  Anderson Acceleration (AA) has been widely used to solve nonlinear
fixed-point problems due to its rapid convergence. This work focuses on a
variant of AA in which multiple Picard iterations are performed between each AA
step, referred to as the Alternating Anderson-Picard (AAP) method. Despite
introducing more 'slow' Picard iterations, this method has been shown to be
efficient and even more robust in both linear and nonlinear cases. However,
there is a lack of theoretical analysis for AAP in the nonlinear case, which
this paper aims to address. We show the equivalence between AAP and a
multisecant-GMRES method that uses GMRES to solve a multisecant linear system
at each iteration. More interestingly, the incorporation of Picard iterations
and AA establishes a deep connection between AAP and the Newton-GMRES method.
This connection is evident in terms of the multisecant matrix, the approximate
Jacobian inverse, search direction, and optimization gain -- an essential
factor in the convergence analysis of AA. We show that these terms converge to
their corresponding terms in the Newton-GMRES method as the residual approaches
zero. Consequently, we build the convergence analysis of AAP. To validate our
theoretical findings, numerical examples are provided.
