TITLE: Computation time of Smith normal form in Maple
QUESTION [8 upvotes]: I am using Maple to compute the Smith normal form (SNF) of a $120 \times 120$ matrix and it seems that I will never get an answer back. I have checked my code for small cases and I believe that it is correct. When I try to compute the SNF for a $24 \times 24$ matrix, the real time and CPU time are about 0.1~0.2 seconds. I don't think it will take more than 3 hours for a $100 \times 100$ matrix. I have also tried it on several operating systems and the results are similar. 
Anyway, what is the approximate time complexity of the computation time of SNF? Is there a limit for the matrix size in Maple to do SNF?
Thank you so much!

REPLY [6 votes]: (Dense) Smith Normal Form is theoretically computable in $O(\|A\| \log \|A\| N^4\log N)$ time (Arne Storjohann, 1996). Storjohann was at Waterloo at the time, so I would not be surprised if that is the algorithm Maple uses. Sparse SNF is much faster.