- 2071 Magh · 6 marks
Romberg: 1/(1+x²) from -1 to 1
Short answer
Use trapezoidal values with halved step sizes and Richardson extrapolation. The Romberg table gives the integral as approximately 1.5712, close to the exact value pi/2 = 1.5708.
Compute integration of the following function using Romberg integration: ∫ from -1 to 1 of dx/(1 + x²)
Romberg integration applies the trapezoidal rule with halved step sizes and then Richardson extrapolation: R(i,j) = [4ʲ·R(i,j−1) − R(i−1,j−1)] / (4ʲ − 1).
Here f(x) = 1/(1+x²), a = −1, b = 1. Take h = 2, 1, 0.5, 0.25 (that is 1, 2, 4, 8 sub-intervals).
Trapezoidal values
T(h=2) = (2/2)[f(−1)+f(1)] = 1.0000
T(h=1) = ½T(2) + 1·f(0) = 1.5000
T(h=0.5) = ½T(1) + 0.5[f(−0.5)+f(0.5)] = 1.5500
T(h=0.25) = ½T(0.5) + 0.25[f(±0.25)+f(±0.75)] = 1.5656
Romberg table
| h | T (order h²) | R1 (order h^4) | R2 (order h^6) | R3 (order h^8) |
|---|---|---|---|---|
| 2.0000 | 1.0000 | |||
| 1.0000 | 1.5000 | 1.6667 | ||
| 0.5000 | 1.5500 | 1.5667 | 1.5600 | |
| 0.2500 | 1.5656 | 1.5708 | 1.5711 | 1.5712 |
The last row entries (1.5708, 1.5711, 1.5712) agree to 3 decimals, so the extrapolation has converged; the best value is the last entry R3.
Answer: ∫₋₁¹ dx/(1+x²) ≈ 1.5712 (exact value π/2 = 1.5708).
✓ Answer checked by Claude Opus 5.5 (high effort)
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