Some Matrix Krylov Subspace Methods For Solving Large And Sparse Linear Systems

Some application problems such as in electromagnetic scattering require the solution of large,sparse and nonsymmetric linear systems of equations with multiple right-hand sides. The global biconjugate gradient (Gl-BCG) method is one of the most important methods for solving large,sparse and nonsymmetric linear systems of equations with multiple right-hand sides. The computational cost and the memory requirement of the Gl-BCG method are small. In many situations, however, the algorithm exhibits a very irregular convergence behavior. To overcome the disadvantage, we develop the global minimal residual smoothing (GMRS) technique for iterative method for solving linear equations. Applying the technique to the Gl-BCG algorithm, we derive the smoothed Gl-BCG (SGl-BCG) algorithm.To ignore adjoint matrix-matrix multiplications, the global conjugate gradient squared algorithm (Gl-CGS) is proposed to solve large,sparse and nonsymmetric linear systems of equations with multiple right-hand sides. The convergence speed of the Gl-CGS algorithm is faster than that of the Gl-BCG algorithm. To smooth the norm of the residual produced by the Gl-CGS algorithm, the GMRS technique is applied to the Gl-BCG algorithm. We derive the smoothed Gl-CGS (SGl-CGS) algorithm.To improve the convergence of the Gl-CGS algorithm, instead of squaring the Gl-BCG polynomial as in Gl-CGS , we proposed to consider products of two nearby Gl-BCG polynomial which leads to generalized Gl-CGS (G-GlCGS) method for solving large,sparse and nonsymmetric linear systems of equations with multiple right-hand sides, of which Gl-CGS and Gl-BiCGSTAB are just particular cases. This approach allows the construction of methods that converge less irregulary than Gl-CGS and that improve on other convergence properties as well. At the same time, a G-GlCGS2 algorithm is presented.We give theoretical analyses on these new algorithm and some numerical experiments. The results show that the new algorithms have better practical performance, significantly faster convergence rate, less computational cost and less storage.