  Hierarchical matrix computations have attracted significant attention in the
science and engineering community as exploiting data-sparse structures can
significantly reduce the computational complexity of many important kernels.
One particularly popular option within this class is the Hierarchical
Off-Diagonal Low-Rank (HODLR) format. In this paper, we show that the
off-diagonal blocks of HODLR matrices that are approximated by low-rank
matrices can be represented in low precision without degenerating the quality
of the overall approximation (with the error growth bounded by a factor of
$2$). We also present an adaptive-precision scheme for constructing and storing
HODLR matrices, and we prove that the use of mixed precision does not
compromise the numerical stability of the resulting HOLDR matrix--vector
product and LU factorization. That is, the resulting error in these
computations is not significantly greater than the case where we use one
precision (say, double) for constructing and storing the HOLDR matrix. Our
analyses further give insight on how one must choose the working precision in
HOLDR matrix computations relative to the approximation error in order to not
observe the effects of finite precision. Intuitively, when a HOLDR matrix is
subject to a high degree of approximation error, subsequent computations can be
performed in a lower precision without detriment. We demonstrate the validity
of our theoretical results through a range of numerical experiments.
