Romberg Integration of Tabulated Data: 04-BS-5, December 2016, Q5

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 data (Q5)
$x$	0	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0
$f(x)$	55	64	76	86	100	110	124	135	145

Given. Nine tabulated $(x,f(x))$ pairs spaced $h=0.5$ apart on $[0,4]$.

Find. $\displaystyle\int_0^4f(x)\,dx$, approximated through the Romberg array $R(4,4)$.

Approach. Build the trapezoidal column $R(k,1)$ by successively halving the step ($H_1=4,H_2=2,H_3=1,H_4=0.5$), then apply Richardson extrapolation across each row using the printed recursion $R(k,j)=R(k,j-1)+\dfrac{R(k,j-1)-R(k-1,j-1)}{4^{j-1}-1}$.

Coarsest trapezoid, $H_1=4$ (endpoints only). $$R(1,1)=\frac{H_1}{2}\bigl[f(0)+f(4)\bigr]=\frac{4}{2}(55+145)=400.0$$
$H_2=2$ (adds $f(2)=100$). $$R(2,1)=\frac12\bigl[R(1,1)+H_1\cdot f(2)\bigr]=\frac12[400+4(100)]=400.0,\qquad R(2,2)=R(2,1)+\frac{R(2,1)-R(1,1)}{3}=400.0$$
$H_3=1$ (adds $f(1)=76,\ f(3)=124$). $$R(3,1)=\frac12\bigl[R(2,1)+H_2(76+124)\bigr]=\frac12[400+2(200)]=400.0$$ $$R(3,2)=400.0+\frac{400.0-400.0}{3}=400.0,\qquad R(3,3)=400.0+\frac{400.0-400.0}{15}=400.0$$
$H_4=0.5$ (adds $f(0.5),f(1.5),f(2.5),f(3.5)=64,86,110,135$) — the row that finally reveals curvature. $$R(4,1)=\frac12\bigl[R(3,1)+H_3(64+86+110+135)\bigr]=\frac12[400+1(395)]=397.5$$ $$R(4,2)=397.5+\frac{397.5-400.0}{3}=396.667,\quad R(4,3)=396.667+\frac{396.667-400.0}{15}=396.444$$ $$\boxed{R(4,4)=396.444+\frac{396.444-400.0}{63}=396.39}$$

$k$	$R(k,1)$	$R(k,2)$	$R(k,3)$	$R(k,4)$
1	400.00
2	400.00	400.00
3	400.00	400.00	400.00
4	397.50	396.67	396.44	396.39

Topic: Romberg integration (repeated Richardson extrapolation of the trapezoidal rule).

Key relations: trapezoidal doubling $R(k,1)=\frac12[R(k-1,1)+H_{k-1}\sum f(\cdots)]$; extrapolation $R(k,j)=R(k,j-1)+\dfrac{R(k,j-1)-R(k-1,j-1)}{4^{j-1}-1}$.

Why this works: the trapezoidal rule's error expands in even powers of $h$; each extrapolation column cancels the next leading error term, so $R(k,k)$ is $O(h^{2k})$ accurate. Here the first three rows stayed exactly flat at $400.0$ because the coarse (2-, 3-, 5-point) trapezoids already captured the near-linear trend in this data; only the finest, $h=0.5$ row exposes the slight curvature that nudges the estimate down to $396.39$.

Common pitfall: using $H_k$ (the new, finer spacing) instead of $H_{k-1}$ when weighting the newly-added function values in the $R(k,1)$ doubling sum — the printed recursion explicitly multiplies by $H_{k-1}$, and swapping the subscript silently shifts every subsequent row.

Source: Kreyszig, Advanced Engineering Mathematics, Ch. 19 (Numerical Integration).

Question 5 of 7: Romberg Integration of Tabulated Data

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