  We present a nonperturbative numerical evaluation of the one-photon electron
self energy for hydrogenlike ions with low nuclear charge numbers Z=1 to 5. Our
calculation for the 1S state has a numerical uncertainty of 0.8 Hz for hydrogen
and 13 Hz for singly-ionized helium. Resummation and convergence acceleration
techniques that reduce the computer time by about three orders of magnitude
were employed in the calculation. The numerical results are compared to results
based on known terms in the expansion of the self energy in powers of (Z
alpha).
