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Inexact rational Krylov subspace method for eigenvalue problems
Author(s) -
Xu Shengjie,
Xue Fei
Publication year - 2022
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.2437
Subject(s) - krylov subspace , mathematics , generalized minimal residual method , subspace topology , eigenvalues and eigenvectors , linear system , solver , iterative method , conjugate residual method , invariant subspace , norm (philosophy) , mathematical optimization , linear subspace , algorithm , computer science , mathematical analysis , pure mathematics , machine learning , artificial neural network , physics , gradient descent , quantum mechanics , political science , law
An inexact rational Krylov subspace method is studied to solve large‐scale nonsymmetric eigenvalue problems. Each iteration (outer step) of the rational Krylov subspace method requires solution to a shifted linear system to enlarge the subspace, performed by an iterative linear solver for large‐scale problems. Errors are introduced at each outer step if these linear systems are solved approximately by iterative methods (inner step), and they accumulate in the rational Krylov subspace. In this article, we derive an upper bound on the errors introduced at each outer step to maintain the same convergence as exact rational Krylov subspace method for approximating an invariant subspace. Since this bound is inversely proportional to the current eigenresidual norm of the target invariant subspace, the tolerance of iterative linear solves at each outer step can be relaxed with the outer iteration progress. A restarted variant of the inexact rational Krylov subspace method is also proposed. Numerical experiments show the effectiveness of relaxing the inner tolerance to save computational cost.