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.

翻译：计算机硬件中低精度浮点算术的出现重新激发了人们对混合精度数值线性代数的兴趣。对于线性方程组，迭代精化的混合精度变体再次引发关注。我们考虑使用不完全分解预条件子对大型稀疏系统进行迭代求解。研究重点在于半精度算术下此类预条件子的鲁棒计算，并将其用于求解对称正定系统以获得更高精度；然而，所提出的思想可更广泛地应用。即使在良态问题中，当分解过程中对角线上出现小元素时，不完全分解也可能崩溃。使用半精度算术时，溢出是额外的崩溃来源。我们研究了如何避免崩溃，并在新的半精度Fortran稀疏不完全乔列斯基分解软件中实现了相关策略。针对实际应用中的一系列问题报告了结果。这些结果表明，即使对于高度病态的问题，半精度预条件子也可能替代双精度预条件子，尽管这可能会以增加Krylov求解器的迭代次数为代价。