How to use the Newton Forward Difference Table
- 1. Enter x and yThe x values must be equally spaced, like 1891, 1901, 1911.
- 2. Pick forward or backwardUse forward near the start of the table and backward near the end.
- 3. Read the termsSee the difference table, p, each term of the formula and the answer.
Newton's forward and backward formulas
For equally spaced x with gap h, build differences: Δyᵢ = yᵢ₊₁ − yᵢ, Δ²yᵢ = Δyᵢ₊₁ − Δyᵢ, and so on.
Forward: with p = (x − x₀)/h, y = y₀ + pΔy₀ + p(p−1)/2! Δ²y₀ + p(p−1)(p−2)/3! Δ³y₀ + …
Backward: with p = (x − xₙ)/h, y = yₙ + p∇yₙ + p(p+1)/2! ∇²yₙ + p(p+1)(p+2)/3! ∇³yₙ + …
The forward formula uses the top diagonal of the table, the backward formula the bottom one.
Worked example
A town's population (thousands) was 46, 66, 81, 93 and 101 in 1891, 1901, 1911, 1921 and 1931. Estimate it for 1895.
Differences: Δy = 20, 15, 12, 8; Δ²y = −5, −3, −4; Δ³y = 2, −1; Δ⁴y = −3. Here h = 10 and p = (1895 − 1891)/10 = 0.4.
y = 46 + 0.4(20) + (0.4)(−0.6)/2 (−5) + (0.4)(−0.6)(−1.6)/6 (2) + (0.4)(−0.6)(−1.6)(−2.6)/24 (−3) = 46 + 8 + 0.6 + 0.128 + 0.1248 = 54.8528 thousand. With the backward formula, 1925 gives 96.8368 thousand.
Which formula to use
Use the forward formula when x is near the start of the table and the backward formula near the end, so the first few terms do most of the work. If the k-th differences are nearly constant, the data behaves like a degree-k polynomial and later terms add little. For uneven spacing, switch to the Lagrange interpolation calculator.
Frequently asked questions
Why must x be equally spaced?
Newton's forward and backward formulas use a single gap h. For uneven gaps, use Lagrange or divided differences.
What is p in Newton's formula?
p = (x − x₀)/h for forward, or (x − xₙ)/h for backward. It measures how many steps x is from the starting point.
Does forward or backward give a different answer?
With all points used, both give the same polynomial. Truncated early, the one closer to x is more accurate.
Is this calculator free?
Yes. You enter your name and WhatsApp number once when you see your result, and other Uncodemy tools then open without the form.