Under suitable conditions,the monotone convergence about the projected iteration method for solving linear complementarity problem is proved and the influence of the involved parameter matrix on the convergence rate o...Under suitable conditions,the monotone convergence about the projected iteration method for solving linear complementarity problem is proved and the influence of the involved parameter matrix on the convergence rate of this method is investigated.展开更多
In this paper, we set up a general framework of parallel matrix mullisplitting relaxation methods for solving large scale system of linear equations. We investigate the convergence properties of this framework and giv...In this paper, we set up a general framework of parallel matrix mullisplitting relaxation methods for solving large scale system of linear equations. We investigate the convergence properties of this framework and give several sufficient conditions ensuring it to converge as well as diverge. At last, we conclude a necessary and sufficient condition for the convergence of this framework when the coefficient matrix is an L-matrix.展开更多
In this paper, by using a new projection, we construct a variant of Zhang’s algorithm and prove its convergence. Specially, the variant of Zhang’s algorithm has quadratic termination and superlinear convergence rale...In this paper, by using a new projection, we construct a variant of Zhang’s algorithm and prove its convergence. Specially, the variant of Zhang’s algorithm has quadratic termination and superlinear convergence rale under certain conditions. Zhang’s algorithm hasn’t these properties.展开更多
A one_step smoothing Newton method is proposed for solving the vertical linear complementarity problem based on the so_called aggregation function. The proposed algorithm has the following good features: (ⅰ) It solve...A one_step smoothing Newton method is proposed for solving the vertical linear complementarity problem based on the so_called aggregation function. The proposed algorithm has the following good features: (ⅰ) It solves only one linear system of equations and does only one line search at each iteration; (ⅱ) It is well_defined for the vertical linear complementarity problem with vertical block P 0 matrix and any accumulation point of iteration sequence is its solution.Moreover, the iteration sequence is bounded for the vertical linear complementarity problem with vertical block P 0+R 0 matrix; (ⅲ) It has both global linear and local quadratic convergence without strict complementarity. Many existing smoothing Newton methods do not have the property (ⅲ).展开更多
In this paper, a class of the stochastic generalized linear complementarity problems with finitely many elements is proposed for the first time. Based on the Fischer-Burmeister function, a new conjugate gradient proje...In this paper, a class of the stochastic generalized linear complementarity problems with finitely many elements is proposed for the first time. Based on the Fischer-Burmeister function, a new conjugate gradient projection method is given for solving the stochastic generalized linear complementarity problems. The global convergence of the conjugate gradient projection method is proved and the related numerical results are also reported.展开更多
In this paper, we present a modified projection method for the linear feasibility problems （LFP）. Compared with the existing methods, the new method adopts a surrogate technique to obtain new iteration instead of th...In this paper, we present a modified projection method for the linear feasibility problems （LFP）. Compared with the existing methods, the new method adopts a surrogate technique to obtain new iteration instead of the line search procedure with fixed stepsize. For the new method, we first show its global convergence under the condition that the solution set is nonempty, and then establish its linear convergence rate. Preliminary numerical experiments show that this method has good performance.展开更多
The extended linear complementarity problem(denoted by ELCP) can be reformulated as the solution of a nonsmooth system of equations. By the symmetrically perturbed CHKS smoothing function, the ELCP is approximated by ...The extended linear complementarity problem(denoted by ELCP) can be reformulated as the solution of a nonsmooth system of equations. By the symmetrically perturbed CHKS smoothing function, the ELCP is approximated by a family of parameterized smooth equations. A one-step smoothing Newton method is designed for solving the ELCP. The proposed algorithm is proved to be globally convergent under suitable assumptions.展开更多
The modified AOR method for solving linear complementarity problem(LCP(M,p))was proposed in literature,with some convergence results.In this paper,we considered the MAOR method for generalized-order linear complementa...The modified AOR method for solving linear complementarity problem(LCP(M,p))was proposed in literature,with some convergence results.In this paper,we considered the MAOR method for generalized-order linear complementarity problem(ELCP(M,N,p,q)),where M,N are nonsingular matrices of the following form:M=[D11H1K1D2],N=[D12H2K2D22],D11,D12,D21 and D22 are square nonsingular diagonal matrices.展开更多
A class of asynchronous nested matrix multisplitting methods for solving large-scale systems of linear equations are proposed, and their convergence characterizations are studied in detail when the coefficient matrice...A class of asynchronous nested matrix multisplitting methods for solving large-scale systems of linear equations are proposed, and their convergence characterizations are studied in detail when the coefficient matrices of the linear systems are monotone matrices and H-matrices, respectively.展开更多
