  The emergence of low precision floating-point arithmetic in computer hardware
has led to a resurgence of interest in the use of mixed precision numerical
linear algebra. For linear systems of equations, there has been renewed
enthusiasm for mixed precision variants of iterative refinement. We consider
the iterative solution of large sparse systems using incomplete factorization
preconditioners. The focus is on the robust computation of such preconditioners
in half precision arithmetic and employing them to solve symmetric positive
definite systems to higher precision accuracy; however, the proposed ideas can
be applied more generally. Even for well-conditioned problems, incomplete
factorizations can break down when small entries occur on the diagonal during
the factorization. When using half precision arithmetic, overflows are an
additional possible source of breakdown. We examine how breakdowns can be
avoided and we implement our strategies within new half precision Fortran
sparse incomplete Cholesky factorization software. Results are reported for a
range of problems from practical applications. These demonstrate that, even for
highly ill-conditioned problems, half precision preconditioners can potentially
replace double precision preconditioners, although unsurprisingly this may be
at the cost of additional iterations of a Krylov solver.
