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Right preconditioned MINRES for singular systems †
Author(s) -
Sugihara Kota,
Hayami Ken,
Zheng Ning
Publication year - 2020
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.2277
Subject(s) - preconditioner , mathematics , iterated function , linear system , norm (philosophy) , residual , coefficient matrix , relaxation (psychology) , matrix (chemical analysis) , positive definite matrix , singular value , iterative method , mathematical analysis , algorithm , eigenvalues and eigenvectors , psychology , social psychology , physics , materials science , quantum mechanics , political science , law , composite material
Summary We consider solving large sparse symmetric singular linear systems. We first introduce an algorithm for right preconditioned minimum residual (MINRES) and prove that its iterates converge to the preconditioner weighted least squares solution without breakdown for an arbitrary right‐hand‐side vector and an arbitrary initial vector even if the linear system is singular and inconsistent. For the special case when the system is consistent, we prove that the iterates converge to a min‐norm solution with respect to the preconditioner if the initial vector is in the range space of the right preconditioned coefficient matrix. Furthermore, we propose a right preconditioned MINRES using symmetric successive over‐relaxation (SSOR) with Eisenstat's trick. Some numerical experiments on semidefinite systems in electromagnetic analysis and so forth indicate that the method is efficient and robust. Finally, we show that the residual norm can be further reduced by restarting the iterations.