On convergence and semi-convergence of SSOR-like methods for augmented linear systems

Abstract In this paper, we analyze the convergence and semi-convergence of a class of SSOR-like methods with four real functions for augmented systems. The class takes most existed SSOR-like methods as its special cases. For nonsingular systems, we obtain the minimum of convergence factors of all the SSOR-like methods in the class, and study when it can be reached by the convergence factors of methods in the class. By considering the equivalence of methods, we show that most of the existed SSOR-like methods have the same minimum of convergence factors.

[1]  Michael K. Ng,et al.  On Inexact Preconditioners for Nonsymmetric Matrices , 2005, SIAM J. Sci. Comput..

[2]  Yi-min Wei,et al.  Fast corrected Uzawa methods for solving symmetric saddle point problems , 2006 .

[3]  David J. Evans,et al.  Generalized AOR method for the augmented system , 2004, Int. J. Comput. Math..

[4]  Junfeng Lu,et al.  A Modified Nonlinear Inexact Uzawa Algorithm with a Variable Relaxation Parameter for the Stabilized Saddle Point Problem , 2010, SIAM J. Matrix Anal. Appl..

[5]  Ting-Zhu Huang,et al.  A modified SSOR iterative method for augmented systems , 2009 .

[6]  Eric de Sturler,et al.  Block-Diagonal and Constraint Preconditioners for Nonsymmetric Indefinite Linear Systems. Part I: Theory , 2005, SIAM J. Sci. Comput..

[7]  Naimin Zhang,et al.  Semi-convergence analysis of GMSSOR methods for singular saddle point problems , 2014, Comput. Math. Appl..

[8]  Ting-Zhu Huang,et al.  Convergence of a generalized MSSOR method for augmented systems , 2012, J. Comput. Appl. Math..

[9]  Zheng Li,et al.  Modified SOR-like method for the augmented system , 2007, Int. J. Comput. Math..

[10]  G. Golub,et al.  Inexact and preconditioned Uzawa algorithms for saddle point problems , 1994 .

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

[12]  E. Sturler,et al.  Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems , 2006 .

[13]  Yimin Wei,et al.  A note on constraint preconditioners for nonsymmetric saddle point problems , 2007, Numer. Linear Algebra Appl..

[14]  M. T. Darvishi,et al.  A modified symmetric successive overrelaxation method for augmented systems , 2011, Comput. Math. Appl..

[15]  Guo-Feng Zhang,et al.  On generalized symmetric SOR method for augmented systems , 2008 .

[16]  Tingzhu Huang,et al.  The Semi-convergence of Generalized SSOR Method for Singular Augmented Systems , 2009, HPCA.

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

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

[19]  Yong-Lin Chen,et al.  Semiconvergence criteria of iterations and extrapolated iterations and constructive methods of semiconvergent iteration matrices , 2005, Appl. Math. Comput..

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

[21]  Nira Dyn,et al.  The numerical solution of equality constrained quadratic programming problems , 1983 .

[22]  Gene H. Golub,et al.  Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices , 2007, Math. Comput..

[23]  Naimin Zhang,et al.  Optimal parameters of the generalized symmetric SOR method for augmented systems , 2014, J. Comput. Appl. Math..

[24]  Jun Zou,et al.  Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems , 2002, Numerische Mathematik.

[25]  Jun Zou,et al.  An Iterative Method with Variable Relaxation Parameters for Saddle-Point Problems , 2001, SIAM J. Matrix Anal. Appl..

[26]  Naimin Zhang,et al.  On the optimal parameters of GMSSOR method for saddle point problems , 2016, Appl. Math. Lett..

[27]  Andrew J. Wathen,et al.  Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations , 2002, Numerische Mathematik.

[29]  Xiaoyan Zhou,et al.  On the minimum convergence factor of a class of GSOR-like methods for augmented systems , 2014, Numerical Algorithms.

[30]  Gene H. Golub,et al.  A Preconditioner for Generalized Saddle Point Problems , 2004, SIAM J. Matrix Anal. Appl..

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

[32]  S. A. Edalatpanah,et al.  On the modified symmetric successive over-relaxation method for augmented systems , 2015 .

[33]  S. A. Edalatpanah,et al.  A new modified SSOR iteration method for solving augmented linear systems , 2014, Int. J. Comput. Math..

[34]  M. Madalena Martins,et al.  A variant of the AOR method for augmented systems , 2012, Math. Comput..

[35]  Michael K. Ng,et al.  Constraint Preconditioners for Symmetric Indefinite Matrices , 2009, SIAM J. Matrix Anal. Appl..

[36]  Tie Zhang,et al.  Two-parameter GSOR method for the augmented system , 2005, Int. J. Comput. Math..

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

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

[39]  Nicholas I. M. Gould,et al.  Constraint Preconditioning for Indefinite Linear Systems , 2000, SIAM J. Matrix Anal. Appl..

[40]  Ke Wang,et al.  SSOR-like methods for saddle point problems , 2009, Int. J. Comput. Math..

[41]  Xu Kong,et al.  Optimal parameters of GSOR-like methods for solving the augmented linear systems , 2008, Appl. Math. Comput..

[42]  Apostol T. Vassilev,et al.  Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems , 1997 .

[43]  Bing Zheng,et al.  On semi-convergence of parameterized Uzawa methods for singular saddle point problems☆ , 2009 .

[44]  Nicholas I. M. Gould,et al.  On the Solution of Equality Constrained Quadratic Programming Problems Arising in Optimization , 2001, SIAM J. Sci. Comput..

[45]  Nikolaos M. Missirlis,et al.  A comparison of the Extrapolated Successive Overrelaxation and the Preconditioned Simultaneous Displacement methods for augmented linear systems , 2015, Numerische Mathematik.

[46]  M. T. Darvishi,et al.  Symmetric SOR method for augmented systems , 2006, Appl. Math. Comput..

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