  Anderson acceleration (AA) is widely used for accelerating the convergence of
an underlying fixed-point iteration $\bm{x}_{k+1} = \bm{q}( \bm{x}_{k} )$, $k =
0, 1, \ldots$, with $\bm{x}_k \in \mathbb{R}^n$, $\bm{q} \colon \mathbb{R}^n
\to \mathbb{R}^n$. Despite AA's widespread use, relatively little is understood
theoretically about the extent to which it may accelerate the underlying
fixed-point iteration. To this end, we analyze a restarted variant of AA with a
restart size of one, a method closely related to GMRES(1). We consider the case
of $\bm{q}( \bm{x} ) = M \bm{x} + \bm{b}$ with matrix $M \in \mathbb{R}^{n
\times n}$ either symmetric or skew-symmetric. For both classes of $M$ we
compute the worst-case root-average asymptotic convergence factor of the AA
method, partially relying on conjecture in the symmetric setting, proving that
it is strictly smaller than that of the underlying fixed-point iteration. For
symmetric $M$, we show that the AA residual iteration corresponds to a
fixed-point iteration for solving an eigenvector-dependent nonlinear eigenvalue
problem (NEPv), and we show how this can result in the convergence factor
strongly depending on the initial iterate, which we quantify exactly in certain
special cases. Conversely, for skew-symmetric $M$ we show that the AA residual
iteration is closely related to a power iteration for $M$, and how this results
in the convergence factor being independent of the initial iterate. Supporting
numerical results are given, which also indicate the theory is applicable to
the more general setting of nonlinear $\bm{q}$ with Jacobian at the fixed point
that is symmetric or skew symmetric.
