- 2068 Magh · 6 marks
Romberg: 1/x² from 1 to 2
Short answer
Applying Richardson extrapolation to trapezoidal values with h=1, 0.5, 0.25, 0.125 yields the Romberg estimate 0.3218.
Use Romberg's method to compute ∫ from 1 to 2 of dx/(1 + x²).
Romberg's method applies Richardson extrapolation, R(i,j) = [4ʲ·R(i,j−1) − R(i−1,j−1)] / (4ʲ − 1), to the trapezoidal values T(h) with h = 1, 0.5, 0.25, 0.125.
f(x) = 1/(1+x²), a = 1, b = 2.
Trapezoidal values
T(1) = (1/2)[f(1)+f(2)] = 0.3500
T(0.5) = ½T(1) + 0.5·f(1.5) = 0.3288
T(0.25) = ½T(0.5) + 0.25[f(1.25)+f(1.75)] = 0.3235
T(0.125) = ½T(0.25) + 0.125[f(1.125)+f(1.375)+f(1.625)+f(1.875)] = 0.3222
Romberg table
| h | T (order h²) | R1 (order h^4) | R2 (order h^6) | R3 (order h^8) |
|---|---|---|---|---|
| 1.0000 | 0.3500 | |||
| 0.5000 | 0.3288 | 0.3218 | ||
| 0.2500 | 0.3235 | 0.3217 | 0.3217 | |
| 0.1250 | 0.3222 | 0.3218 | 0.3218 | 0.3218 |
Answer: ∫₁² dx/(1+x²) ≈ 0.3218 (exact value tan⁻¹2 − tan⁻¹1 = 0.3218).
✓ Answer checked by Claude Opus 5.5 (high effort)
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