On symmetric block triangular splitting iteration method for a class of complex symmetric system of linear equations

Abstract For solving a class of complex symmetric linear system, we first transform the system into a block two-by-two real formulation and construct a symmetric block triangular splitting (SBTS) iteration method based on two splittings. Then, eigenvalues of iterative matrix are calculated, convergence conditions with relaxation parameter are derived, and two optimal parameters are obtained. Besides, we present the optimal convergence factor and test two numerical examples to confirm theoretical results and to verify the high performances of SBTS iteration method compared with two classical methods.

[1]  Owe Axelsson,et al.  Real valued iterative methods for solving complex symmetric linear systems , 2000, Numer. Linear Algebra Appl..

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

[3]  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..

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

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

[6]  D. Bertaccini.  EFFICIENT PRECONDITIONING FOR SEQUENCES OF PARAMETRIC COMPLEX SYMMETRIC LINEAR SYSTEMS , 2004 .

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

[8]  Owe Axelsson,et al.  A comparison of iterative methods to solve complex valued linear algebraic systems , 2014, Numerical Algorithms.

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

[10]  Zhong-Zhi Bai.  On the convergence of additive and multiplicative splitting iterations for systems of linear equations , 2003 .

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

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

[13]  V. Simoncini,et al.  Iterative system solvers for the frequency analysis of linear mechanical systems , 2000 .

[14]  M. Benzi,et al.  Block preconditioning of real-valued iterative algorithms for complex linear systems , 2007 .

[15]  Changfeng Ma,et al.  The parameterized upper and lower triangular splitting methods for saddle point problems , 2017, Numerical Algorithms.

[16]  Zhong-zhi,et al.  A SHIFT-SPLITTING PRECONDITIONER FOR NON-HERMITIAN POSITIVE DEFINITE MATRICES , 2006 .

[17]  Qingqing Zheng,et al.  A shift-splitting preconditioner for a class of block two-by-two linear systems , 2017, Appl. Math. Lett..

[18]  Qingqing Zheng,et al.  A class of triangular splitting methods for saddle point problems , 2016, J. Comput. Appl. Math..

[19]  Zhong-Zhi Bai,et al.  On SSOR‐like preconditioners for non‐Hermitian positive definite matrices , 2016, Numer. Linear Algebra Appl..

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

[21]  A. Bunse-Gerstner,et al.  On a conjugate gradient-type method for solving complex symmetric linear systems , 1999 .