Iterative Refinement with Low-Precision Posits

TL;DR

This paper explores a mixed-precision iterative refinement approach using low-precision posit numbers for solving large sparse linear systems, showing potential for efficiency gains without sacrificing accuracy.

Contribution

It introduces a novel iterative refinement method employing 16-bit posit numbers and matrix equilibration techniques for improved linear system solutions.

Findings

16-bit posit with equilibration achieves accuracy comparable to fp16.

Low-precision posit-based refinement reduces computational cost.

Method maintains accuracy in large sparse systems.

Abstract

This research investigates using a mixed-precision iterative refinement method using posit numbers instead of the standard IEEE floating-point format. The method is applied to solve a general linear system represented by the equation $Ax = b$, where $A$ is a large sparse matrix. Various scaling techniques, such as row and column equilibration, map the matrix entries to higher-density regions of machine numbers before performing the $O(n^3)$ factorization operation. Low-precision LU factorization followed by forward/backward substitution provides an initial estimate. The results demonstrate that a 16-bit posit configuration combined with equilibration produces accuracy comparable to IEEE half-precision (fp16), indicating a potential for achieving a balance between efficiency and accuracy.