A class of preconditioned iterative methods, i.e., preconditioned generalized accelerated overrelaxation （GAOR） methods, is proposed to solve linear systems based on a class of weighted linear least squares problems...A class of preconditioned iterative methods, i.e., preconditioned generalized accelerated overrelaxation （GAOR） methods, is proposed to solve linear systems based on a class of weighted linear least squares problems. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned iterative methods is better than that of the original methods. Furthermore, the effectiveness of the proposed methods is shown in the numerical experiment.展开更多
We establish the convergence theories of the symmetric relaxation methods for the system of linear equations with symmetric positive definite coefficient matrix, and more generally, those of the unsymmetric relaxation...We establish the convergence theories of the symmetric relaxation methods for the system of linear equations with symmetric positive definite coefficient matrix, and more generally, those of the unsymmetric relaxation methods for the system of linear equations with positive definite matrix.展开更多
A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the...A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the regularization solution are proven;a convergence estimate of H?lder type for the regularization method is obtained under the a-priori bound assumption for the exact solution. An iterative scheme is proposed to calculate the regularization solution;some numerical results show that this method works well.展开更多
In this paper, for general linear methods applied to strictly dissipative initial value problem in Hilbert spaces, we prove that algebraic stability implies B-convergence, which extends and improves the existing resul...In this paper, for general linear methods applied to strictly dissipative initial value problem in Hilbert spaces, we prove that algebraic stability implies B-convergence, which extends and improves the existing results on Runge-Kutta methods. Specializing our results for the case of multi-step Runge-Kutta methods, a series of B-convergence results are obtained.展开更多
Abstract In this paper,a class of generalized parallel matrix multisplitting relaxation methods for solving linear complementarity problems on the high speed multiprocessor systems is set up.This class of methods not ...Abstract In this paper,a class of generalized parallel matrix multisplitting relaxation methods for solving linear complementarity problems on the high speed multiprocessor systems is set up.This class of methods not only includes all the existing relaxation methods for the linear complementarity problems,but also yields a lot of novel ones in the sense of multisplitting.We establish the convergence theories of this class of generalized parallel multisplitting relaxation methods under the condition that the system matrix is an H matrix with positive diagonal elements.展开更多
The relaxation methods have served as very efficient tools for solving linear system and have many important applications in the field of science and engineering.In this paper,we study an efficient relaxation method b...The relaxation methods have served as very efficient tools for solving linear system and have many important applications in the field of science and engineering.In this paper,we study an efficient relaxation method based on the well-known Gauss-Seidel iteration method.Theoretical analysis shows our method can converge to the unique solution of the linear system.In addition,our method is applied to solve the saddle point problem and Page Rank problem,and the numerical results show our method is more powerful than the existent relaxation methods.展开更多
Recently,the projected Jacobi(PJ)and projected Gauss-Seidel(PGS)iteration methods have been studied for solving the horizontal linear complementarity problems(HLCPs).To further improve the convergence rates of the PJ ...Recently,the projected Jacobi(PJ)and projected Gauss-Seidel(PGS)iteration methods have been studied for solving the horizontal linear complementarity problems(HLCPs).To further improve the convergence rates of the PJ and PGS iteration methods,by using the successive overrelaxation(SOR)matrix splitting technique,a projected SOR iteration method is introduced in this paper to solve the HLCP.Convergence analyses are carefully studied when the system matrices are strictly diagonally dominant and irreducibly diagonally dominant.The newly obtained convergence results greatly extend the current convergence theory.Finally,two numerical examples are given to show the effectiveness of the proposed PSOR iteration method and its advantages over the recently proposed PJ and PGS iteration methods.展开更多
In this paper,we present a new convergence upper bound for the greedy Gauss-Seidel(GGS)method proposed by Zhang and Li[38].The new convergence upper bound improves the upper bound of the GGS method.In addition,we also...In this paper,we present a new convergence upper bound for the greedy Gauss-Seidel(GGS)method proposed by Zhang and Li[38].The new convergence upper bound improves the upper bound of the GGS method.In addition,we also propose a novel greedy block Gauss-Seidel(RDBGS)method based on the greedy strategy of the GGS method for solving large linear least-squares problems.It is proved that the RDBGS method converges to the unique solution of the linear least-squares problem.Numerical experiments demonstrate that the RDBGS method has superior performance in terms of iteration steps and computation time.展开更多
Some convergence results are given for A(a)-stable linear multistep methods applied to two classes of two-parameter singular perturbation problems, which extend the existing relevant results about one-parameter proble...Some convergence results are given for A(a)-stable linear multistep methods applied to two classes of two-parameter singular perturbation problems, which extend the existing relevant results about one-parameter problems by Lubich~[1]. Some numerical examples confirm our results.展开更多
文摘Under suitable conditions,the monotone convergence about the projected iteration method for solving linear complementarity problem is proved and the influence of the involved parameter matrix on the convergence rate of this method is investigated.
