  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.
