Moulus-based Synchronous Block Multi-splitting Iteration Methods For Linear Complementarity Problems And Convergence Analysis

Many problems in scientific computing and engineering applications require to compute solutions of linear complementarity problems. Such class of problems includes, for example, the convex quadratic programming, the bimatrix game, the free boundary problems of fluid dynamics, the network equilibrium problems, the contact problems, and so on.This kind of problem of matrix tend to be involved in the large sparse matrix, to solve the LCP(q,A) fast and economically by iterative methods, Bai recently proposed a class of modulus-based splitting iteration methods essentially based on an equivalent transformation of the LCP(q, A) into a system of fixed-point equations involving only absolute value of certain vector. O’Leary and White set up the definite of matrix multi-splitting iteration methods for supplying a feasibility method of solve the LCP(q,A). However, there is few study for the modern higher-speed parallel multiprocessor, for example, the coefficient matrix of the linear complementarity problems is a block matrix, and so on.We constructed the modulus-based synchronous block multi-splitting iteration algorithms and block two-stage multi-splitting iteration algorithms base on the definition of multi-splitting and the technique of block accelerated over-relaxation iteration method, including the multi-splitting block Jacobi, block Gauss-Seidel. block SOR and block AOR, the algorithm is simple, highly parallel computing capabilities, we also can improve the convergence by appropriately adjust relaxation parameters. When the coefficient matrix is the block H matrix of the block diagonal matrix is symmetric positive definite matrix, in reasonable fragmentation of relaxation parameters and multiple constraints, we discuss deeply and in detail the convergence condition of the new method and convergence theory is established.In this paper we study the convergence property of the linear complementarity problems of modulus-based synchronous block multi-splitting iteration methods and block two-stage multi-splitting iteration methods. Firstly, constructing modulus-based synchronous block multi-splitting iteration methods and bock two-stage multi-splitting base on block relaxation iteration methods and block two-stage multi-splitting and given the analysis of the theory; Finally, establish the convergence theory of these modulus-based synchronous block multi-splitting iteration methods. This dissertation includes four chapters, which is organized as follows:Firstly, the research background and research status of the linear complementarit-y problem are given, as well as the preliminary knowledge. Furthermore, the main contents of this paper are briefed.In the second chapter, we are interested in constructing modulus-based synchro-nous block multi-splitting iteration methods based on multi-splitting of the coeffici-ent matrix is the point form and block relaxation iteration methods. Furthermore, we establish the convergence theory of these modulus-based synchronous block multi-splitting iteration methods when the coefficient matrix is a block H matrix which block diagonal is symmetric positive-definite matrix.In the third chapter, we are interested in constructing modulus-based synchro-nous block multi-splitting iteration methods based on two-stage multi-splitting of the coefficient matrix is the point form and block relaxation iteration methods. Further-more, we establish the convergence theory of these modulus-based synchronous block multi-splitting iteration methods when the coefficient matrix is a block H matrix which block diagonal is symmetric positive-definite matrix.Finally, the research work of this dissertation is summarized and the future research directions based on this work are discussed.