A scalar field is evaluated on a square grid of side

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Computational Physics, Ch Arrays: a field is a rectangle of numbers

Problem

A scalar field

f(x,y) = e^{-x^2-y^2}

is evaluated on a square grid of side

N

, once with a double loop over the two indices and once with the equivalent broadcasting array expression. Each timing below is the smallest of seven repeats from

\texttt{timeit.repeat}

on a stated machine:

$N$	loop time	array time
$51$	$0.83\,\mathrm{ms}$	$0.021\,\mathrm{ms}$
$201$	$13.2\,\mathrm{ms}$	$0.089\,\mathrm{ms}$
$801$	$209\,\mathrm{ms}$	$1.31\,\mathrm{ms}$
$2001$	$1310\,\mathrm{ms}$	$12.9\,\mathrm{ms}$

(a) For each

N

, compute the ratio of the loop time to the array time. (b) Compute the loop time and the array time per grid node (i.e. divided by

N^2

), in nanoseconds, at each of the four sizes. (c) State, in one or two sentences, what the per-node figures show about why the ratio rises and then falls, without assuming that a grid size beyond those given behaves the same way.

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