Massively parallel Poisson and QR factorization solvers

The paper brings a massively parallel Poisson solver for rectangle domain and parallel algorithms for computation of QR factorization of a dense matrix A by means of Householder reflections and Givens rotations. The computer model under consideration is a SIMD mesh-connected toroidal n × n processor array. The Dirichlet problem is replaced by its finite-difference analog on an M × N (M + 1, N are powers of two) grid. The algorithm is composed of parallel fast sine transform and cyclic odd-even reduction blocks and runs in a fully parallel fashion. Its computational complexity is O(M N log L/n2), where L = max(M + 1, N). A parallel proposal of QR factorization by the Householder method zeros all subdiagonal elements in each column and updates all elements of the given submatrix in parallel. For the second method with Givens rotations, the parallel scheme of the Sameh and Kuck was chosen where the disjoint rotations can be computed simultaneously. The algorithms were coded in MPF and MPL parallel programming languages and results of computational experiments on the MasPar MP-1 system are also presented.