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If all processors execute their part of the preconditioner solve without further communication, the overall method is technically a block Jacobi preconditioner (see §). While their parallel execution is very efficient, they may not be as effective as more complicated, less parallel preconditioners, since improvement in the number of iterations may be only modest. To get a bigger improvement while retaining the efficient parallel execution, Radicati di Brozolo and Robert  suggest that one construct incomplete decompositions on slightly overlapping domains. This requires communication similar to that for matrix--vector products.