Work in the float32 arithmetic of the harmonic-series example

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

Work in the float32 arithmetic of the harmonic-series example, with

\epsilon_{\rm mach}=2^{-23}

and every operation rounded to nearest. ((a)) For three terms, write the recursively computed sum

\mathrm{fl}(\mathrm{fl}(x_1+x_2)+x_3)

with one rounding factor

(1+\delta_k)

per addition, and show that to first order in

\epsilon_{\rm mach}

the effective perturbations on

x_1

x_2

and

x_3

are

\delta_2+\delta_3

\delta_2+\delta_3

and

\delta_3

. Explain how this pattern gives the bound

(N-1)\epsilon_{\rm mach}/2

for

N

terms, and evaluate it for

N=2^{20}

. ((b)) The float32 value stored for

0.1

x=0.100\,000\,001\,490\,116\,12

. Add

x

to itself

N=2^{20}

times in float32, once with a plain running total and once with Kahan's update, and give the relative error of each against the exact sum

Nx

. Explain why

Nx=104\,857.601\,562\,5

is exact. ((c)) State whether each error respects its bound, and what the comparison shows about growth with

N

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