Here is a simple algorithm implementing the multiplication of an 
matrix and an 
vector. We use only scalar operations here for maximum transparency.
matrix and an vector. We use only scalar operations here for maximum transparency. n = 6;
A = magic(n);
x = ones(n,1);
y = zeros(n,1);
for i = 1:n
for j = 1:n
y(i) = y(i) + A(i,j)*x(j); % 2 flops
end
end
Each of the loops implies a summation of flops. The total flop count for this algorithm is
Since the matrix 
has 
elements, all of which have to be involved in the product, it seems unlikely that we could get a flop count that is smaller than 
.
has elements, all of which have to be involved in the product, it seems unlikely that we could get a flop count that is smaller than .Let's run an experiment with the built-in matrix-vector multiplication. We use tic and toc to start and end timing of the computation.
n_ = (400:400:4000)';
t_ = 0*n_;
for i = 1:length(n_)
n = n_(i);
A = randn(n,n); x = randn(n,1);
tic % start a timer
for j = 1:10 % repeat ten times
A*x;
end
t = toc; % read the timer
t_(i) = t/10; % seconds per instance
end
The reason for doing multiple repetitions at each value of n is to avoid having times so short that the resolution of the timer is a factor.
table(n_,t_,'variablenames',{'size','time'})