Why is that bad? Are there any cases where Chebyshev iteration is the best option? It seems like acquiring the necessary information about the spectrum would be prohibitive in practice, although I have never tried it out.
I’ve only really played with it on a single-node machine, which isn’t where it ought to shine.
IMO it sits at a really interesting spot as a sort of “more robust” (hand-waves) iterative solver that doesn’t require inner products. You need to know something about the spectrum sure, but sneakily figuring out things about the spectrum is somewhere where people can show off their expertise I think.
I actually spent a little time digging into this, and I'm not sure if this method is actually due to Chebyshev! This link has the most extensive references I found:
