I have been trying to compute eigenvalues of a rather sizable matrix
A, about $500 \times 500$ (but sparse). I asked Mathematica to compute
Eigenvalues[A], and left it to work. After a night of computation, Mathematica still failed to produce the answer. Just out of curiousity, I tried to see what happens if I replace
A by it's numeric approximation (before,
A had integer entries, hence exact). (I didn't quite expect it to make a difference, but I was trying some random things). Much to my surprise, the answer appeared immediately. Hence my question: How does it happen that Mathematica produces the answer so much faster for inexact input? To what extent can this output be trusted?