Q$20$: Complexity calculation

The command flops used to be available in early versions of Matlab. Currently, cputime, etime, tic$/$toc are the only available commands if you want to indirectly evaluate complexity.

Let $Ain{R}^{mn},n{R}^n.$ Consider the matrix vector product $y=Ax.$ By definition,

$y_i={\displaystyle \underset{j=1}{\overset{n}{}}}{a}_{ij}{x}_j,i=1,$dots,$m.$

Modify this expression for the case where $A$ is a dense lower$-$triangular matrix, so that operations with the zero entries in A are excluded.

Implement your modified expression using the nested loop below so that it computes the matrix$-$vector product, while at the same time it updates a variable flops that counts the number of operations to termination.

rng$(20)$; m$=$randi$([1120])$; $n=$ randi$(10)$;$\%--->$ SETUP: do not edit

$A=$tril$(randn(m,n))$;$x=$randn$(n,1)$;$\%-$cdots SETUP: do not edit

$y=$zeros$(m,1)$;

flops $=0$;

for $i=1$:$1$

for $j=1$:$1$

$y=y$;

flops$=$flops;

end