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GPBi‐CGstab( L ): A Lanczos‐type product method unifying Bi‐CGstab( L ) and GPBi‐CG
Author(s) -
Aihara Kensuke
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.2298
Subject(s) - lanczos resampling , mathematics , generalization , product (mathematics) , type (biology) , residual , algebra over a field , pure mathematics , mathematical analysis , algorithm , physics , geometry , quantum mechanics , eigenvalues and eigenvectors , ecology , biology
Summary Lanczos‐type product methods (LTPMs), in which the residuals are defined by the product of stabilizing polynomials and the Bi‐CG residuals, are effective iterative solvers for large sparse nonsymmetric linear systems. Bi‐CGstab( L ) and GPBi‐CG are popular LTPMs and can be viewed as two different generalizations of other typical methods, such as CGS, Bi‐CGSTAB, and Bi‐CGStab2. Bi‐CGstab( L ) uses stabilizing polynomials of degree L , while GPBi‐CG uses polynomials given by a three‐term recurrence (or equivalently, a coupled two‐term recurrence) modeled after the Lanczos residual polynomials. Therefore, Bi‐CGstab( L ) and GPBi‐CG have different aspects of generalization as a framework of LTPMs. In the present paper, we propose novel stabilizing polynomials, which combine the above two types of polynomials. The resulting method is referred to as GPBi‐CGstab( L ). Numerical experiments demonstrate that our presented method is more effective than conventional LTPMs.