The increased degree of concurrent operations by lower precision arithmetic enables high performance for iterative refinement. Most of related work present statically defined mixed precision arithmetic approaches, while adapting a level of arithmetic precision dynamically in a loop with one-bit granularity can further improve the performance. This paper presents Arbitrary Dynamic Precision Iterative Refinement algorithm (AIR) that minimizes the total significand bit-width to solve iterative refinement. AIR detects the number of cancellation bits dynamically per iteration and uses the information to provide the least sufficient significand bit-width for the next iteration. We prove that AIR is a backward stable algorithm and can bring up to $2-3\times$ speedups over a mixed precision iterative refinement depending on the characteristics of hardware. Our software demonstration shows that AIR requires only 83% of the significand bits required by mixed precision iterative refinement that solve linear systems for double precision accuracy for backward error with 32 × 32 standard normally distributed matrices.

较低精度的算术提高了并行操作的程度，从而为迭代优化提供了高性能。大多数相关工作提出了静态定义的混合精度算术方法，而在具有一位粒度的循环中动态调整算术精度水平可以进一步提高性能。本文提出了一种任意动态精度迭代细化算法（AIR），该算法将总有效位和最小位宽最小化以解决迭代细化问题。AIR在每次迭代中动态检测取消位的数量，并使用该信息为下一次迭代提供最少有效的位宽。我们证明AIR是一种向后稳定的算法，可以提出$2-3\times$取决于硬件的特性，可以加快混合精度迭代细化的速度。我们的软件演示表明，AIR仅需要混合精度迭代优化所需要的有效位的83％，该精度可以用32×32标准正态分布矩阵解决线性系统的双精度精度，以实现向后误差。