The Toda lattice, beyond being a completely integrable dynamical system, has many important properties.  Classically, the Toda flow is seen as acting on a specific class of bi-infinite Jacobi matrices.  Depending on the boundary conditions imposed for finite matrices, it is well known that the flow can be used as an eigenvalue algorithm. It was noticed by P. Deift, G. Menon and C. Pfrang that the fluctuations in the time it takes to compute eigenvalues of a random symmetric matrix with the Toda, QR and matrix sign algorithms are universal. In this talk, I will present a proof of such universality for the Toda algorithm.  I will also discuss empirical and rigorous results for other algorithms from numerical analysis.  This is joint work with P. Deift.