Polynomial Preconditioning Methods For Nonsymmetric Positive Definite Linear System

Many problems in scientific computing and engineering applications require solutions of large sparse linear equations.Krylov subspace methods,as a class of important iterative methods,are often used to solve such problems.Efficient preconditioners can significantly speed up the convergence rate of Krylov methods,hence the research on preconditioning techniques is of great value,both theoretically and practically.In this paper,in order to speed up the convergence rate of Krylov subspace methods,a type of polynomial preconditioner,based on splitting iterative methods,is designed.The inverse coefficient matrix can be expressed by power series of the iterative matrix when the splitting iteration is convergent.By truncating the Neumann series in the expression,the polynomial preconditioner is obtained.Based on different splitting methods,we construct several m-step polynomial preconditioners correspondingly.Theoretical results and numerical experiments demonstrate that this type of preconditioner is effective in accelerating GMRES method for solving nonsymmetric positive definite linear systems.