  We study the convergence rate of the alternating projection method (APM)
applied to the intersection of an affine subspace and the second-order cone. We
show that when they intersect non-transversally, the convergence rate is
$O(k^{-1/2})$, where $k$ is the number of iterations of the APM. In particular,
when the intersection is not at the origin or forms a half-line with the origin
as the endpoint, the obtained convergence rate can be exact because a lower
bound of the convergence rate is evaluated. These results coincide with the
worst-case convergence rate obtained from the error bound discussed in [Borwein
et al., SIOPT, 2014] and [Drusvyatskiy et al., Math. Prog., 2017]. Moreover, we
consider the convergence rate of the APM for the intersection of an affine
subspace and the product of two second-order cones. We provide an example that
the worst-case convergence rate of the APM is better than the rate expected
from the error bound for the example.
