A charge-integrating amplifier samples a current every few microseconds and digitally…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Mathematical Methods for Physics, Ch Floating-point arithmetic, conditioning and numerical stability

Problem

A charge-integrating amplifier samples a current every few microseconds and digitally sums the resulting charge increments to obtain the total charge delivered by a discharge pulse. The sum has N=1.2×106N = 1.2\times10^6 increments, each of size about 3.0×10−6 C3.0\times10^{-6}\,\mathrm{C}, so the running total grows steadily to Q≈3.6 CQ \approx 3.6\,\mathrm{C} and never exceeds it. The summation is carried out in double precision, ϵM=2.2×10−16\epsilon_{\mathrm{M}} = 2.2\times10^{-16}. ((a)) Estimate the worst-case absolute error in the computed total charge, treating every rounding error as adding with the same sign. ((b)) Estimate the typical absolute error expected from random rounding, treating the rounding at each addition as an independent small contribution.

Browse the problem library