On SSOR iteration method for a class of block two-by-two linear systems

In this paper, the optimal iteration parameters of the symmetric successive overrelaxation (SSOR) method for a class of block two-by-two linear systems are obtained, which result in optimal convergence factor. An accelerated variant of the SSOR (ASSOR) method is presented, which significantly improves the convergence rate of the SSOR method. Furthermore, a more practical way to choose iteration parameters for the ASSOR method has also been proposed. Numerical experiments demonstrate the efficiency of the SSOR and ASSOR methods for solving a class of block two-by-two linear systems with the optimal parameters.

[1]  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 .

[2]  O. Axelsson,et al.  Real valued iterative methods for solving complex symmetric linear systems , 2000 .

[3]  Guo-Feng Zhang,et al.  A parameterized splitting iteration method for complex symmetric linear systems , 2014 .

[4]  Zhong-Zhi Bai,et al.  Structured preconditioners for nonsingular matrices of block two-by-two structures , 2005, Math. Comput..

[5]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

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

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

[8]  Oliver Lass,et al.  Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems , 2009, Computing.

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

[10]  A. Frommer,et al.  Numerical Challenges in Lattice Quantum Chromodynamics , 2000 .

[11]  Fang Chen,et al.  Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices , 2013, Numerical Algorithms.

[12]  A. Sommerfeld.  Partial Differential Equations in Physics , 1949 .

[13]  Shen Wang,et al.  Modified HSS iteration methods for a class of non-Hermitian positive-definite linear systems , 2012, Appl. Math. Comput..

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

[15]  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..

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

[17]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

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

[19]  David Day,et al.  Solving Complex-Valued Linear Systems via Equivalent Real Formulations , 2001, SIAM J. Sci. Comput..

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

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

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

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

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

[25]  Zhi-Ru Ren,et al.  Inexact rotated block triangular preconditioners for a class of block two-by-two matrices , 2015 .

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

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

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

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

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

[31]  Shi-Liang Wu,et al.  Several variants of the Hermitian and skew‐Hermitian splitting method for a class of complex symmetric linear systems , 2015, Numer. Linear Algebra Appl..