TY - JOUR
T1 - Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for Hermitian-Definite Generalized Eigenvalue Problems
AU - Cai , Yunfeng
AU - Bai , Zhaojun
AU - Pask , John E.
AU - Sukumar , N.
JO - Journal of Computational Mathematics
VL - 5
SP - 739
EP - 760
PY - 2018
DA - 2018/06
SN - 36
DO - http://doi.org/10.4208/jcm.1703-m2016-0580
UR - https://global-sci.org/intro/article_detail/jcm/12455.html
KW - Eigenvalue problem, Steepest descent method, Preconditioning, Superlinear convergence.
AB - <p style="text-align: justify;">By extending the classical analysis techniques due to Samokish, Faddeev and Faddeeva,
and Longsine and McCormick among others, we prove the convergence of the preconditioned
steepest descent with implicit deflation (PSD-id) method for solving Hermitian-definite
generalized eigenvalue problems. Furthermore, we derive a nonasymptotic estimate of the
rate of convergence of the PSD-id method. We show that with a proper choice of the
shift, the indefinite shift-and-invert preconditioner is a locally accelerated preconditioner,
and is asymptotically optimal that leads to superlinear convergence. Numerical examples
are presented to verify the theoretical results on the convergence behavior of the PSD-id
method for solving ill-conditioned Hermitian-definite generalized eigenvalue problems
arising from electronic structure calculations. While rigorous and full-scale convergence
proofs 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 improved
understanding of the convergence behavior of these block methods.</p>