Gradient Flow Approach to Geometric Convergence Analysis of Preconditioned Eigensolvers

Preconditioned eigenvalue solvers (eigensolvers) are gaining popularity, but their convergence theory remains sparse and complex. We consider the simplest preconditioned eigensolver— the gradient iterative method with a fixed step size—for symmetric generalized eigenvalue problems, where we use the gradient of the Rayleigh quotient as an optimization direction. A sharp convergence rate bound for this method has been obtained in 2001–2003. It still remains the only known such bound for any of the methods in this class. While the bound is short and simple, its proof is not. We extend the bound to Hermitian matrices in the complex space and present a new self-contained and significantly shorter proof using novel geometric ideas.