A stress field on a two-dimensional grid is stored as a float64 array of shape in C order

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Scientific Computing for Physicists, Ch Arrays in memory: layout, cost and the data structures of physical problems

Problem

A stress field on a two-dimensional grid is stored as a float64 array

\sigma

of shape

(500, 800, 3, 3)

in C order, so

\sigma[p,q]

is the

3\times3

stress tensor at grid cell

(p,q)

. ((a)) Is

\mathrm{diag} = \sigma[\ldots,0,0]

(the

xx

-component field) a view or a copy? Give its shape and byte strides. ((b)) Using the

64\,\mathrm{B}

cache line,

8\,\mathrm{B}

-element model, predict the ratio of the memory traffic of

\mathrm{diag}.\mathrm{sum}()

to that of summing a contiguous array of the same

500\times800

float64 values, and hence the ratio of their times if both sums stream from main memory at the same rate. ((c)) A stress-invariant loop computes trace = sigma[...,0,0] + sigma[...,1,1] + sigma[...,2,2] at every one of

2000

time steps, each call allocating a new temporary array. Predict whether accumulating into one preallocated buffer with in-place addition changes the bytes read from

\sigma

each step. ((d)) Measurement: summing a contiguous array of

500\times800 = 4\times10^{5}

float64 values takes

0.5\,\mathrm{ms}

; summing

\mathrm{diag}

takes

3.9\,\mathrm{ms}

. Over the full

2000

-step loop the original code takes

25.3\,\mathrm{s}

and the preallocated in-place version takes

25.0\,\mathrm{s}

. State the measured ratio and the measured percentage reduction in loop time, and account for how each compares with your predictions in (b) and (c).

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