There are basically two approaches for solving linear systems: one is to exactly solve the linear sytem such as Gaussian-elimination. The other approximates the solution in the Krylov spaces; Conjugate-gradient and General minimum residual method are typical examples. For sparse and large-scaled, like 10000x10000, matrices, the latter is much more efficient.

Those basic subjects will be briefly reviewed including incomplete LU-preconditioning, and a recent research of parallel ILU-PCG algorithm will be introduced.