Observations 1) the Wb Is

15] W. Hackbush; On the computation of approximate eigenvalues and eigenfunctions of elliptic operators by means of a multi-grid method; SIAM J. 17 12) Divergence can be treated using diierent relaxations, e.g., which amplify most speci-ed components. 13) The computation of W and the optimization of the algorithm using W can be eeciently performed on coarse levels. The computation of W on ne levels is usualy a very expensive task but generally not needed, a coarse version of W being suucient for optimization. 14) The relative separation factors may be important in an FMG algorithm, and not as much the asymptotic factors. In an FMG algorithm for computing U 1 , the amplitudes B come already close to (1; 0; :::; 0) so that only the relative separation of certain components may be relevant, usually the ones which mix with U 1. 15) The separation analysis can be useful also for cases when W is not linear. Two such cases are when a projection is used in the algorithm, and when W is a composition of separation of operators some of which can be analysed. 16) The computation of W can be performed in diierent ways. The presented way, i.e., using its analytic form, may not be the best one. A way to compute easily W for diierent changes of the algorithm is required. One way is to compute directly W using the action of the algorithm on a basis, e.g., on the columns of U or on the columns of the identity matrix. Another way is to compute W as a composition of simpler mappings. 17) The optimization can be performed by an MG procedure. This is an optimization problem with several local minima usually. See 11] for an outline and application of an MG optimization approach usable for several local minima. 18) The optimization of the algorithm using W leads to robust algorithms. The asymptotic convergence rates can be accurately predicted. Worst cases can be found. The behavior predicted by W was accurately reproduced by a program implementing the two level algorithm. References 1] R. E. Bank; Analysis of a multi-level inverse iteration procedure for eigenvalue prob-16 This can be done by changing the value of , say close to 2, to damp more eeciently the frequencies close to =2; 3=2, which are multiplied with 1/2 by transfers. 2) The relaxation can be improved also using a combination …