Keywords: reduced system, Schur-complement, Greedy algorithm, approximate inverse, preconditioning, PC cluster.

During the last ten years, numerous advances in the development of Krylov subspace solvers with preconditioning are proposed. This new development includes the reduced systems of equations, and different approximate inverse preconditioners. This chapter how shows to reduce the order of the large and sparse linear systems of equations and to make the approximate inverse preconditioner. Such that the linear systems can be transformed into the reduced systems with the Schur-complement by using the independent set, which are based on the greedy algorithm. However, the Schur complement becomes less sparse, and its condition number is often worse. Therefore, it is indispensable to use preconditioners for the reduced systems.

We now study the parallel preconditioning technique of minimum residual (MR), Newton and approximate inverse with the Sherman-Morrison formula (AISM) schemes. In this chapter, we apply them to the generalised minimal residual method GMRES(m) algorithm on a PC cluster machine with eight CPUs to solve the reduced system. We also compute the preconditioner by using MR and Newton schemes, and show the effectiveness of the Newton scheme. By avoiding the matrix-matrix products and without using the preconditioner in explicit form, the Newton scheme using only a matrix-vector form is much less costly than the MR scheme.

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