Accelerated GPMHSS Method for Solving Complex Systems of Linear Equations

Preconditioned modified Hermitian and skew-Hermitian splitting method (PMHSS) is an unconditionally convergent iteration method for solving large sparse complex symmetric systems of linear equations, and uses one parameter α. Adding another parameter β , the generalized PMHSS method (GPMHSS) is essentially a twoparameter iteration method. In order to accelerate the GPMHSS method, using an unexpected way, we propose an accelerated GPMHSS method (AGPMHSS) for large complex symmetric linear systems. Numerical experiments show the numerical behavior of our new method. AMS subject classifications: 65F10, 65W50

[1]  Yujiang Wu,et al.  Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices , 2012 .

[2]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[3]  Hong-Xiu Zhong,et al.  On preconditioned modified Newton-MHSS method for systems of nonlinear equations with complex symmetric jacobian matrices , 2014, Numerical Algorithms.

[4]  Yang Li,et al.  Multi-step modified Newton-HSS methods for systems of nonlinear equations with positive definite Jacobian matrices , 2017, Numerical Algorithms.

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

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

[7]  Minhong Chen,et al.  Convergence analysis of the modified Newton-HSS method under the Hölder continuous condition , 2014, J. Comput. Appl. Math..

[8]  Minhong Chen,et al.  Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations , 2013, Numerical Algorithms.

[9]  Gene H. Golub,et al.  Block Triangular and Skew-Hermitian Splitting Methods for Positive-Definite Linear Systems , 2005, SIAM J. Sci. Comput..

[10]  Bill Poirier,et al.  Efficient preconditioning scheme for block partitioned matrices with structured sparsity , 2000, Numer. Linear Algebra Appl..

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

[12]  Iain S. Duff,et al.  Semilocal and global convergence of the Newton‐HSS method for systems of nonlinear equations , 2011, Numer. Linear Algebra Appl..

[13]  Wei-wei Xu,et al.  A generalization of preconditioned MHSS iteration method for complex symmetric indefinite linear systems , 2013, Appl. Math. Comput..

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

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

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

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

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

[19]  F. M. Toyama,et al.  Accurate numerical solutions of the time-dependent Schrödinger equation. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Gene H. Golub,et al.  On successive‐overrelaxation acceleration of the Hermitian and skew‐Hermitian splitting iterations , 2007, Numer. Linear Algebra Appl..

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

[22]  Zhong-zhi,et al.  ON NEWTON-HSS METHODS FOR SYSTEMS OF NONLINEAR EQUATIONS WITH POSITIVE-DEFINITE JACOBIAN MATRICES , 2010 .

[23]  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 .

[24]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

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

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