis polynomial Lᵢ(x) depends on all nodes and must be recomputed when any node changes.
Question 5 Short Answer
Why is Newton's divided difference form computationally superior to Lagrange interpolation when data arrives incrementally?
Think about your answer, then reveal below.
Model answer: In Lagrange interpolation, each basis polynomial Lᵢ(x) is defined as a product over all n+1 nodes, so adding a new data point forces complete recomputation of all n+1 basis polynomials. Newton's form builds the polynomial incrementally: each coefficient is a divided difference computed from a triangular subtable, and existing coefficients are unchanged when a new point is added. Adding the (n+1)th point requires only computing one new column in the divided difference table and appending one new term to the polynomial — O(n) work instead of O(n²).
This incremental property makes Newton's form the natural choice for applications where data arrives one point at a time: adaptive ODE solvers that add mesh points, interpolation in iterative algorithms, or real-time systems receiving sensor readings. It also explains why divided difference tables are organized diagonally — the diagonal is precisely the sequence of coefficients, built up one at a time.