section_path: Properties of series/Evaluation of truncation errors/Matrix exponential
section_name: Matrix exponential
paragraph_text: Matrix exponential
  
  For the matrix exponential:
  
  exp (X):= ∑ k = 0 ∞ 1 k ! X k, X ∈ C n × n, {\displaystyle \exp(X):=\sum _{k=0}^{\infty }{\frac {1}{k!}}X^{k},\quad X\in \mathbb {C} ^{n\times n},}
  
  the following error evaluation holds (scaling and squaring method):
  
  T r, s (X):= s, ‖ exp (X) − T r, s (X) ‖ ≤ ‖ X ‖ r + 1 s r (r + 1) ! exp (‖ X ‖). {\displaystyle T_{r,s}(X):=\left^{s},\quad \|\exp(X)-T_{r,s}(X)\|\leq {\frac {\|X\|^{r+1}}{s^{r}(r+1)!}}\exp(\|X\|).}