A tremendous impact will be made if the performance of these sparse matrix solvers can be improved. A current approach to solving large sparse matrix equations is to use direct methods. This approach is often preferred in industry because direct solvers are robust and effective for moderate size problems. However, the unprecedented pace of the advance in technology has caused the size of the matrices to be solved to grow dramatically. The storage requirement for three-dimensional simulations makes the direct methods prohibitively expensive.
Iterative techniques become mandatory.
Unfortunately, iterative methods lack the robustness of direct methods. They often fail when the matrix is very ill-conditioned. The performance of these methods is eventually related to the condition number of the coefficient matrix of the system. Constructing a preconditioner to improve the condition number of a matrix was proposed a few decades ago. These techniques did not have much impact initially due to the simplicity of their heuristics and the relatively small size of the matrices to be solved.