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

Solving sparse linear systems lies at the core of numerous computational applications. Consequently, understanding the numerical 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 pre-conditioning 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 unprece-dented evaluation based on the SuiteSparse Matrix Collection-a dataset of real-world matrices with di-verse sizes and condition numbers. A key contribution is the faithful reproduction of SuiteSparse's UMF-PACK multifrontal LU factorization and SPQR mul-tifrontal 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 increased stability, even at low precision.