What kind of computer do you need to do adaptive optics of big telescopes? Are we talking supercomputer clusters or can my desktop do it, or my smartphone? I suppose it depends on the size of the telescope and the wavelength that is targeted. I'm interested in both the kind of computers that are used and the actual number of FLOPs per second that is required.
If you're only trying to correct a small field, using a single guide star (or a single laser guide star), then the requirements aren't very arduous. The MACAO adaptive-optics module used on the Very Large Telescope at the European Southern Observatory dates from the early 2000s and is based around two 400 MHz PowerPC 604 chips (one for the "supervisory computer", the other for the "real time computer"). You can read more about it here (PDF file).
For next-generation multi-conjugate adaptive optics, which tries to correct a larger field using multiple guide stars, the requirements are bit more stringent: my impression is that you need several GFLOPs in the real time computer (some of this may be handled by using specialized DSP chips instead of general CPUs), and additional processing power for external monitoring and analysis.
So, crudely speaking, your smartphone could probably handle single-guide-star AO (except it would need to be re-engineered a bit and running a real-time operating system, and you'd need an extra computer for external monitoring); for multi-guide-star AO, you'd end up needing the equivalent of a powerful (multi-CPU) server or (small) cluster.
If you look around, you should be able to find home-user AO systems which run, albeit at slightly lower bandwidth than you might like, on your PC (or an equivalent chip in the vendor's box). It depends a lot on the number of actuators on your deformable mirror, as do most matrix operations.