Newton Interpolation Calculator
Newton’s form builds the polynomial one divided difference at a time: through the cubes of 1–4 it gives P(2.5) = 15.625.
15.625
Full answer, then cut to fewer decimals (not rounded):
- 15.625
- 15.62
- 15.6
- Interpolating polynomial
- P(x) = x^3
- Leading coefficients f[x₀], f[x₀,x₁], …
- 1, 7, 6, 1
| xᵢ | f[xᵢ] | order 1 | order 2 | order 3 |
|---|---|---|---|---|
| 1 | 1 | 7 | 6 | 1 |
| 2 | 8 | 19 | 9 | |
| 3 | 27 | 37 | ||
| 4 | 64 |
Step-by-step solution
- f[xᵢ,…,xᵢ₊ₖ] = (f[xᵢ₊₁,…,xᵢ₊ₖ] − f[xᵢ,…,xᵢ₊ₖ₋₁]) ÷ (xᵢ₊ₖ − xᵢ)
- P(x) = f[x₀] + f[x₀,x₁](x − x₀) + f[x₀,x₁,x₂](x − x₀)(x − x₁) + …
- P(2.5) = 15.625
Newton Interpolation Formula
P(x) = f[x₀] + f[x₀,x₁](x − x₀) + f[x₀,x₁,x₂](x − x₀)(x − x₁) + …
f[xᵢ,…,xᵢ₊ₖ] = (f[xᵢ₊₁,…] − f[xᵢ,…]) ÷ (xᵢ₊ₖ − xᵢ)
Worked Example
The points (1, 1), (2, 8), (3, 27), (4, 64) at x = 2.5:
- f[xᵢ,…,xᵢ₊ₖ] = (f[xᵢ₊₁,…,xᵢ₊ₖ] − f[xᵢ,…,xᵢ₊ₖ₋₁]) ÷ (xᵢ₊ₖ − xᵢ)
- P(x) = f[x₀] + f[x₀,x₁](x − x₀) + f[x₀,x₁,x₂](x − x₀)(x − x₁) + …
- P(2.5) = 15.625
Leading coefficients f[x₀], f[x₀,x₁], …: 1, 7, 6, 1
| xᵢ | f[xᵢ] | order 1 | order 2 | order 3 |
|---|---|---|---|---|
| 1 | 1 | 7 | 6 | 1 |
| 2 | 8 | 19 | 9 | |
| 3 | 27 | 37 | ||
| 4 | 64 |
GATE tip
Use the difference table to spot the degree of the data: the column that becomes constant tells you the polynomial degree, a common two-mark check.
More Numerical Methods Calculators
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Lagrange Interpolation Calculator
P(x) = Σ yᵢ Π (x − xⱼ)/(xᵢ − xⱼ)
degree ≤ n − 1 through n points
Example: The points (1, 1), (2, 8), (3, 27), (4, 64) at x = 2.5: P(2.5) = 15.625.
-
Numerical Differentiation Calculator
f′(x) ≈ [f(x + h) − f(x − h)] ÷ 2h
forward and backward differences have O(h) error; central O(h²)
Example: f(x) = sin x at x = 1 with h = 0.1: Central difference f′(x) ≈ 0.53940225217.
-
Newton-Raphson Method Calculator
xₙ₊₁ = xₙ − f(xₙ) ÷ f′(xₙ)
quadratic convergence near a simple root
Example: f(x) = x³ − 2x − 5 starting at x₀ = 2: Root x ≈ 2.09455148154.
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Bisection Method Calculator
c = (a + b)/2; regula falsi c = (a·f(b) − b·f(a)) ÷ (f(b) − f(a))
keep the half where f changes sign
Example: f(x) = x³ − x − 2 on [1, 2]: Root x ≈ 1.5213804245.
Frequently asked questions
How are forward differences related to divided differences?
For equal spacing h, f[x₀,…,xₖ] = Δᵏy₀ ÷ (k! hᵏ), so the forward-difference formula is a special case.
What does a constant k-th difference mean?
The data come from a polynomial of degree k; for x³ the third differences are all 6.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Newton Interpolation Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/newton-interpolation-calculator/
IEEE
Scientific Converters, "Newton Interpolation Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/newton-interpolation-calculator/