标题： Bfloat16、Posit 和 Takum 算术在稀疏线性求解器中的评估 显示英文标题

标题： Evaluation of Bfloat16, Posit, and Takum Arithmetics in Sparse Linear Solvers

摘要： 求解稀疏线性系统是许多计算应用的核心。 因此，了解最近提出的替代传统IEEE 754浮点数的方案的性能，例如bfloat16以及渐变精度的posit和takum机器数格式，具有重要意义。 本文在广泛使用的求解器（即LU、QR和GMRES）背景下研究这些格式，包括不完全LU预处理和混合精度迭代细化（MPIR）。 这与目前侧重于为新算术格式设计专用算法的趋势形成对比。 本文基于SuiteSparse Matrix Collection进行了广泛且前所未有的评估——这是一个包含各种尺寸和条件数的真实世界矩阵的数据集。 一个关键贡献是忠实再现了SuiteSparse的UMFPACK多前端LU分解和SPQR多前端QR分解，适用于超出单精度和双精度IEEE 754的机器数格式。 渐变精度的posit和takum格式在直接求解器中表现出更好的精度，在间接求解器中减少了迭代次数。 特别是takum算术在低精度下也表现出极高的稳定性。 显示英文摘要

摘要： 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.