Numerical analysis and random matrix theory have long been coupled, going (at least) back to the seminal work of Goldstine and von Neumann (1951) on the condition number of random matrices. The works of Trotter (1984) and Silverstein (1985) incorporate "numerical techniques" to assist in the analysis of random matrices. One can also consider the problem of computing distributions (e.g. Tracy-Widom) from random matrix theory. In this talk, I will discuss a different numerical analysis problem: using random matrices to analyze the runtime (or halting time) of numerical algorithms.  I will focus on recent results for the conjugate gradient algorithm including a proof that the halting time is almost deterministic.