We present a comprehensive convergence analysis for the self-consistent field (SCF) iteration to solve a class of nonlinear eigenvalue problems with eigenvector dependency (NEPvs). Using the tangent-Angle matrix as an intermediate measure for approximation error, we establish new formulas for two fundamental quantities that characterize the local convergence behavior of the plain SCF: The local contraction factor and the local asymptotic average contraction factor. In comparison with previously established results, new convergence rate estimates provide much sharper bounds on the convergence speed. As an application, we extend the convergence analysis to a popular SCF variant-The level-shifted SCF. The effectiveness of the convergence rate estimates is demonstrated numerically for NEPvs arising from solving the Kohn-Sham equation in electronic structure calculation and the Gross-Pitaevskii equation for modeling of the Bose-Einstein condensation.
|Number of pages||27|
|Journal||SIAM Journal on Matrix Analysis and Applications|
|State||Published - 2022|
Bibliographical noteFunding Information:
˚Received by the editors September 10, 2020; accepted for publication (in revised form) by K. Meerbergen October 25, 2021; published electronically February 28, 2022. https://doi.org/10.1137/20M136606X Funding: The first author was supported by NSF grant DMS-1913364. The second author was supported in part by NSF grants DMS-1719620 and DMS-2009689. The third author was supported by NSF grant DMS-2110731. :Department of Computer Science, University of California, Davis, Davis, CA 95616 USA (firstname.lastname@example.org). ;Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019 USA (email@example.com). \S Department of Mathematics, University of Kentucky, Lexington, KY 40506 USA (ding. firstname.lastname@example.org).
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- convergence factor
- level-shifted SCF
- nonlinear eigenvalue problem
- self-consistent field iteration
ASJC Scopus subject areas