Research On Preconditioning Iteration Technique For Sparse Matrix Equations

The solving of large sparse linear systems is one of the most important technologies for carrying out numerical simulation to many of the problems in natural and social sciences,and GMRES is one of the most popular algorithms for solving large nonsymmetrical linear systems. Unfortunately, the computational cost of full GMRES usually becomes unacceptable as the iteration number increases. To overcome the difficulty, restarting scheme or hybrid scheme of GMRES can be employed. Recently, the complementary behavior of restarted GMRES has attracted a wide interest. In particular, based on the study of complementary behavior,a product hybrid GMRES algorithm has been proposed, which improves the efficiency of hybrid scheme significantly.To implement the product hybrid GMRES algorithm ,a couple of GMRES polynomials are constructed, and then the product of the polynomials are used repeatedly via a Richardson iteration.However, if the restarting frequency is not small,the GMRES polynomials may undergo a great instability. In consequence, the Richardson iteration may diverge. Polynomial preconditioned hybrid generalized minimal residual algorithm is proposed to accelerate its convergence rate and improve its stability.Firstly, the principle of Krylov methods for solving large sparse linear systems is introduced. Then, GMRES algorithm and CG algorithm are introduced, which are based on the principle of Krylov methods.Secondly, we discuss the complementary behavior of restarted GMRES and the product hybrid GMRES algorithm which is based on the study of complementary behavior.Then, calculating the polynomial preconditioned matrix, and using it as preconditioned matrix for the product hybrid GMRES algorithm to improving spectral nature of the coefficient, accelerating its convergence rate and improving its stability.This paper gives some numerical experiments simulation and analysis for new preconditioned methods with classical algorithm of mature. The results show that the new algorithms is more suitable for large sparse matrix linear systems,computing and storage has a corresponding improvement. The product hybrid GMRES algorithm for solving large sparse matrix have been further improved.