On modified Newton-DGPMHSS method for solving nonlinear systems with complex symmetric Jacobian matrices

Abstract This paper aims to give an efficient iterative method for solving large sparse nonlinear system with complex symmetric Jacobian matrix. Employing the double-parameter generalized preconditioned MHSS (DGPMHSS) method as the inner iteration, and using the modified Newton method as the outer iteration , we establish a modified Newton–DGPMHSS method for solving nonlinear system with complex symmetric Jacobian matrix. For the new presented method, we provide the local convergence analysis under Holder condition, which is weaker than Lipschitz condition. Furthermore, we compare our new method with the modified Newton–PMHSS method, which is a considerable method for dealing with large sparse nonlinear system with complex symmetric Jacobian matrix, and the numerical results show the efficiency of our new method.

[1]  Jing Wang,et al.  MN-DPMHSS iteration method for systems of nonlinear equations with block two-by-two complex Jacobian matrices , 2017, Numerical Algorithms.

[2]  Ali Barati,et al.  A third-order Newton-type method to solve systems of nonlinear equations , 2007, Appl. Math. Comput..

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

[4]  Cui-Xia Li,et al.  A Double-Parameter GPMHSS Method for a Class of Complex Symmetric Linear Systems from Helmholtz Equation , 2014 .

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

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

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

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

[9]  I. Aranson,et al.  The world of the complex Ginzburg-Landau equation , 2001, cond-mat/0106115.

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

[11]  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 .

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

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

[14]  Hengbin An,et al.  A choice of forcing terms in inexact Newton method , 2007 .

[15]  Wang Yang.  Newton-LHSS with Backtracking Methods and the Global Convergence , 2012 .

[16]  Guo-Feng Zhang,et al.  A class of iteration methods based on the HSS for Toeplitz systems of weakly nonlinear equations , 2015, J. Comput. Appl. Math..

[17]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

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

[19]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

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

[21]  Yujiang Wu,et al.  Accelerated Newton-GPSS methods for systems of nonlinear equations , 2014 .

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

[23]  Z. Bai,et al.  A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations , 2007 .

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

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

[26]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .

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

[28]  Xiang Wang,et al.  Efficient single-step preconditioned HSS iteration methods for complex symmetric linear systems , 2017, Comput. Math. Appl..

[29]  M. Benzi,et al.  Regularized HSS iteration methods for saddle-point linear systems , 2017 .

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

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