Richardson Extrapolation - Numerical Methods Step by Step

Improve derivative accuracy by combining two central-difference estimates with different step sizes: R = (4·D(h/2) - D(h)) / 3. Example: f(x)=x³, x=2, h=1 gives exact result R=12, error=0.

The result has O(h⁴) accuracy.

richardson-extrapolation Richardson extrapolation cancels the leading O(h²) error term by combining two central-difference approximations: D1 = (f(x+h) - f(x-h)) / (2h) D2 = (f(x+h/2) - f(x-h/2)) / h R = (4·D2 - D1) / 3

Example

Combine two estimates to cancel the leading error term. The remaining error is exactly zero here only because the sampled function is a cubic; for a general smooth function a nonzero higher-order error term remains.

highlighted = computed this step

Step 1 — Set up

Set up Richardson extrapolation.

f(x)= x^{3} \quad x= 2 \quad h= 1

Step 2 — Estimate D(h)

Compute the first central-difference estimate.

D_{ 1 }=( 27 - 1 )/ 2 = \hl{13}

Step 3 — Estimate D(h/2)

Compute the half-step central estimate.

D_{ 2 }=( \frac{125}{8} - \frac{27}{8} )/ 1 = \hlmath{\frac{49}{4}}

Step 4 — Richardson combine

Combine 4 times D2 minus D1 over 3.

R=( 4 \cdot \frac{49}{4} - 13 )/ 3 = 36 / 3 = \hl{12}

Step 5 — Result

State the Richardson estimate.

R= \hl{12}