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Iterative methods play an important role in science and engineeringapplications, with uses ranging from linear system solvers in finite elementmethods to optimization solvers in model predictive control. Recently, a newcomputational strategy for iterative methods called ARCHITECT was proposed byLi et al. in [1] that uses the redundant number representation to create"stable digits" in the Most-significant Digits (MSDs) of an iterate, allowingthe future iterations to assume the stable MSDs have not changed their value,eliminating the need to recompute them. In this work, we present a theoreticalanalysis of how these "stable digits" arise in iterative methods by showingthat a Fejer monotone sequence in the redundant number representation candevelop stable MSDs in the elements of the sequence as the sequence grows inlength. This property of Fejer monotone sequences allows us to expand the classof iterative methods known to have MSD stability when using the redundantnumber representation to include any fixed-point iteration of a contractiveLipschitz continuous function. We then show that this allows for thetheoretical guarantee of digit stability not just in the Jacobi method that waspreviously analyzed by Li et al. in [2], but also in other commonly usedmethods such as Newton's method.

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