If I encounter a zero pivot, and there is no non-zero candidate in any of the rows below it, can I make a column swap to select a replacement diagonal element?
The procedure I have implemented will reduce an ${N}\times{N}$ matrix to an ${(N-1)}\times{(N-1)}$ system to remove the problem pivot if a row exchange cannot be made. Then, once the system has be solved, I revisit any elements corresponding to deleted rows/columns and guess their values. This works, but for 4 unknown elements I only find that there are two valid solution vectors, and this leads me to believe that my procedure is not optimal.