A distributed fault-tolerant Convergence Function (CF) is proposed to resynchronize clocks with high precision. The CF can be used in conjunction with drift corrections which significantly increases the period between resynchronizations. This is important when clock drifts are large, because clock synchronization overhead then becomes intolerable. Clock differences are disseminated by a reliable message protocol after the clock valves are exchanged. All correct clocks then have identical copies of a matrix of clock differences. Matrix elements of correct clocks satisfy several relations. A subset of clocks that satisfy these relations defines an average clock. The influence of incorrect clocks on the average clock is shown to be bounded. The resynchronization is precise if clock reading errors are small, including uncertainties in transmission delays. Incorrect clocks may run too slow or too fast, exhibit omission failures or report inconsistent values
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