Newton Interpolation Calculator

Construct Newton's divided-difference interpolating polynomial and evaluate at a given x.

About Newton Interpolation Calculator

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)

Worked example

With the default values below, Newton Interpolation Calculator returns P(x) of 6.25000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

x values (comma-separated): 1, 2, 3, 4
y values (comma-separated): 1, 4, 9, 16
Evaluate at x: 2.5

Results

P(x): 6.25000000
Divided differences: c0=1, c1=3, c2=1, c3=0

Step by step

1Values used
2Newton form
3Divided difference
4P(x)= 6.25000000
5Divided differences= c0=1, c1=3, c2=1, c3=0

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

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.