  We study the asymptotic convergence of AA($m$), i.e., Anderson acceleration
with window size $m$ for accelerating fixed-point methods $x_{k+1}=q(x_{k})$,
$x_k \in R^n$. Convergence acceleration by AA($m$) has been widely observed but
is not well understood. We consider the case where the fixed-point iteration
function $q(x)$ is differentiable and the convergence of the fixed-point method
itself is root-linear. We identify numerically several conspicuous properties
of AA($m$) convergence: First, AA($m$) sequences $\{x_k\}$ converge
root-linearly but the root-linear convergence factor depends strongly on the
initial condition. Second, the AA($m$) acceleration coefficients $\beta^{(k)}$
do not converge but oscillate as $\{x_k\}$ converges to $x^*$. To shed light on
these observations, we write the AA($m$) iteration as an augmented fixed-point
iteration $z_{k+1} =\Psi(z_k)$, $z_k \in R^{n(m+1)}$ and analyze the continuity
and differentiability properties of $\Psi(z)$ and $\beta(z)$. We find that the
vector of acceleration coefficients $\beta(z)$ is not continuous at the fixed
point $z^*$. However, we show that, despite the discontinuity of $\beta(z)$,
the iteration function $\Psi(z)$ is Lipschitz continuous and directionally
differentiable at $z^*$ for AA(1), and we generalize this to AA($m$) with $m>1$
for most cases. Furthermore, we find that $\Psi(z)$ is not differentiable at
$z^*$. We then discuss how these theoretical findings relate to the observed
convergence behaviour of AA($m$). The discontinuity of $\beta(z)$ at $z^*$
allows $\beta^{(k)}$ to oscillate as $\{x_k\}$ converges to $x^*$, and the
non-differentiability of $\Psi(z)$ allows AA($m$) sequences to converge with
root-linear convergence factors that strongly depend on the initial condition.
Additional numerical results illustrate our findings.
