The seed method is used for solving multiple linear systems A (i)x (i) =b (i) for 1≤i≤s, where the coefficient matrix A (i) and the right-hand side b (i) are different in general. It is known that the CG method is an effective method for symmetric coefficient matrices A (i). In this paper, the FOM method is employed to solve multiple linear sy stems when coefficient matrices are non-symmetric matrices. One of the systems is selected as the seed system which generates a Krylov subspace, then the resi duals of other systems are projected onto the generated Krylov subspace to get t he approximate solutions for the unsolved ones. The whole process is repeated u ntil all the systems are solved.
We are interested in the numerical solution of the large nonsymmetric shifted linear system, （A ＋ αI）x -= b, for many different values of the shift a in a wide range. We apply the Saad＇s flexible preconditioning technique to the solution of the shifted systems. Such flexible preconditioning with a few parameters could probably cover all the shifted systems with the shift in a wide range. Numerical experiments report the effectiveness of our approach on some problems.
We consider using seed projection methods for solving unsymmetric shifted systems with multiple right-hand sides (A - σjI)x^(j) = b^(j) for 1 ≤ j ≤ p. The methods use a single Krylov subspace corresponding to a seed system as a generator of approximations to the nonseed systems. The residual evaluates of the methods are given. Finally, numerical results are reported to illustrate the effectiveness of the methods.
In this paper, a global quasi-minimal residual （QMR） method was presented for solving the Sylvester equations. Some properties were investigated with a new matrix product for the global QMR method. Numerical results with the global QMR and GMRES methods compared with the block GMRES method were given. The results show that the global QMR method is less time-consuming than the global GMRES （generalized minimal residual） and block GMRES methods in some cases.
