We present the Iterated-Tverberg algorithm, the first deterministic algorithm for computing an approximate centerpoint of a set $S\in R^d$ with running time sub-exponential in $d$. The algorithm is a derandomization of the Iterated-Radon algorithm of Clarkson et al and is guaranteed to terminate with an $O(1/d^2)$-center. Moreover, it returns a polynomial-time checkable proof of the approximation guarantee, despite the coNP-Completenes of testing centerpoints in general. We also explore the use of higher order Tverberg partitions to improve the runtime of the deterministic algorithm and improve the approximation guarantee for the randomized algorithm. In particular, we show how to improve the $O(1/d^2)$- center of the Iterated-Radon algorithm to $O(1/d^{\frac{r}{r-1}})$ for a cost of $O((rd)^{d})$ in time for any integer $r$.