For at , the following table gives the error in the central-difference estimate of…

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

Computational Physics, Ch Differentiating and integrating numbers

Problem

For f(x)=ln⁡xf(x)=\ln x at x0=2x_0=2, the following table gives the error in the central-difference estimate Dhf(x0)D_hf(x_0) of f′(x0)f'(x_0) predicted by the chapter's total-error model E(h)=16h2∣f′′′∣+ϵmach∣f∣/hE(h)=\tfrac{1}{6}h^2|f'''|+\epsilon_{\rm mach}|f|/h, at a sequence of steps:
hh10−210^{-2}10−310^{-3}10−410^{-4}10−510^{-5}10−610^{-6}10−710^{-7}
error4.17×10−64.17\times10^{-6}4.17×10−84.17\times10^{-8}4.18×10−104.18\times10^{-10}1.96×10−111.96\times10^{-11}1.54×10−101.54\times10^{-10}1.54×10−91.54\times10^{-9}
(a) Use hopt=(3ϵmach∣f/f′′′∣)1/3h_{\rm opt}=(3\epsilon_{\rm mach}|f/f'''|)^{1/3} to predict the optimal step, taking ϵmach=2.22×10−16\epsilon_{\rm mach}=2.22\times10^{-16}. (b) Read the step giving the smallest tabulated error, and state whether it agrees with the prediction to within an order of magnitude.

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