  We present a randomized, inverse-free algorithm for producing an approximate
diagonalization of any $n \times n$ matrix pencil $(A,B)$. The bulk of the
algorithm rests on a randomized divide-and-conquer eigensolver for the
generalized eigenvalue problem originally proposed by Ballard, Demmel, and
Dumitriu [Technical Report 2010]. We demonstrate that this divide-and-conquer
approach can be formulated to succeed with high probability provided the input
pencil is sufficiently well-behaved, which is accomplished by generalizing the
recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni, and
Srivastava [Foundations of Computational Mathematics 2022]. In particular, we
show that perturbing and scaling $(A,B)$ regularizes its pseudospectra,
allowing divide-and-conquer to run over a simple random grid and in turn
producing an accurate diagonalization of $(A,B)$ in the backward error sense.
The main result of the paper states the existence of a randomized algorithm
that with high probability (and in exact arithmetic) produces invertible $S,T$
and diagonal $D$ such that $||A - SDT^{-1}||_2 \leq \varepsilon$ and $||B -
ST^{-1}||_2 \leq \varepsilon$ in at most $O \left(\log^2 \left(
\frac{n}{\varepsilon} \right) T_{\text{MM}}(n) \right)$ operations, where
$T_{\text{MM}}(n)$ is the asymptotic complexity of matrix multiplication. This
not only provides a new set of guarantees for highly parallel generalized
eigenvalue solvers but also establishes nearly matrix multiplication time as an
upper bound on the complexity of inverse-free, exact arithmetic matrix pencil
diagonalization.
