Past exam of the numerical analysis course of the University of Oxford / 2015 / 1 / d / Solution

Write $E(h)$ for the supplied approximation. The asymptotic expansion iswhere $O$ is Big O notation. At the refined parameter $h/3$,Multiply this identity by nine and subtract the first one. The unknown leading truncation error coefficient cancels:Thus the Richardson extrapolation estimate isThe two supplied values giveThe correction to the finer approximation is $(3-2)/8=1/8$. Neglecting the higher-order remainder would also give $c=9/8$.

The conclusion is asymptotic: the leading error order improves from two to at least three when both parameter values are in the range of the stated asymptotic expansion. With no bound on the remainder constant, the two finite observations alone do not guarantee that $25/8$ is closer to $Y$. Indeed, for any proposed value $L$, the polynomial modelsatisfies $E(1)=2$, $E(1/3)=3$ and the required asymptotic expansion with true limit $L$. This is why finite observations do not certify Richardson extrapolation accuracy: asymptotic cancellation provides an improved-order estimator, while a numerical error certificate needs quantitative remainder information.