Adaptive Preconditioned Gmres(m) Algorithms

This paper presents a new preconditioning for the restarted GMRES algorithm. It is based on an invariant subspace approximation which is updated at each cycle. By choosing the appropriate preconditioner, the condition number of the preconditioning matrix is greatly reduced. GMRES to solve the preconditioning equations is compared with exisiting methods.The first chapter introduces the background of GMRES algorithm, and the content of this paper to study.The second chapter, for the new method proposed, gives a brief review about GMRES algorithm and GMRES(m) algorithm, and the implementation of these algorithms.The third chapter describes the theory of adaptively preconditioned GMRES(m) algorithm, gives the process of implementation of the algorithm, and proves the convergence of the algorithm, finally, by the numerical experiments shows that the algorithm is more effective.The fourth chapter summaries the full next, and future research prospects.