Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for Hermitian-Definite Generalized Eigenvalue Problems (Journal of Computational Mathematics, 2018)
By extending the classical analysis techniques due to Samokish, Faddeev and Faddeeva,and Longsine and McCormick among others, we prove the convergence of the preconditionedsteepest descent with implicit deflation (PSD-id) method for solving Hermitian-definitegeneralized eigenvalue problems. Furthermore, we derive a nonasymptotic estimate of therate of convergence of the PSD-id method. We show that with a proper choice of theshift, the indefinite shift-and-invert preconditioner is a locally accelerated preconditioner,and is asymptotically optimal that leads to superlinear convergence. Numerical examplesare presented to verify the theoretical results on the convergence behavior of the PSD-idmethod for solving ill-conditioned Hermitian-definite generalized eigenvalue problemsarising from electronic structure calculations. While rigorous and full-scale convergenceproofs of the preconditioned block steepest descent methods in practical use still largely elude us, we believe the theoretical results presented in this paper shed light on an improvedunderstanding of the convergence behavior of these block methods.