Matrix$-$vector multiplication, denoted as $y=Ax,$ is a very common operation in linear algebra and has a

variety of real$-$world applications $($e$.$g$.$ graph theory, deep learning, quantum mechanics, etc.$).$ Each $y_i$ entry

is computed as $A_{i0}{x}_0+A_{i1}{x}_1+$dots$+A_{ij}{x}_j.$

Suppose you are a researcher with access to high$-$performance computing $($HPC$)$ systems at ARCH $($Advanced

Research Computing at Hopkins$).$ You are tasked to implement an optimized the matrix$-$vector routine in C$.$

An example declaration of your function follows below:$//$ y is an m$-$length vector $($array$)$

$//$ A is an m by n matrix $(2$d array$)$

$//$ x is an n$-$length vector $($array$)$

void matrix$\_$vector$\_$multiply$($double$*$ y$,$ double$*$ A$,$ double$*$ x$,$ int m$,$ int n$)$;An efficient routine will make good use of pipelineing and caches. Assume for this task that all matrix$-$vector

multiplication routines considered have equivalent algorithmic efficiency.

$($a$)$ Assume $A$ is too big to fit into cache at once, but $x$ and $y$ could potentially fit into cache simultaneously

with some room left over $($i$.$e$.m+n$ is much less than $mn).$ What is the best order to iterate over the

entries of $A?{?}^1$ Why? Give your answer in two to four sentences long.

$($b$)$ Does matrix$-$vector multiplication present any pipelining concerns $($e$.$g$.$ data hazards, control hazards$)?$

Why or why not? Give your answer in two to four sentences long.