Conditions for Digit Stability in Iterative Methods Using the Redundant Number Representation

TL;DR

This paper provides a theoretical foundation for the stability of significant digits in iterative methods using redundant number representation, extending previous results to a broader class of methods like Newton's method.

Contribution

It introduces a theoretical analysis showing stable digits arise in iterative sequences via Fejer monotone properties, broadening the class of methods with guaranteed digit stability.

Findings

Stable digits develop in Fejer monotone sequences.

Digit stability extends to fixed-point iterations of contractive Lipschitz functions.

Guarantees are provided for methods like Newton's method.

Abstract

Iterative methods play an important role in science and engineering applications, with uses ranging from linear system solvers in finite element methods to optimization solvers in model predictive control. Recently, a new computational strategy for iterative methods called ARCHITECT was proposed by Li et al. in [1] that uses the redundant number representation to create "stable digits" in the Most-significant Digits (MSDs) of an iterate, allowing the future iterations to assume the stable MSDs have not changed their value, eliminating the need to recompute them. In this work, we present a theoretical analysis of how these "stable digits" arise in iterative methods by showing that a Fejer monotone sequence in the redundant number representation can develop stable MSDs in the elements of the sequence as the sequence grows in length. This property of Fejer monotone sequences allows us to expand the class of iterative methods known to have MSD stability when using the redundant number representation to include any fixed-point iteration of a contractive Lipschitz continuous function. We then show that this allows for the theoretical guarantee of digit stability not just in the Jacobi method that was previously analyzed by Li et al. in [2], but also in other commonly used methods such as Newton's method.