  A novel method which is called the Chebyshev inertial iteration for
accelerating the convergence speed of fixed-point iterations is presented. The
Chebyshev inertial iteration can be regarded as a valiant of the successive
over relaxation or Krasnosel'ski\v{\i}-Mann iteration utilizing the inverse of
roots of a Chebyshev polynomial as iteration dependent inertial factors. One of
the most notable features of the proposed method is that it can be applied to
nonlinear fixed-point iterations in addition to linear fixed-point iterations.
Linearization around the fixed point is the key for the analysis on the local
convergence rate of the proposed method. The proposed method appears effective
in particular for accelerating the proximal gradient methods such as ISTA. It
is also proved that the proposed method can successfully accelerate almost any
fixed-point iterations if all the eigenvalues of the Jacobian at the fixed
point are real.
