  A one-step analysis of Anderson acceleration with general algorithmic depths
is presented. The resulting residual bounds within both contractive and
noncontractive settings reveal the balance between the contributions from the
higher and lower order terms, which are both dependent on the success of the
optimization problem solved at each step of the algorithm. The new residual
bounds show the additional terms introduced by the extrapolation produce terms
that are of a higher order than was previously understood. In the contractive
setting, these bounds sharpen previous convergence and acceleration results.
The bounds rely on sufficient linear independence of the differences between
consecutive residuals, rather than assumptions on the boundedness of the
optimization coefficients, allowing the introduction of a theoretically sound
safeguarding strategy. Several numerical tests illustrate the analysis
primarily in the noncontractive setting, and demonstrate the use of the method,
the safeguarding strategy, and theory-based guidance on dynamic selection of
the algorithmic depth, on a p-Laplace equation, a nonlinear Helmholtz equation,
and the steady Navier-Stokes equations with high Reynolds number in three
spatial dimensions.
