An Integer Arithmetic-Based Sparse Linear Solver Using a GMRES Method and Iterative Refinement

TL;DR

This paper introduces an integer arithmetic-based GMRES solver with iterative refinement, demonstrating comparable convergence to floating-point methods and emphasizing the importance of preconditioning for stability and overflow prevention.

Contribution

The paper presents a novel integer arithmetic implementation of GMRES with iterative refinement, including data formats and operand shift techniques to prevent overflow.

Findings

Integer arithmetic GMRES solver achieves convergence similar to floating-point methods.

Preconditioning improves convergence and reduces overflow risk.

Numerical tests validate the effectiveness of the proposed approach.

Abstract

In this paper, we develop a (preconditioned) GMRES solver based on integer arithmetic, and introduce an iterative refinement framework for the solver. We describe the data format for the coefficient matrix and vectors for the solver that is based on integer or fixed-point numbers. To avoid overflow in calculations, we introduce initial scaling and logical shifts (adjustments) of operands in arithmetic operations. We present the approach for operand shifts, considering the characteristics of the GMRES algorithm. Numerical tests demonstrate that the integer arithmetic-based solver with iterative refinement has comparable solver performance in terms of convergence to the standard solver based on floating-point arithmetic. Moreover, we show that preconditioning is important, not only for improving convergence but also reducing the risk of overflow.