Anything that can't be solved analytically, I suppose. That's a pretty huge range of problems!

For sure it is!

The odd thing with what I listed is that these methods (nowadays) are really mostly useful for massive sparse problems, which wouldn’t really be practical without computing machines.

I’m pretty sure the Chebyschev semi-iterative method for solving linear systems is just named after his polynomials (and you can use his polynomials by hand for other stuff), but I really am at a loss as to what Krylov was up to.