Study On Preconditioning Techniques For Generalized Minimal Residual Algorithm

In this paper, the sparse approximation inverses and eigenvalue translation based preconditioners for Krylov subspace methods are thoroughly researched by situation of research at home and abroad, their internal mechanism are studied also. A new sparse approximate inverse algorithm and a preconditioning based on matrix correction are presented.Firstly, based on Sparse Approximate Inverse(SPAI) algorithm,Minimal Residual (MR) algorithm and Self-Preconditioned Minimal Residual algorithm, the Approximate Inverse (AI) algorithm based on sparsity pattern update of the approximate inverse is presented.Secondly, the method utilizing eigenvectors corresponding the smallest eigenvalues in magnitude to construct a matrix correction is presented, based on research on eigenvalue translation based preconditioners.Finally, numerical experiments are performed with this algorithm, specially the comparative study between AI algorithm and SPAI, MR and Self-Preconditioned algorithm is presented, then show the efficiency of the two suggested preconditioning strategy.These algorithms are proved to be highly parallel and efficiency for krylov subspace methods for the solution of linear systems by theoretical analysis and numerical results. It is reasonable on theory and effective on computational practice.