I'm interested in computing of the product $M^n v$, where $M$ is an $m\times m$ matrix (over a semiring) and $v$ is column-vector, with the smallest number of multiplications in the underlying semiring and in the regime of $n=\Theta(m)$.
If we first compute $M^n$ using fast exponentiation, this would take $\Theta(\log n)$ matrix multiplications, totaling in $\Theta(m^3\log n)$ semiring multiplications (assuming naive matrix multiplication).
On the other hand, if we just iteratively compute $M^k v$ for $k=1,2,\dots,n$, each time multiplying a vector by $M$, then it'd take us $\Theta(m^2n)$ semiring multiplications.
So, fast exponentiation appears to be useless. Is there any other way to speed up things here?
