Past exam of the options I course of the University of Oxford / 2012 / P / P2

The leading error term can be removed by comparing the two step sizes $h$ and $h/2$:Multiply the second expression by four and subtract the first. The $h^2$ terms cancel, so the first Richardson extrapolation step isRepeat with $R_1(h/2)$ and $R_1(h)$, now cancelling the fourth-order error using the factor $16$. In general, with $R_0(h)=I_h$,Under the stated even-power expansion, successive columns have errors of orders $h^4,h^6,h^8,\dots$. Each step uses the known scaling of the leading error to eliminate its unknown coefficient.

For one interval, the trapezoidal rule givesFor two intervals of width $h=1/2$, the composite trapezoidal rule gives half weights at the endpoints and full weight at the internal node:

Apply the first Richardson extrapolation step to the two computed approximations:The exact integral is $I={\int }_0^1{x}^{4}dx=1/5$, so the actual absolute error isThus the requested strict error bound is verified directly. This first extrapolation of trapezoidal approximations is also the one-panel Simpson's rule value on the three nodes $0,1/2,1$.