Solving sparse linear systems lies at the core of numerous computational applications. Consequently, understanding the performance of recently proposed alternatives to the established IEEE 754 floating-point numbers, such as bfloat16 and the tapered-precision posit and takum machine number formats, is of significant interest. This paper examines these formats in the context of widely used solvers, namely LU, QR, and GMRES, with incomplete LU preconditioning and mixed precision iterative refinement (MPIR). This contrasts with the prevailing emphasis on designing specialized algorithms tailored to new arithmetic formats. This paper presents an extensive and unprecedented evaluation based on the SuiteSparse Matrix Collection -- a dataset of real-world matrices with diverse sizes and condition numbers. A key contribution is the faithful reproduction of SuiteSparse's UMFPACK multifrontal LU factorization and SPQR multifrontal QR factorization for machine number formats beyond single and double-precision IEEE 754. Tapered-precision posit and takum formats show better accuracy in direct solvers and reduced iteration counts in indirect solvers. Takum arithmetic, in particular, exhibits exceptional stability, even at low precision.

翻译：求解稀疏线性系统是众多计算应用的核心任务。因此，理解近期提出的替代现有IEEE 754浮点数格式的新算术格式（如bfloat16以及锥形精度posit与takum机器数格式）的性能表现具有重要研究意义。本文在广泛使用的求解器（即LU、QR和GMRES）框架下，结合不完全LU预处理与混合精度迭代优化（MPIR）技术，对这些格式进行系统考察。这与当前主要聚焦于为新型算术格式设计专用算法的研究取向形成对比。本文基于SuiteSparse矩阵集合——一个包含不同规模与条件数的真实世界矩阵数据集——开展了全面且前所未有的评估。关键贡献在于：针对超越IEEE 754单双精度范围的机器数格式，精确复现了SuiteSparse中UMFPACK多波前LU分解与SPQR多波前QR分解的运算过程。锥形精度posit与takum格式在直接求解器中表现出更优精度，在迭代求解器中实现更少迭代次数。特别值得注意的是，takum算术即使在低精度条件下仍展现出卓越的数值稳定性。