The amplitude reaching a detector from the -th ring of a model diffraction pattern is…

PHYSICS · Multi-part · 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 amplitude reaching a detector from the

i

-th ring of a model diffraction pattern is

A_i = 1/i

(dimensionless), and the

N = 2\times10^5

ring amplitudes

A_1,\dots,A_N

are added in binary32 (

u = 5.96\times10^{-8}

). The exact sum, from the harmonic-series asymptotic formula, is

S = \sum_{i=1}^N A_i = 12.7832908

to the precision quoted. ((a)) Added by a plain left-to-right (recursive) loop, the binary32 result is

12.7827568

. Added by pairwise (cascade) summation, with adjacent pairs and a zero appended whenever a level has an odd count, the binary32 result is

12.7832947

. State the absolute error of each. ((b)) Compute the recursive bound

(N-1)u\sum_i|A_i|

and the pairwise bound

\lceil\log_2 N\rceil\,u\sum_i|A_i|

, using

\sum_i|A_i| = S

since every term is positive. ((c)) Compare each measured error with its own bound, and state which order of summation has the tighter (more informative) guarantee.

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