Romberg Integration

Numerical Analysis · Axiom Academy

Accelerating convergence through Richardson extrapolation The trapezoid rule error has a special structure: E = c₁h² + c₂h⁴ + c₃h⁶ + ... By computing with h and h/2, we can eliminate the h² term! We build a triangular table where each column has higher-order accuracy: Each entry combines two from the previous column: For the first extrapolation (k=1): R(j,1) = (4R(j,0) - R(j-1,0))/3 Each column eliminates another error term: Column 1: Eliminates h² → O(h⁴) (equivalent to Simpson's!) Column 2: Eliminates h⁴ → O(h⁶) Column 3: Eliminates h⁶ → O(h⁸) Computing ∫₀¹ e^x dx (exact value = e - 1 ≈ 1.718282):

This is the written version of the interactive lesson above. See the full Numerical Analysis course.