  A high-precision numerical calculation is reported for the self-energy
correction to the hyperfine splitting and to the bound-electron g factor in
hydrogenlike ions with low nuclear charge numbers. The binding nuclear Coulomb
field is treated to all orders, and the nonperturbative remainder beyond the
known $Z\alpha$-expansion coefficients is determined. For the $^3{\rm He}^+$
ion, the nonperturbative remainder yields a contribution of -450 Hz to the
normalized difference of the 1S and 2S hyperfine-structure intervals, to be
compared with the experimental uncertainty of 71 Hz and with the theoretical
error of 50 Hz due to other contributions. In the case of the g factor, the
calculation provides the most stringent test of equivalence of the perturbative
and nonperturbative approaches reported so far in the bound-state QED
calculations.
