Formula
With equally spaced points x₀, x₁ = x₀ + h, …, xₙ, the forward differences are Δyᵢ = yᵢ₊₁ − yᵢ, Δ²yᵢ = Δyᵢ₊₁ − Δyᵢ, and so on. Newton’s forward formula starts from the first point:
y = y₀ + p·Δy₀ + p(p − 1)/2!·Δ²y₀ + p(p − 1)(p − 2)/3!·Δ³y₀ + …, p = (x − x₀)/h
The backward formula starts from the last point and uses the bottom diagonal of the same table (∇ᵏyₙ = Δᵏyₙ₋ₖ):
y = yₙ + p·∇yₙ + p(p + 1)/2!·∇²yₙ + p(p + 1)(p + 2)/3!·∇³yₙ + …, p = (x − xₙ)/h
Example
For the default (illustrative) table, h = 10 and x = 14 gives p = 0.4. The top diagonal of the table is 46, 20, −5, 2, −3, so
y = 46 + 0.4·20 + (−0.12)(−5) + 0.064·2 + (−0.0416)(−3) = 46 + 8 + 0.6 + 0.128 + 0.1248 = 54.8528
Each term is smaller than the last, which is the sign that the table is smooth enough for the formula to converge quickly.
Reading the difference table
Each column is the difference of neighbouring entries in the column to its left: subtract the upper value from the lower one. Unlike divided differences there is no division, because the spacing is constant and moves into the factor p.
If a column is constant, every later column is zero and the data lies exactly on a polynomial of that degree. For y = x³ the third differences are all 6h³.
Choosing the degree
Stop adding terms when the next one is negligible compared with the accuracy you need; the first omitted term is a good estimate of the error. If the differences stop shrinking or start alternating in sign with growing size, the data is noisy or too widely spaced and higher terms only add noise.
For a step-by-step treatment with a second worked example, see the Newton forward and backward interpolation guide.
FAQ
When should I use forward and when backward?
Use the forward formula when x is near the start of the table and the backward formula when it is near the end. Each one puts the most weight on the points closest to x, so the first few terms already give a good estimate.
Do forward and backward give different answers?
Only when you stop early. With every point included, both formulas describe the same polynomial and give exactly the same value. Set a lower degree to see them differ.
What if my x values are not equally spaced?
Finite differences need a constant step h. For uneven spacing use the divided difference calculator, which gives the same polynomial without that restriction.