New Variants of Divide & Conquer Method Arisingfrom Block Cyclic Reduction Type FormulationTuomo

The Divide & Conquer fast direct method for solving the Poisson equation in rectangular domains is formulated in the classical block cyclic reduction manner, demonstrating the close relationship between the two approaches. Diierent ways of solving the arising reduced systems are considered. The partial solution approach leads to the standard Divide & Conquer method, while the new variants are obtained by using the matrix rational polynomial factorization technique , including the partial fraction expansions, the FFT{approach, and the FACR{technique. Such techniques have originally been considered in the standard cyclic reduction framework. The equivalence of the partial solution and the partial fraction techniques is shown. The computational cost of the considered variants is O(N log N) operations, except for the FACR{technique, for which it is O(N log log N). The stability estimate for the Divide & Conquer method is constructed, and the stability is demonstrated by numerical experiments.