Global GMERR Algorithm For Solving AX=B

This thesis is concerned with Krylov subspace methods for solving large linear systems with multiple right-hand sides. The generalized minimal error method (GMERR method) for solving large linear systems minimizes the Euclidean norm of the error in the related generalized Krylov subspace, and is one of Krylov subspace methods for solving large linear system Ax = b. GMERR method may be used to solve large linear systems with multiple right-hand sides AX = B by solving each one separately. However, it is very expensive and the information of solving one linear system can not be used to solve another linear system. In this paper, by projecting the initial residual matrix onto a matrix Krylov subspace, we present a new method -- global generalized minimal error method (GLGMERR method). The numerical results show that this new algorithm is more effective than GMERR method. To reduce the computational cost and storage requirement of GLGMERR method, we improve GLGMERR method, and present a truncated version of GLGMERR method--incomplete global generalized minimal error method. This method uses only a few rather than all the previously computed matrix in recurrences to get next matrix, and a quasi-minimum error solution is obtained over the Krylov subspace. The GLGMERR algorithm is also applied to solve Sylvester matrix equation, and the related algorithm is given.We have given theoretical analyses and numerical experiments for the new algorithms. The results show that the algorithms have better practical performance and less CPU time.