Stirling formula: y(35)

There are four equally spaced points (h = 10) and x = 35 lies in the middle, so Stirling's central difference formula is used with x₀ = 30 (p = (35 − 30)/10 = 0.5).

Difference table

Central differences about x₀ = 30: Δy₋₁ = −73 (from x = 20), Δy₀ = −93, Δ²y₋₁ = −20, and the only third difference available is Δ³y₋₁ = 10.

Stirling's formula:

y(x₀ + ph) = y₀ + p·(Δy₋₁ + Δy₀)/2 + p²/2!·Δ²y₋₁ + p(p²−1)/3!·(Δ³y₋₂ + Δ³y₋₁)/2 + …

Only four points exist, so Δ³y₋₂ is not available; the single third difference is used in its place.

y(35) = 439 + 0.5(−73 − 93)/2 + (0.5)²/2 (−20) + 0.5(0.25 − 1)/6 (10)
      = 439 − 41.5 − 2.5 − 0.625
      = 394.375

Without the third-difference term the value is 395.000; with it, 394.375. The cubic through all four points gives exactly 394.375, so the correction is justified.

Answer: y(35) ≈ 394.375