Digit Stability Inference for Iterative Methods Using Redundant Number Representation

TL;DR

This paper introduces a method to reliably determine when the most significant digits in iterative linear solvers stabilize, enabling more efficient hardware implementations with significant speedups.

Contribution

It provides a formal proof of digit stability using interval and error analysis, improving upon prior work by guaranteeing digit stabilization in redundant number representations.

Findings

Achieves up to 2.2x speedup in hardware linear solvers

Proves stability of high-significance digits using formal analysis

Links matrix conditioning to digit stability rate

Abstract

In our recent work on iterative computation in hardware, we showed that arbitrary-precision solvers can perform more favorably than their traditional arithmetic equivalents when the latter's precisions are either under- or over-budgeted for the solution of the problem at hand. Significant proportions of these performance improvements stem from the ability to infer the existence of identical most-significant digits between iterations. This technique uses properties of algorithms operating on redundantly represented numbers to allow the generation of those digits to be skipped, increasing efficiency. It is unable, however, to guarantee that digits will stabilize, i.e., never change in any future iteration. In this article, we address this shortcoming, using interval and forward error analyses to prove that digits of high significance will become stable when computing the approximants of systems of linear equations using stationary iterative methods. We formalize the relationship between matrix conditioning and the rate of growth in most-significant digit stability, using this information to converge to our desired results more quickly. Versus our previous work, an exemplary hardware realization of this new technique achieves an up-to 2.2x speedup in the solution of a set of variously conditioned systems using the Jacobi method.