Research On The Convergence Behavior Of Galerkin Method For Linear Systems

In this paper some convergence properties of the Galerkin method for large scale linear systems are studied, including the complementary behavior of restarted FOM algorithm and the error-reducing property of preconditioned CG algorithm.Because the computational cost per step grows drastically as the number of steps increases, the full FOM algorithm is impractical in practice. Usually, the restarted FOM algorithm is employed. It is traditionally thought that the information of the previous FOM cycles, for example, the orthogonality of the basis of the Krylov subspace, is lost at the time of a restart. In Chapter 2 a new point of view is present. It is shown that some important information of the previous cycles may be saved by the iteration approximates automatically, with which successive FOM cycles can complement one another harmoniously in reducing the residual vector. Based on the complementary behavior of restarted FOM, a product hybrid FOM algorithm for solving linear systems and an Arnoldi-FOM algorithm for solving eigenvalue problems are proposed in Chapter 3.The CG algorithm has been known as one of the most famous method for solving large symmetric positive definite linear systems, due to its coding simplicity and its error-minimizing property with respect to the A-norm. This algorithm can also be applied to nonsymmetric linear systems when combined with the NR/NE techniques. It has been shown in a recent paper that the CGNR algorithm is error-reducing with respect to the Euclidean norm. However, in practice the simple CGNR algorithm is seldom used because of the squared condition number of its iteration matrix. In Chapter 4 a much richer result concerning the error-reducing property of the CG procedure is presented. Assume that the preconditioner M is also symmetric positive definite. It is shown that the preconditioned CG method is error-reducing with respect to the M-norm.