Abstract

We give an overview of numerical linear algebra applied directly to infinite-dimensional operators – as opposed to finite-dimensional linear algebra applied to discretizations – which is made possible by exploiting structure in the operators, e.g., operators that are perturbations of Toeplitz operators. Algorithms discussed include the adaptive QR decomposition, infinite QR algorithm and infinite QL algorithm. These methods allow for the systematic solution of equations, calculation of continuous and discrete spectrum, and spectral measures. The benefit over the traditional discretize-and-solve approach is that the algorithm is rigorous (in applications, this translates to reliability), fast and extremely accurate. Having a hands-on representation of the spectral transformation allows for a robust implementation of functional calculus, used for example to solve the time-dependent Schrodinger equation.