  Anderson mixing (AM) is a classical method that can accelerate fixed-point
iterations by exploring historical information. Despite the successful
application of AM in scientific computing, the theoretical properties of AM are
still under exploration. In this paper, we study the restarted version of the
Type-I and Type-II AM methods, i.e., restarted AM. With a multi-step analysis,
we give a unified convergence analysis for the two types of restarted AM and
justify that the restarted Type-II AM can locally improve the convergence rate
of the fixed-point iteration. Furthermore, we propose an adaptive mixing
strategy by estimating the spectrum of the Jacobian matrix. If the Jacobian
matrix is symmetric, we develop the short-term recurrence forms of restarted AM
to reduce the memory cost. Finally, experimental results on various problems
validate our theoretical findings.
