Let $J$ be the following $k-by-k$ Jordan block: $ J:= \begin{bmatrix} e^{i \theta} & 1 & \\ & e^{i \theta} & 1 \\ & & \ddots & \ddots \\ & & & \ddots & 1 \\ & & & & e^{i \theta} \end{bmatrix}. $
Is there an efficient way to compute the action of large powers of $J$, ie: $J^n x$ to within tolerance $\epsilon$, when $n$ is extremely large?
Notes,
- The goal is to find a computational procedure $f_n$, so that $||f_n(x)-J^nx||<\epsilon$ for all $x$, and computing $f_n(x)$ takes (significantly) less work than iteratively applying $J$ over and over.
- Feel free to make $n$ depend on $k$ and $\epsilon$ and let $n$ it be as large as you want - I'm interested in the asymptotic behavior.
- $\theta/2\pi$ could be irrational