Preconditioned accelerated generalized successive overrelaxation method for solving complex symmetric linear systems

Abstract In this paper, by adopting the preconditioned technique for the accelerated generalized successive overrelaxation method (AGSOR) proposed by Edalatpour et al. (2015), we establish the preconditioned AGSOR (PAGSOR) iteration method for solving a class of complex symmetric linear systems. The convergence conditions, optimal iteration parameters and corresponding optimal convergence factor of the PAGSOR iteration method are determined. Besides, we prove that the spectral radius of the PAGSOR iteration method is smaller than that of the AGSOR one under proper restrictions, and its optimal convergence factor is smaller than that of the preconditioned symmetric block triangular splitting (PSBTS) one put forward by Zhang et al. (2018) recently. The spectral properties of the preconditioned PAGSOR matrix are also proposed. Numerical experiments illustrate the correctness of the theories and the effectiveness of the proposed iteration method and the preconditioner for the generalized minimal residual (GMRES) method.

[1]  Mehdi Dehghan,et al.  A Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems , 2013 .

[2]  Gene H. Golub,et al.  Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems , 2002, SIAM J. Matrix Anal. Appl..

[3]  Gene H. Golub,et al.  Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems , 2004, Numerische Mathematik.

[4]  Zhong-Zhi Bai,et al.  On preconditioned iteration methods for complex linear systems , 2015 .

[5]  Jing Zhao,et al.  Preconditioned symmetric block triangular splitting iteration method for a class of complex symmetric linear systems , 2018, Appl. Math. Lett..

[6]  Davod Khojasteh Salkuyeh,et al.  Two-parameter TSCSP method for solving complex symmetric system of linear equations , 2017, Calcolo.

[7]  Teng Wang,et al.  Alternating-directional PMHSS iteration method for a class of two-by-two block linear systems , 2016, Appl. Math. Lett..

[8]  Zhong-Zhi Bai,et al.  Motivations and realizations of Krylov subspace methods for large sparse linear systems , 2015, J. Comput. Appl. Math..

[9]  Zeng-Qi Wang,et al.  On parameterized inexact Uzawa methods for generalized saddle point problems , 2008 .

[10]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[11]  Davod Khojasteh Salkuyeh,et al.  A new iterative method for solving a class of complex symmetric system of linear equations , 2016, Numerical Algorithms.

[12]  Davod Khojasteh Salkuyeh,et al.  Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations , 2015, Int. J. Comput. Math..

[13]  X. Wang,et al.  A new single-step iteration method for solving complex symmetric linear systems , 2018, Numerical Algorithms.

[14]  Min-Li Zeng,et al.  A parameterized SHSS iteration method for a class of complex symmetric system of linear equations , 2016, Comput. Math. Appl..

[15]  Zhong-Zhi Bai,et al.  Block alternating splitting implicit iteration methods for saddle‐point problems from time‐harmonic eddy current models , 2012, Numer. Linear Algebra Appl..

[16]  Dario Fasino,et al.  An inverse Robin problem for Laplace's equation: theoretical results and numerical methods , 1999 .

[17]  Z. Bai,et al.  Rotated block triangular preconditioning based on PMHSS , 2013 .

[18]  Fang Chen,et al.  On preconditioned MHSS iteration methods for complex symmetric linear systems , 2011, Numerical Algorithms.

[19]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[20]  Davod Khojasteh Salkuyeh,et al.  Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations , 2014, Numer. Linear Algebra Appl..

[21]  Fang Chen,et al.  Modified HSS iteration methods for a class of complex symmetric linear systems , 2010, Computing.

[22]  Gene H. Golub,et al.  SOR-like Methods for Augmented Systems , 2001 .

[23]  Gene H. Golub,et al.  Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems , 2007 .

[24]  D. K. Salkuyeh,et al.  Generalized SOR iterative method for a class of complex symmetric linear system of equations , 2014, 1403.5902.

[25]  S. Arridge.  Optical tomography in medical imaging , 1999 .

[26]  Beresford N. Parlett,et al.  On generalized successive overrelaxation methods for augmented linear systems , 2005, Numerische Mathematik.

[27]  Stephen A. Vavasis,et al.  An Iterative Method for Solving Complex-Symmetric Systems Arising in Electrical Power Modeling , 2005, SIAM J. Matrix Anal. Appl..

[28]  D. K. Salkuyeh,et al.  Accelerated generalized SOR method for a class of complex systems of linear equations , 2015 .

[29]  Yu-Jiang Wu,et al.  On symmetric block triangular splitting iteration method for a class of complex symmetric system of linear equations , 2018, Appl. Math. Lett..

[30]  Xu Li,et al.  Lopsided PMHSS iteration method for a class of complex symmetric linear systems , 2013, Numerical Algorithms.

[31]  M. Benzi,et al.  Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems , 2013 .

[32]  Zhong Xu,et al.  An efficient two-step iterative method for solving a class of complex symmetric linear systems , 2018, Comput. Math. Appl..