On a machine slower than the one timed in this chapter

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Computational Physics, Ch Spectra of sampled signals

Problem

On a machine slower than the one timed in this chapter, the direct sum

\tilde{y}_k=\sum_n y_ne^{-2\pi\mathrm{i}kn/N}

takes

8.5\,\mathrm{ms}

for a record of

N=256

samples and

0.136\,\mathrm{s}

for

N=1024

. The library's fast transform, on the same machine, takes

12\,\mu\mathrm{s}

N=1024

and

56\,\mu\mathrm{s}

N=4096

. ((a)) Fit a power law

t\propto N^{\alpha}

to each pair of timings and give the two exponents. ((b)) Predict from its fitted law how long the direct sum takes at

N=4096

. ((c)) Extrapolate the same law to

N=10^6

and give the direct sum's time there in hours.

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