Solving Band Diagonally Dominant Linear Systems Using Gaussian Elimination: Shared-Memory Parallel Programming with OpenMP

The Gaussian elimination (GE) is an important direct method that transforms the initial linear system into an equivalent upper triangular system, which is straightforward to solve. To ensure numerical stability and reduce the effects of round-off error that can overcome the solution, most direct methods include a pivoting strategy. Diagonally dominant (DD) matrices are numerically stable during the GE method. Thus, there is no need to incorporate pivoting. In this paper, we propose a new scheduling GE approach for band DD systems which is based on allocating tasks to suitable cores. All cores carry out tasks assigned to them in parallel with dependencies under consideration. Our purposes are to obtain a high degree of parallelism by balancing the load among cores and decreasing the parallel execution time. The effectiveness of our investigation is proved by several experiments which are carried out on a shared-memory multicore architecture using OpenMP.