基金Supported by Natural Science Fundations of China and Shanghai.
文摘In this paper, we set up a general framework of parallel matrix mullisplitting relaxation methods for solving large scale system of linear equations. We investigate the convergence properties of this framework and give several sufficient conditions ensuring it to converge as well as diverge. At last, we conclude a necessary and sufficient condition for the convergence of this framework when the coefficient matrix is an L-matrix.
基金The subject is supported by Natural Science Foundation of China and Natural Science Foundation of Shandong Province.
文摘In this paper, by using a new projection, we construct a variant of Zhang’s algorithm and prove its convergence. Specially, the variant of Zhang’s algorithm has quadratic termination and superlinear convergence rale under certain conditions. Zhang’s algorithm hasn’t these properties.
文摘A one_step smoothing Newton method is proposed for solving the vertical linear complementarity problem based on the so_called aggregation function. The proposed algorithm has the following good features: (ⅰ) It solves only one linear system of equations and does only one line search at each iteration; (ⅱ) It is well_defined for the vertical linear complementarity problem with vertical block P 0 matrix and any accumulation point of iteration sequence is its solution.Moreover, the iteration sequence is bounded for the vertical linear complementarity problem with vertical block P 0+R 0 matrix; (ⅲ) It has both global linear and local quadratic convergence without strict complementarity. Many existing smoothing Newton methods do not have the property (ⅲ).
文摘In this paper, a class of the stochastic generalized linear complementarity problems with finitely many elements is proposed for the first time. Based on the Fischer-Burmeister function, a new conjugate gradient projection method is given for solving the stochastic generalized linear complementarity problems. The global convergence of the conjugate gradient projection method is proved and the related numerical results are also reported.
基金supported by National Natural Science Foundation of China (No. 10771120)Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry
文摘In this paper, we present a modified projection method for the linear feasibility problems （LFP）. Compared with the existing methods, the new method adopts a surrogate technique to obtain new iteration instead of the line search procedure with fixed stepsize. For the new method, we first show its global convergence under the condition that the solution set is nonempty, and then establish its linear convergence rate. Preliminary numerical experiments show that this method has good performance.
基金Supported by the NNSF of China(11071041, 11171257)
文摘The extended linear complementarity problem(denoted by ELCP) can be reformulated as the solution of a nonsmooth system of equations. By the symmetrically perturbed CHKS smoothing function, the ELCP is approximated by a family of parameterized smooth equations. A one-step smoothing Newton method is designed for solving the ELCP. The proposed algorithm is proved to be globally convergent under suitable assumptions.
文摘The modified AOR method for solving linear complementarity problem(LCP(M,p))was proposed in literature,with some convergence results.In this paper,we considered the MAOR method for generalized-order linear complementarity problem(ELCP(M,N,p,q)),where M,N are nonsingular matrices of the following form:M=[D11H1K1D2],N=[D12H2K2D22],D11,D12,D21 and D22 are square nonsingular diagonal matrices.
