The Reflection Method for the Numerical Solution of Linear Systems

We present Cimmino's reflection algorithm for the numerical solution of linear systems, which starts with an arbitrary point in Rn that gets reflected with respect to the system's hyperplanes. The centroid of the ensuing collection of points becomes the starting point of the next iteration. We provide error estimates for the convergence at each step. A probabilistic argument is also devised to improve this elegant geometrical algorithm. This subject is an opportunity to show students how linear algebra can interact fruitfully not only with algebra, geometry, and numerical analysis, but also with probability theory and methods.