Several Preconditioned Algorithms Analysis For Large-scale Sparse Linear Equations

Solving large scale linear equations discretized through differential equations obtained from the scientific computations and engineering applications has become an important subject recently.At present,the CG iteration algorithm and the GMRES(m)iteration algorithm established based on the Galerkin principle are assumed to be effective methods to solve such linear systems.However,numerical examples show that the computation and storage capacity increase with the rise of the number of iteration and these methods are not acceptable.Therefore,several algorithms with incomplete decomposition preconditioning technology are proposed in this paper.Then,the iterative steps of the algorithm are derived and the convergence of the algorithm is proved by the theoretical analysis.Finally,the numerical solution,exact solution and the absolute error are given by numerical experiments.This paper is organized as follows.Firstly,the background,development and research significance of the CG and GMRES(m)iteration algorithms are introduced.Then,the related basic theories of these iteration algorithms and preconditioning technologies are studied.Secondly,based on the fully study about the above theoretical knowledge,SSOR preconditioning is applied to CG iteration algorithm and a new SSOR-ICCG iteration algorithm are proposed.And the iterative steps of the algorithm are derived.Then the convergence of the new algorithm is proved by the theoretical analysis.Then,by the Matlab program,the numerical solution of the original equations is given and ti is accurate.Thirdly,combining incomplete LU decomposition with the VRP-GMRES(m)algorithm,an iterative method named ILU-VRP-GMRES(m)algorithm is proposed to solve linear equations.And the feasibility and the convergence of the new algorithm are demonstrated by the theoretical analysis and numerical examples.Then the factors that have an effect on the algorithm are analyzed.Comparing the new algorithm with the GMRES(m)algorithm,the VRP-GMRES(m)algorithm,and the numerical results show that the new algorithm is accurate.Thus the new algorithm will play a key role in the calculation of actual problems.Finally,the WGMRES(m)algorithm are introduced and the iteration steps for this WGMRES(m)are given.By combining the incomplete LU decomposition with WGMRES(m)algorithm,a new iterative method named ILU-WGMRES(m)algorithm are proposed.Numerical example again shows the high efficiency and precision of the ILU-WGMRES(m) algorithm.