文摘A class of asynchronous nested matrix multisplitting methods for solving large-scale systems of linear equations are proposed, and their convergence characterizations are studied in detail when the coefficient matrices of the linear systems are monotone matrices and H-matrices, respectively.
基金supported by the National Natural Science Foundation of China (No. 11071033)the Fundamental Research Funds for the Central Universities (No. 090405013)
文摘A class of preconditioned iterative methods, i.e., preconditioned generalized accelerated overrelaxation （GAOR） methods, is proposed to solve linear systems based on a class of weighted linear least squares problems. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned iterative methods is better than that of the original methods. Furthermore, the effectiveness of the proposed methods is shown in the numerical experiment.
文摘We establish the convergence theories of the symmetric relaxation methods for the system of linear equations with symmetric positive definite coefficient matrix, and more generally, those of the unsymmetric relaxation methods for the system of linear equations with positive definite matrix.
文摘A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the regularization solution are proven;a convergence estimate of H?lder type for the regularization method is obtained under the a-priori bound assumption for the exact solution. An iterative scheme is proposed to calculate the regularization solution;some numerical results show that this method works well.
基金The project supported by the National Natural Science Foundation of China
文摘In this paper, for general linear methods applied to strictly dissipative initial value problem in Hilbert spaces, we prove that algebraic stability implies B-convergence, which extends and improves the existing results on Runge-Kutta methods. Specializing our results for the case of multi-step Runge-Kutta methods, a series of B-convergence results are obtained.
文摘Abstract In this paper,a class of generalized parallel matrix multisplitting relaxation methods for solving linear complementarity problems on the high speed multiprocessor systems is set up.This class of methods not only includes all the existing relaxation methods for the linear complementarity problems,but also yields a lot of novel ones in the sense of multisplitting.We establish the convergence theories of this class of generalized parallel multisplitting relaxation methods under the condition that the system matrix is an H matrix with positive diagonal elements.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11871136,11801382,11971092)the Fundamental Research Funds for the Central Universities(Grant No.DUT19LK06)。
文摘The relaxation methods have served as very efficient tools for solving linear system and have many important applications in the field of science and engineering.In this paper,we study an efficient relaxation method based on the well-known Gauss-Seidel iteration method.Theoretical analysis shows our method can converge to the unique solution of the linear system.In addition,our method is applied to solve the saddle point problem and Page Rank problem,and the numerical results show our method is more powerful than the existent relaxation methods.
基金National Natural Science Foundation of China(No.11771225)Qinglan Project of Jiangsu Province of China.
文摘Recently,the projected Jacobi(PJ)and projected Gauss-Seidel(PGS)iteration methods have been studied for solving the horizontal linear complementarity problems(HLCPs).To further improve the convergence rates of the PJ and PGS iteration methods,by using the successive overrelaxation(SOR)matrix splitting technique,a projected SOR iteration method is introduced in this paper to solve the HLCP.Convergence analyses are carefully studied when the system matrices are strictly diagonally dominant and irreducibly diagonally dominant.The newly obtained convergence results greatly extend the current convergence theory.Finally,two numerical examples are given to show the effectiveness of the proposed PSOR iteration method and its advantages over the recently proposed PJ and PGS iteration methods.
基金National Natural Science Foundation of China(No.11871444).
文摘In this paper,we present a new convergence upper bound for the greedy Gauss-Seidel(GGS)method proposed by Zhang and Li[38].The new convergence upper bound improves the upper bound of the GGS method.In addition,we also propose a novel greedy block Gauss-Seidel(RDBGS)method based on the greedy strategy of the GGS method for solving large linear least-squares problems.It is proved that the RDBGS method converges to the unique solution of the linear least-squares problem.Numerical experiments demonstrate that the RDBGS method has superior performance in terms of iteration steps and computation time.
基金the National Natural Science Foundation of China (No.19871070), Wang Kuancheng Foundation for Rewarding the Postdoctors of Chine
文摘Some convergence results are given for A(a)-stable linear multistep methods applied to two classes of two-parameter singular perturbation problems, which extend the existing relevant results about one-parameter problems by Lubich~[1]. Some numerical examples confirm our results.
