A sample decays as

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Computational Physics, Ch Numbers a machine can hold, and the two errors that follow

Problem

A sample decays as N(t)=N0e−5tN(t) = N_0 e^{-5t}, with N0=1N_0 = 1 (a dimensionless fraction of the initial number of nuclei) and tt in seconds, and its instantaneous decay rate at t=0t=0 is estimated by the forward-difference formula, using ϵmach=2.220446×10−16\epsilon_{\rm mach} = 2.220446\times10^{-16}. (a) Using hopt≈2ϵmach∣f/f′′∣h_{\rm opt} \approx 2\sqrt{\epsilon_{\rm mach}|f/f''|}, predict, before looking at any table, the decade of hh at which the forward-difference error should be smallest. (b) A one-decade-per-row run, in binary64, recorded the following errors of the forward-difference estimate against the exact derivative at t=0t=0: h=10−6 sh=10^{-6}\,\mathrm{s}: 1.2×10−51.2\times10^{-5}; h=10−7 sh=10^{-7}\,\mathrm{s}: 1.3×10−61.3\times10^{-6}; h=10−8 sh=10^{-8}\,\mathrm{s}: 1.3×10−71.3\times10^{-7}; h=10−9 sh=10^{-9}\,\mathrm{s}: 3.0×10−83.0\times10^{-8}; h=10−10 sh=10^{-10}\,\mathrm{s}: 4.1×10−74.1\times10^{-7}; h=10−11 sh=10^{-11}\,\mathrm{s}: 4.1×10−74.1\times10^{-7}. State which row turns over, and say whether it agrees with the prediction of part (a) to the precision that an order-of-magnitude argument can support.

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