Parameterized QHSS Iteration Method and Its Variants for Non-Hermitian Positive Definite Linear Systems of Strong Skew-Hermitian Parts

Abstract

Based on the quasi-Hermitian and skew-Hermitian splitting (QHSS) iteration method proposed by Bai for solving the large sparse non-Hermitian positive definite linear systems of strong skew-Hermitian parts, this paper introduces a parameterized QHSS (PQHSS) iteration method. The PQHSS iteration is essentially a two-parameter iteration which covers the standard QHSS iteration and can further accelerate the iterative process. In addition, two practical variants, viz., inexact and extrapolated PQHSS iteration methods are established to further improve the computational efficiency. The convergence conditions for the iteration parameters of the three proposed methods are presented. Numerical results illustrate the effectiveness and robustness of the PQHSS iteration method and its variants when used as linear solvers, as well as the PQHSS preconditioner for Krylov subspace iteration methods.

Keywords

System of linear equations / Quasi-Hermitian and skew-Hermitian splitting (QHSS) iteration method / Inexact iteration / Extrapolation / Convergence analysis / 65F10 / 65F20 / 65K05 / 90C25 / 15A06

References

[1] BaiZ-Z. Several splittings for non-Hermitian linear systems. Sci. China Ser. A, 2008, 51(8): 1339-1348

[2] BaiZ-Z. Quasi-HSS iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts. Numer. Linear Algebra Appl., 2018, 254e2116

[3] BaiZ-Z. A two-step matrix splitting iteration paradigm based on one single splitting for solving systems of linear equations. Numer. Linear Algebra Appl., 2024, 31e2510

[4] BaiZ-Z, BenziM, ChenF. Modified HSS iteration methods for a class of complex symmetric linear systems. Computing, 2010, 87(3–4): 93-111

[5] BaiZ-Z, BenziM, ChenF. On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithms, 2011, 56(2): 297-317

[6] BaiZ-Z, GolubGH. Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal., 2007, 27(1): 1-23

[7] BaiZ-Z, GolubGH, LiC-K. Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices. Math. Comput., 2007, 76(257): 287-298

[8] BaiZ-Z, GolubGH, LuL-Z, YinJ-F. Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM J. Sci. Comput., 2005, 26(3): 844-863

[9] BaiZ-Z, GolubGH, NgMK. Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl., 2003, 24(3): 603-626

[10] BaiZ-Z, GolubGH, NgMK. On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations. Numer. Linear Algebra Appl., 2007, 14(4): 319-335

[11] BaiZ-Z, PanJ-YMatrix Analysis and Computations, 2021, Philadelphia. SIAM.

[12] BaiZ-Z, ParlettBN, WangZ-Q. On generalized successive overrelaxation methods for augmented linear systems. Numer. Math., 2005, 102(1): 1-38

[13] BaiZ-Z, RozložníkM. On the numerical behavior of matrix splitting iteration methods for solving linear systems. SIAM J. Numer. Anal., 2015, 53(4): 1716-1737

[14] BaiZ-Z, WangZ-Q. On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl., 2008, 428(11–12): 2900-2932

[15] CaoY, TanW-W, JiangM-Q. A generalization of the positive-definite and skew-Hermitian splitting iteration. Numer. Algebra Control Optim., 2012, 2(4): 811-821

[16] CaoZ-H. A convergence theorem on an extrapolated iterative method and its applications. Appl. Numer. Math., 1998, 27(3): 203-209

[17] ChenF, LiT-Y, LuK-Y, MuratovaGV. Modified QHSS iteration methods for a class of complex symmetric linear systems. Appl. Numer. Math., 2021, 164: 3-14

[18] LiB, CuiJ, HuangZ, XieX. On preconditioned MQHSS iterative method for solving a class of complex symmetric linear systems. Comput. Appl. Math., 2022, 416250

[19] LiX, YangA-L, WuY-J. Lopsided PMHSS iteration method for a class of complex symmetric linear systems. Numer. Algorithms, 2014, 66(3): 555-568

[20] SaadYIterative Methods for Sparse Linear Systems, 2003, Philadelphia. SIAM.

[21] VargaRSMatrix Iterative Analysis, 2000, Berlin. Springer.

[22] WuW-T. On minimization of upper bound for the convergence rate of the QHSS iteration method. Commun. Appl. Math. Comput., 2019, 1(2): 263-282

[23] WuY-J, LiX, YuanJ-Y. A non-alternating preconditioned HSS iteration method for non-Hermitian positive definite linear systems. Comput. Appl. Math., 2017, 36(1): 367-381

[24] YangA-L, AnJ, WuY-J. A generalized preconditioned HSS method for non-Hermitian positive definite linear systems. Appl. Math. Comput., 2010, 216(6): 1715-1722

[25] YinJ-F, DouQ-Y. Generalized preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems. J. Comput. Math., 2012, 30(4): 404-417

[26] ZengM-L. Inexact modified QHSS iteration methods for complex symmetric linear systems of strong skew-Hermitian parts. IAENG Int. J. Appl. Math., 2021, 51(1): 109-115

Funding

China Scholarship Council(202208625004)

Natural Science Foundation of Gansu Province(20JR5RA464)

National Natural Science Foundation of China(11501272)

RIGHTS & PERMISSIONS

Shanghai University