An extended GSGS method for dense linear systems

Davey and Rosindale [K. Davey, I. Rosindale, An iterative solution scheme for systems of boundary element equations, Internat. J. Numer. Methods Engrg. 37 (1994) 1399–1411] derived the GSOR method, which uses an upper triangular matrix ΩΩ in order to solve dense linear systems. By applying functional analysis, the authors presented an expression for the optimum ΩΩ. Moreover, Davey and Bounds [K. Davey, S. Bounds, A generalized SOR method for dense linear systems of boundary element equations, SIAM J. Comput. 19 (1998) 953–967] also introduced further interesting results. In this note, we employ a matrix analysis approach to investigate these schemes, and derive theorems that compare these schemes with existing preconditioners for dense linear systems. We show that the convergence rate of the Gauss–Seidel method with preconditioner PGPG is superior to that of the GSOR method. Moreover, we define some splittings associated with the iterative schemes. Some numerical examples are reported to confirm the theoretical analysis. We show that the EGS method with preconditioner PG(γopt) produces an extremely small spectral radius in comparison with the other schemes considered.