Dynamical Systems and Non-Hermitian Iterative Eigensolvers

Simple preconditioned iterations can provide an efficient alternative to more elaborate eigenvalue algorithms. We observe that these simple methods can be viewed as forward Euler discretizations of well-known autonomous differential equations that enjoy appealing geometric properties. This connection facilitates novel results describing convergence of a class of preconditioned eigensolvers to the leftmost eigenvalue, provides insight into the role of orthogonality and biorthogonality, and suggests the development of new methods and analyses based on more sophisticated discretizations. These results also highlight the effect of preconditioning on the convergence and stability of the continuous-time system and its discretization.