Parallel WZ factorization on mesh multiprocessors

We present a parallel algorithm for the QIF (Quadrant Interlocking Factorization) method, which solves linear equation systems using WZ factorization. The parallel algorithm we developed is general in the sense that it does not impose any restrictions on the size of the problem and that it is independent from the dimensions oJ the mesh. The result is an efficient algorithm using half the messages of the equivalent parallel LU algorithm. Finally, we have compared the QIF algorithm in multiprocessor architectures with mesh topology and hypercube topology, obtaining similar calculation times for both architectures. This last aspect confirms the communication redundancy of the hypercube topology, which raises hardware costs without any significant improvement in the efficiency of the algorithms. The solution of a linear equation system is a problem which arises in many situations, whether scientific (simulations, solution of differential equation systems), economic (resource assignment, econometric models in general) and engineering (passive electronic circuits), etc. The QIF (Quadrant Interlocking Factorization) algorithm, introduced by Evans and Hatzopoulos in 1979 [EvHa79], is a numerical method for finding a solution for systems of the type Ax=b, where A is a non singular matrix of dimensions (N×N), x is an unknown column vector, and b is the independent term vector provided. The QIF method is based on the WZ factorization of the system matrix, A = WZ. The main advantage of this factorization is that it presents a complexity order half of the one in the LU decomposition, due to the fact that it performs the simultaneous evaluation of two columns or two rows. A detailed description of this algorithm can be found in [EvHa79], [EvHa80], [EHN81], [Evan82] and [Hatz82]. We will just summarize its basic steps. Matrices W and Z into which we are going to factor matrix A, are: 1 0 0 wl, 0 1 0 Wl,u_ 1 w2, 0 w2, t 1 0 W2,N_ 2 W2,N_ 1 wN_3, 0 WN_3, l 0 1 WN_3~V_ 2 WN_3,N_ 1 WN_2, 0 0 1 WN_2,N_ 1 0 0 1 Z-(1)