$1.$ Answer items a$-$f below, assuming a single iteration $$i$$ of the following loop on an IBM BlueGene$/$Q processor $($Peak $204\mathrm{.}8$ GFLOP$/$node$)$

for $($i $=0$; i $<$ N; $++$i$)$ y$[$i$]=$ a$*$x$[$i$]+$y$[$i$]$

Note $1$: Loads and stores are $8$ bytes each. Note $2$: The Roofline model for the IBM BlueGene$/$Q processor is given in Fig. $1.$

Figure $1-$ Roofline model for the IBM BlueGene$/$Q processor

a$.[0\mathrm{.}5$ point$]$ Calculate the memory bandwidth of the IBM BlueGene$/$Q processor.

b$.[0\mathrm{.}5$ point$]$ Calculate the arithmetic intensity of the loop execution.

c$.[0\mathrm{.}5$ point$]$ Estimate the maximum performance that can be achieved for the loop execution.

d$.[0\mathrm{.}5$ point$]$ A student performed some experiments on the system to assess to its performance when executing the loop with single and multiple threads. The measurements from the student$$s experiment are depicted in Fig. $2.$ Discuss these results and explain how they agree$/$disagree with your result from part $($c$)$ and present some suggestions on how to improve the performance of the program.

$1$

Figure $2-$ Experiment Results

e$.[0\mathrm{.}5$ point$]$ Suppose you run another program on this processor, and its performance falls on the diagonal line of the roofline plot. What can you say about this program?

f$.[0\mathrm{.}5$ point$]$ Suppose you run another program on this processor, and its data points fall above the roofline limit of the roofline plot. What can you say about this program?