  Recently, circumcentering reflection method (CRM) has been introduced for
solving the feasibility problem of finding a point in the intersection of
closed constraint sets. It is closely related with Douglas--Rachford method
(DR). We prove local convergence of CRM in the same prototypical settings of
most theoretical analysis of regular nonconvex DR, whose consideration is made
natural by the geometry of the phase retrieval problem. For the purpose, we
show that CRM is related to the method of subgradient projections. For many
cases when DR is known to converge to a feasible point, we establish that CRM
locally provides a better convergence rate. As a root finder, we show that CRM
has local convergence whenever Newton--Raphson method does, has quadratic rate
whenever Newton--Raphson method does, and exhibits superlinear convergence in
many cases when Newton--Raphson method fails to converge at all. We also obtain
explicit regions of convergence. As an interesting aside, we demonstrate local
convergence of CRM to feasible points in cases when DR converges to fixed
points that are not feasible. We demonstrate an extension in higher dimensions,
and use it to obtain convergence rate guarantees for sphere and subspace
feasibility problems. Armed with these guarantees, we experimentally discover
that CRM is highly sensitive to compounding numerical error that may cause it
to achieve worse rates than those guaranteed by theory. We then introduce a
numerical modification that enables CRM to achieve the theoretically guaranteed
rates. Any future works that study CRM for product space formulations of
feasibility problems should take note of this sensitivity and account for it in
numerical implementations.
