The generalized residual cutting method and its convergence characteristics

Iterative methods and especially Krylov subspace methods (KSM) are a very useful numerical tool in solving for large and sparse linear systems problems arising in science and engineering modeling. More recently, the nested loop KSM have been proposed that improve the convergence of the traditional K...

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Bibliographic Details
Published in:Numerical linear algebra with applications 2023-12, Vol.30 (6)
Main Authors: Abe, T., Chronopoulos, A. T.
Format: Article
Language:English
Subjects:
Algorithms
Convergence
Cutting
Iterative methods
Linear systems
Nested loops
Preconditioning
Robustness (mathematics)
Schmidt method
Sparse matrices
Subspace methods
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Summary:Iterative methods and especially Krylov subspace methods (KSM) are a very useful numerical tool in solving for large and sparse linear systems problems arising in science and engineering modeling. More recently, the nested loop KSM have been proposed that improve the convergence of the traditional KSM. In this article, we review the residual cutting (RC) and the generalized residual cutting (GRC) that are nested loop methods for large and sparse linear systems problems. We also show that GRC is a KSM that is equivalent to Orthomin with a variable preconditioning. We use the modified Gram–Schmidt method to derive a stable GRC algorithm. We show that GRC presents a general framework for constructing a class of “hybrid” (nested) KSM based on inner loop method selection. We conduct numerical experiments using nonsymmetric indefinite matrices from a widely used library of sparse matrices that validate the efficiency and the robustness of the proposed methods.
ISSN:1070-5325
1099-1506
DOI:10.1002/nla.2517