Newton Interpolation Calculator
Construct Newton's divided-difference interpolating polynomial and evaluate at a given x.
Inputs
At least 2 distinct x values.
Corresponding y values.
P(x)
6.25000000
Divided differences
c0=1, c1=3, c2=1, c3=0
Step by step
Values used
x values (comma-separated) = 1, 2, 3, 4; y values (comma-separated) = 1, 4, 9, 16; Evaluate at x = 2.50
Newton form
P(x) = f[x₀] + f[x₀,x₁](x-x₀) + f[x₀,x₁,x₂](x-x₀)(x-x₁) + ...
Divided difference
f[xᵢ,...,xⱼ] = (f[xᵢ₊₁,...,xⱼ] - f[xᵢ,...,xⱼ₋₁]) / (xⱼ - xᵢ)
P(x)
= 6.25000000
Divided differences
= c0=1, c1=3, c2=1, c3=0
How it works
Newton's divided-difference interpolation builds the polynomial incrementally: P(x) = c₀ + c₁(x-x₀) + c₂(x-x₀)(x-x₁) + .... The coefficients cₖ are divided differences computed from the data. This form is computationally efficient and makes it easy to add new points without recomputing everything.
Formulas
Newton form
P(x) = f[x₀] + f[x₀,x₁](x-x₀) + f[x₀,x₁,x₂](x-x₀)(x-x₁) + ...
- f[x₀,...,xₖ]
- k-th divided difference
- (x-xⱼ)
- Newton basis terms
Divided difference
f[xᵢ,...,xⱼ] = (f[xᵢ₊₁,...,xⱼ] - f[xᵢ,...,xⱼ₋₁]) / (xⱼ - xᵢ)
Frequently Asked Questions
How is Newton interpolation different from Lagrange?
Both produce the same polynomial, but Newton's form is more efficient when adding new points incrementally and uses Horner's scheme for fast evaluation.
What are divided differences?
They generalize finite differences to non-uniform spacing. The k-th divided difference f[x₀,...,xₖ] equals the leading coefficient of the degree-k interpolating polynomial through those k+1 points.
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