Question 5 of 7: Romberg Integration from Tabulated Data

Question 5: Romberg Integration from Tabulated Data (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Nine tabulated points of an unknown curve $F(x)$ over $x\in[0,4]$ at uniform spacing $h=0.5$ (eight intervals $=2^{3}$, exactly enough for a 4-level Romberg triangle).

Find. A four-level Romberg estimate $R(4,4)$ of $\int_0^4 F(x)\,dx$.

Approach. Compute composite-trapezoidal estimates at $h=4,2,1,0.5$ (reusing previously tabulated points at each coarser level), place them in column $R(k,1)$, then apply Richardson extrapolation $R(k,j)=R(k,j-1)+\dfrac{R(k,j-1)-R(k-1,j-1)}{4^{\,j-1}-1}$ across the triangle.

1. Level 1, $h=4$ (2 points: $x=0,4$). $$R(1,1)=\frac{h}{2}\left[F(0)+F(4)\right]=\frac{4}{2}(10+90)=\boxed{200.00000}$$
2. Level 2, $h=2$ (3 points: $x=0,2,4$). $$R(2,1)=\frac{2}{2}\left[10+2(80)+90\right]=260.00000;\qquad R(2,2)=R(2,1)+\frac{R(2,1)-R(1,1)}{3}=\boxed{280.00000}$$
3. Level 3, $h=1$ (5 points: $x=0,1,2,3,4$). $$R(3,1)=\frac{1}{2}\left[10+2(70+80+60)+90\right]=260.00000$$ $$R(3,2)=R(3,1)+\frac{R(3,1)-R(2,1)}{3}=260.00000;\qquad R(3,3)=R(3,2)+\frac{R(3,2)-R(2,2)}{15}=\boxed{258.66667}$$
4. Level 4, $h=0.5$ (all 9 points). $$R(4,1)=\frac{0.5}{2}\left[10+2(63.75+70+86.25+80+68.75+60+61.25)+90\right]=270.00000$$ $$R(4,2)=273.33333,\qquad R(4,3)=274.22222,\qquad R(4,4)=R(4,3)+\frac{R(4,3)-R(3,3)}{63}=\boxed{274.46914}$$ Each successive column narrows the estimate (200 → 280 → 258.67 → 274.47), converging as the extrapolation cancels higher-order error terms.