Random Perturbation of Toeplitz Matrices

In 1947 John Von Neumann and Herman Goldstine, while developing the IAS computing machines, wrote a seminal paper on numerical errors in matrix computations. They suggested modeling the "computing noise" (coming from rounding errors, transcendental operations,...) by "probabilistic noise".  Nearly 80 years later, this pioneering idea remains an insightful paradigm in mathematical programming and numerical analysis. 

In this lecture, we will explain how this perspective of Von Neumann and Goldstine can help us to understand outrageous numerical errors made by our current computers when computing eigenvalues of badly conditioned matrices. 

We will take the example of Toeplitz matrices. These matrices form a rich class of matrices which appear in many areas of mathematics, physics and engineering. Toeplitz matrices are notoriously highly sensitive to small perturbations. They also come with a century-old rich theory that allows for a remarkably precise description of their spectral behavior under random perturbations.

The talk is based on a joint work with Mireille Capitaine and François Chapon.