[betterunix]

I suspect that the real-world app using bubble sort would be fine...until the day it was not fine. By that point, you'll have a bunch of people who depend on the software, while you scramble to figure out why things became so slow.

Constant factors should not be ignored, but neither should scalability. That is the reason that production-grade sorting algorithms switch from quicksort to bubblesort/selection sort when the sequence is short enough. That is also the reason we typically use quicksort rather than heapsort -- quicksort usually has a lower constant factor on modern architectures, which is what matters when the asymptotics are the same.

The problem with focusing on the performance of your system for particular input sizes is that you are usually not guaranteed that the input size will not increase. It is not premature optimization if you have a good reason to believe that the input size will not grow, or that it will not grow enough to matter. Such is the case with matrix multiplication. If you have no evidence of that, though, you are almost certainly better off choosing the asymptotically better algorithm first, and coming back to tune that / switch to difference algorithms on smaller inputs / etc. later on.