  We give a self-contained randomized algorithm based on shifted inverse
iteration which provably computes the eigenvalues of an arbitrary matrix
$M\in\mathbb{C}^{n\times n}$ up to backward error $\delta\|M\|$ in
$O(n^4+n^3\log^2(n/\delta)+\log(n/\delta)^2\log\log(n/\delta))$ floating point
operations using $O(\log^2(n/\delta))$ bits of precision. While the $O(n^4)$
complexity is prohibitive for large matrices, the algorithm is simple and may
be useful for provably computing the eigenvalues of small matrices using
controlled precision, in particular for computing Ritz values in shifted QR
algorithms as in (Banks, Garza-Vargas, Srivastava, 2022).
