The Toda lattice is the prototypical discrete-space, continuous-time completely integrable Hamiltonian system.  It was introduced by Morikazu Toda in 1967 and analyzed in detail by Flaschka in 1974.  The bi-infinite Toda lattice can be solved with its associated inverse scattering transform (IST).  The IST is closely tied to the interpretation of the flow as an isospectral deformation of a bi-infinite tridiagonal matrix.  The Toda lattice has a completely integrable counterpart for finite symmetric and Hermitian (dense) matrices.  And due the the isospectral nature of the flow, it can be used as an eigenvalue algorithm. This talk has two parts. First, I will discuss the numerical computation of the IST for the Toda lattice by solving Riemann--Hilbert problems numerically.  Second, I will show that the time, called the halting time, it takes for the Toda lattice to compute the largest eigenvalue of a random matrix is universal --- the rescaled halting time converges to a universal distribution.