  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.
