  In this paper we develop convergence and acceleration theory for Anderson
acceleration applied to Newton's method for nonlinear systems in which the
Jacobian is singular at a solution. For these problems, the standard Newton
algorithm converges linearly in a region about the solution; and, it has been
previously observed that Anderson acceleration can substantially improve
convergence without additional a priori knowledge, and with little additional
computation cost. We present an analysis of the Newton-Anderson algorithm in
this context, and introduce a novel and theoretically supported safeguarding
strategy. The convergence results are demonstrated with the Chandrasekhar
H-equation and some standard benchmark examples.
