Richardson Extrapolation
Combine two central differences to cancel their leading error and see when a higher-order error remains.
Free theory articleCombine two central differences to cancel their leading error and see when a higher-order error remains.
Concept & interpretation
Richardson extrapolation combines two central-difference derivative approximations evaluated at nested step sizes h and h/2 to eliminate the leading O(h2) truncation error term:
Interactive Experiment
Test stencil cancellation and fourth-order error scaling in the Richardson Extrapolation interactive workspace:
- Select Function = Cubic, Probe x = 1, and Step h = 1. Verify that Central h = 4 and Central h over2 = 3.25 combine into Extrapolated derivative = 3 with Absolute error = 0.
- Switch Function to Quintic with Probe x = 1 and Step h = 1. Observe that Extrapolated derivative = 4.75 underestimates Exact derivative = 5 by
1/4 = 0.25. - Set Function = Quintic, Probe x = 0, and Step h = 1/4. Note that Extrapolated derivative is negative (
-1/1024 = -0.0009765625) even though Exact derivative = 0. - Select Function = Sine at Probe x = 0 (
h = 1/2) and Probe x = 1 (h = 1/4) to see how Richardson extrapolation reduces trigonometric derivative error by orders of magnitude.