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Pseudoeigenvector bases and deflated GMRES for highly nonnormal matrices
Author(s) -
Morgan Ronald B.,
Yang Zhao,
Zhong Baojiang
Publication year - 2016
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.2067
Subject(s) - generalized minimal residual method , eigenvalues and eigenvectors , mathematics , convergence (economics) , matrix (chemical analysis) , stability (learning theory) , basis (linear algebra) , algorithm , iterative method , computer science , geometry , chemistry , physics , quantum mechanics , machine learning , economics , economic growth , chromatography
Summary Pseudoeigenvalues have been extensively studied for highly nonnormal matrices. This paper focuses on the corresponding pseudoeigenvectors. The properties and uses of pseudoeigenvector bases are investigated. It is shown that pseudoeigenvector bases can be much better conditioned than eigenvector bases. We look at the stability and the varying quality of pseudoeigenvector bases. Then applications are considered including the exponential of a matrix. Several aspects of GMRES convergence are looked at, including why using approximate eigenvectors to deflate eigenvalues can be effective even when there is not a basis of eigenvectors.