Augmented And Deflated Global Minimizing Method

This dissertation studies the augmented and deflated global Krylov subpace method for solving large linear equations with multiple right-hand sides. We derive some algorithms on the basis of global generalized minimum residual method (Global GMRES method) and global generalized minimal error method(Global GMERR method).To improve the convergence of the Global GMRES method, we construct the approximate extreme eigenmatrices of A corresponding to a few of the extreme eigenvalues at each beginning and add them to the new matrix Krylov subspaces, then the restarted global GMERES method augmented with extreme eigenmatrix is proposed.To improve the convergence of the Global GMERR method, we construct the approximate extreme eigenmatrices of AT corresponding to a few of the extreme eigenvalues at each beginning and add them to the next cycle, then we present the restarted global GMERR method augmented with extreme eigenmatrix which deflates the extreme eigenvalues .Combining the implicit restarting technique with global GMERR method,we derive the implicited restarted global GMERR method. With the restarted global GMERES method augmented with extreme eigenmatrix ,the approximate eigenmatrices do not fit into the matrix subspace naturally,but are forced on at the end .However,in this algorithm we select the approximate eigenmatrices and then they fit more naturally into the subspace.We have given theoretical analyses and numerical experiments for the new algorithms. The results show that the new algorithms improve the convergence of the Global GMRES method and the Global GMERR method efficiently.