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Improving projection‐based eigensolvers via adaptive techniques
Author(s) -
Galgon Martin,
Krämer Lukas,
Lang Bruno
Publication year - 2018
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2124
Subject(s) - mathematics , eigenvalues and eigenvectors , subspace topology , hermitian matrix , projection (relational algebra) , convergence (economics) , algorithm , positive definite matrix , projector , eigendecomposition of a matrix , matrix (chemical analysis) , mathematical optimization , mathematical analysis , pure mathematics , computer science , physics , materials science , quantum mechanics , economics , composite material , computer vision , economic growth
Summary We consider subspace iteration (or projection‐based) algorithms for computing those eigenvalues (and associated eigenvectors) of a Hermitian matrix that lie in a prescribed interval. For the case that the projector is approximated with polynomials, we present an adaptive strategy for selecting the degree of these polynomials such that convergence is achieved with near‐to‐optimum overall work without detailed a priori knowledge about the eigenvalue distribution. The idea is then transferred to the approximation of the projector by numerical integration, which corresponds to FEAST algorithm proposed by E. Polizzi in 2009. [E. Polizzi: Density‐matrix‐based algorithm for solving eigenvalue problems. Phys. Rev. B 2009; 79 :115112]. Here, our adaptation controls the number of integration nodes. We also discuss the interaction of the method with search space reduction methods.