Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics and probability resit / 6 / d

Write $f^{(4)}(x)=24(5-4x)/x^6$. Its derivative is $f^{(5)}(x)=240(2x-3)/x^7$, so on $[1,2]$ it decreases to $x=3/2$ and then increases. Its endpoint and turning-point values areThus $M_4=24$, and the preceding bound becomesFor the rounding estimate, standard binary64 has spacing $ϵ=2^{-52}$ above one. Balancing this nominal absolute accuracy with the bound givesThus a few thousand even strips, roughly $5000$, reach the double-precision scale according to this bound. A scale-aware estimate leads to the same order. Indeed,lies in magnitude between $4$ and $8$, where the binary64 spacing is $2^{-50}$. Balancing the error bound against half that spacing gives about $4.16\times 10^3$ strips.

These are saturation estimates from balancing quadrature truncation and round-off error, not sharp guarantees of the actual rounding plateau. The error bound can be conservative, and evaluation and summation errors can limit accuracy earlier. Merely storing the final result does not specify every source of rounding error.