  In recent years, a new kind of accelerated hardware has gained popularity in
the Artificial Intelligence (AI) and Machine Learning (ML) communities which
enables extremely high-performance tensor contractions in reduced precision for
deep neural network calculations. In this article, we exploit Nvidia Tensor
cores, a prototypical example of such AI/ML hardware, to develop a mixed
precision approach for computing a dense matrix factorization of the inverse
overlap matrix in electronic structure theory, $S^{-1}$. This factorization of
$S^{-1}$, written as $ZZ^T=S^{-1}$, is used to transform the general matrix
eigenvalue problem into a standard matrix eigenvalue problem. Here we present a
mixed precision iterative refinement algorithm where $Z$ is given recursively
using matrix-matrix multiplications and can be computed with high performance
on Tensor cores. To understand the performance and accuracy of Tensor cores,
comparisons are made to GPU-only implementations in single and double
precision. Additionally, we propose a non-parametric stopping criteria which is
robust in the face of lower precision floating point operations. The algorithm
is particularly useful when we have a good initial guess to $Z$, for example,
from previous time steps in quantum-mechanical molecular dynamics simulations
or from a previous iteration in a geometry optimization.
