A data-logger records successive current readings from a circuit and a program sums…

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

Error, Chance and the Machine, Ch Rounding, backward error and the sensitivity of a computation

Problem

A data-logger records

n = 500

successive current readings from a circuit and a program sums them in binary64 by plain left-to-right recursive addition to obtain the total. Using the derived bound that

\mathrm{fl}\left(\sum_{i=1}^n x_i\right)

equals the exact sum of the

x_i

each perturbed by at most

(n-1)u

in relative terms, find the numerical bound on the backward error (the largest relative change any one reading would need in order to make the computed total exact).

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