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

TL;DR

This paper evaluates the performance and accuracy of bfloat16, posit, and takum arithmetic formats in sparse linear solvers, demonstrating their potential advantages over traditional IEEE 754 formats in real-world applications.

Contribution

It provides the first comprehensive evaluation of these alternative formats in the context of popular sparse solvers using real-world datasets, including faithful implementation of multifrontal factorizations.

Findings

Tapered-precision posit and takum formats improve accuracy in direct solvers.

Takum arithmetic shows exceptional stability at low precision.

Alternative formats reduce iteration counts in iterative solvers.

Abstract

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.