The Matrix-Multilevel approach is based on a purely matrix dependent description of Multigrid and related methods. The formulation of Multilevel methods as singular matrix extensions leads to the description of the Multilevel method as a preconditioned iterative scheme, and illuminates the significance of the used prolongation, resp. restriction operator for the related preconditioner. As matrix dependent black box restriction C we introduce a shifted form of the original matrix A, namely C = B(:,1:2:n) with B = aI - A, where a is a good upper bound for the largest eigenvalue of A. This mapping is chosen in such a way that via the related preconditioner the small eigenvalues are enlarged while the maximum eigenvalue remains nearly unchanged. If the components of each eigenvector of A can be seen as a discretization of a continuous function, then we derive estimates on the improved condition number after one step. We mainly consider symmetric positive definite matrices related to 1D-problems, but the results can be directly generalized to nonsymmetric and higher dimensional problems.     «

The Matrix-Multilevel approach is based on a purely matrix dependent description of Multigrid and related methods. The formulation of Multilevel methods as singular matrix extensions leads to the description of the Multilevel method as a preconditioned iterative scheme, and illuminates the significance of the used prolongation, resp. restriction operator for the related preconditioner. As matrix dependent black box restriction C we introduce a shifted form of the original matrix A, namely C = B(...     »