Question 5 of 7: Romberg Integration of Tabulated Data

Question 5: Romberg Integration of 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 spanning $x=-2$ to $x=2$ at uniform step $h=0.5$.

Find. The Romberg estimate $R(4,4)$ of $\displaystyle\int_{-2}^{2}y\,dx$.

Approach. Build trapezoidal estimates $R(k,1)$ for step sizes $H=4,2,1,0.5$ by sub-sampling the table (using every 8th, 4th, 2nd, and every point respectively), then Richardson-extrapolate the triangular array using the recursion.

1. Base trapezoidal estimates. $R(1,1)$ uses only the two endpoints ($H_1=4$); $R(2,1)$ adds the midpoint ($H_2=2$, nodes $-2,0,2$); $R(3,1)$ uses every second table point ($H_3=1$, nodes $-2,-1,0,1,2$); $R(4,1)$ uses the full table ($H_4=0.5$): $$R(1,1)=\dfrac{4}{2}[10.0+90.0]=\boxed{200}$$ $$R(2,1)=\dfrac{2}{2}[10.0+2(80.0)+90.0]=\boxed{260}$$ $$R(3,1)=\dfrac{1}{2}[10.0+2(70.0+80.0+60.0)+90.0]=\boxed{260}$$ $$R(4,1)=\dfrac{0.5}{2}[10.0+2(63.75+70.0+86.25+80.0+68.75+60.0+61.25)+90.0]=\boxed{270}$$
2. Richardson-extrapolate column 2 ($j=2$, kills the $O(H^2)$ error). $R(k,2)=R(k,1)+\dfrac{R(k,1)-R(k-1,1)}{3}$: $$R(2,2)=260+\dfrac{260-200}{3}=\boxed{280},\quad R(3,2)=260+\dfrac{260-260}{3}=260,\quad R(4,2)=270+\dfrac{270-260}{3}=273.\overline3$$
3. Extrapolate column 3 ($j=3$). $R(k,3)=R(k,2)+\dfrac{R(k,2)-R(k-1,2)}{15}$: $$R(3,3)=260+\dfrac{260-280}{15}=258.\overline6,\qquad R(4,3)=273.\overline3+\dfrac{273.\overline3-260}{15}=274.2\overline2$$
4. Extrapolate column 4 ($j=4$, the final answer). $R(4,4)=R(4,3)+\dfrac{R(4,3)-R(3,3)}{63}=274.2\overline2+\dfrac{274.2\overline2-258.\overline6}{63}$ $$\boxed{R(4,4)=274.4691\ (=22232/81)}$$