How many counting operations are for matrix exponential?

Question

Let A be a m x m matrix and

$$\sum_{k=0}^n \frac{A^k}{k!}$$

I have this following flops count:

1. For $A^k$, $(n-1)(2m^3-m^2)$ flops
2. $n$ multiplications for $k!$
3. $n$ divisions for $\frac{A^k}{k!}$
4. $n-1$ additions for the summation

So the total is $(3n - 1) + (n-1)(2m^3-m^2)$ flops. Is it correct?

Answer 1

Let's slow down a bit and count summand by summand: \begin{array}{lll} \mbox{summand} & \mbox{multiplications required, using prev. results} & \mbox{some prev. results}\\ \hline {1 \over 2!} A^2 & 1 + m^3 + m^2 & \\ \\ {1 \over 3!} A^3 & 1 + m^3 + m^2 \; \mbox{(as $A^3 = A^2 A$, i.e., $m^3$ multiplications)} & {1 \over 2!}, \; A^2 \\ \\ {1 \over 4!} A^4 & 1 + m^3 + m^2 \; \mbox{(as $A^4 = A^3 A$, i.e., $m^3$ multiplications)} & {1 \over 3!}, \; A^3 \\ \\ \end{array} Am I missing something?