  This article introduces new acceleration methods for fixed point iterations.
Speed and stability are achieved by alternating the number of mappings to
compute step lengths and using them multiple times by cycling. A new type of
step length is also proposed with good properties for nonlinear mappings. The
methods require no specific adaptation and are especially efficient for
high-dimensional problems. Computation uses few objective function evaluations,
no matrix inversion and little extra memory. A convergence analysis is followed
by seven applications, including gradient descent acceleration for
unconstrained optimization. Performances are on par or better than
alternatives.
  The algorithm is available as a stand-alone Julia package and may be
downloaded at https://github.com/NicolasL-S/ACX.
