The work done by a varying force over discrete displacement segments is computed as…

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

The work done by a varying force over

n = 300

discrete displacement segments is computed as the dot product

W = \sum_{i=1}^{n} F_i\,\Delta x_i

, evaluated in binary64 by forming each product

F_i\Delta x_i

and then summing left to right. Each product satisfies

\mathrm{fl}(F_i\Delta x_i) = F_i\Delta x_i(1+\delta_i)

|\delta_i|\le u

, and the subsequent summation of the

n

products carries the recursive summation backward error

(n-1)u

derived for a sum of

n

terms. Combining these two roundings to first order in

u

, estimate the backward error bound on the computed

W

: the largest relative perturbation that would need to be applied to each

F_i\Delta x_i

for the computed total to be exact.

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