A detector's overall efficiency is formed by multiplying positive calibration factors

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

Scientific Computing for Physicists, Ch The numerical toolkit closed: backward error and the audit of a library routine

Problem

A detector's overall efficiency is formed by multiplying N=64N = 64 positive calibration factors, one after another, in a fixed order, in binary64 arithmetic: p^=fl(⋯fl(fl(x1x2)x3)⋯xN)\hat p = \mathrm{fl}(\cdots\mathrm{fl}(\mathrm{fl}(x_1x_2)x_3)\cdots x_N). Each of the N−1N-1 multiplications rounds its result, introducing a relative error of magnitude at most ϵmach/2\epsilon_{\rm mach}/2, exactly as each addition does in recursive summation, so the rounded product is the exact product multiplied by one factor (1+δk)(1+\delta_k) for each multiplication. ((a)) By collecting the roundings factor by factor, as for recursive summation, give the bound on the relative forward error of p^\hat p. ((b)) The observed relative error of p^\hat p, found by comparison with a higher-precision recomputation, is 2.8×10−152.8\times10^{-15}. Is this within the bound?

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