  Many problems reduce to the fixed-point problem of solving $x=T(x)$. To this
problem, we apply the coordinate-update algorithms, which update only one or a
few components of $x$ at each step. When each update is cheap, these algorithms
are faster than the full fixed-point iteration (which updates all the
components).
  In this paper, we focus on the coordinate-update algorithms based on the
cyclic selection rules, where the ordering of coordinates in each cycle is
arbitrary. These algorithms are fast, but their convergence is unknown in the
fixed-point setting.
  When $T$ is a nonexpansive operator and has a fixed point, we show that the
sequence of coordinate-update iterates converges to a fixed point under proper
step sizes. This result applies to the primal-dual coordinate-update
algorithms, which have applications to optimization problems with nonseparable
nonsmooth objectives, as well as global linear constraints.
  Numerically, we apply coordinate-update algorithms with the cyclic, shuffled
cyclic, and random selection rules to $\ell_1$ robust least squares, a CT image
reconstruction problem, as well as nonnegative matrix factorization. They
converge much faster than the standard fixed-point iteration. Among the three
rules, cyclic and shuffled cyclic rules are overall faster than the random
rule.
