  Deterministic and randomized, row-action and column-action linear solvers
have become increasingly popular owing to their simplicity, low computational
and memory complexities, and ease of composition with other techniques.
Moreover, in order to achieve high-performance, such solvers must often be
adapted to the given problem structure and to the hardware platform on which
the problem will be solved. Unfortunately, determining whether such adapted
solvers will converge to a solution has required equally unique analyses. As a
result, adapted, reliable solvers are slow to be developed and deployed. In
this work, we provide a general set of assumptions under which such adapted
solvers are guaranteed to converge with probability one, and provide worst case
rates of convergence. As a result, we can provide practitioners with guidance
on how to design highly adapted, randomized or deterministic, row-action or
column-action linear solvers that are also guaranteed to converge.
