A Two-Sweep Iteration Algorithm for Non-Hermitian Positive Definite Linear Systems

In order to solve the problem of large-scale sparse non-Hermitian positive definite linear systems, the single-step iterative algorithm has received some research attention. Although single-step iterative algorithm shows certain advantages in dealing with such linear systems, it still has the problem of slow convergence speed or harsh convergence conditions. To deal with this issue, this paper proposes a two-sweep iteration algorithm. On the basis of retaining the advantages of single-step iterative algorithm, this method adds a scanning process, and carries out a forward scan first in each iteration. Then, a backward scan is performed again, and the iterative value is updated by the results of the two scans, so as to accelerate the convergence process and improve the convergence speed and stability of the algorithm. A double-scan iterative algorithm with large sparse non-Hermitian positive numbers is constructed and theoretically proved to converge to the unconditional unique solution of linear systems. Meanwhile, the experimental results show that the algorithm shows better convergence speed and convergence conditions than the one-step iterative algorithm.